2020 Mathematics subject classification : 17B69, 18M20, 11E12, 11F03, 81T40.
Key words : Haagerup fusion category, Drinfeld center, vertex operator algebra, vector-valued modular form, lattice, genus of vertex operator algebras, Heisenberg commutant.
Introduction
The Haagerup subfactor of index 5 + 13 2 \frac{5+\sqrt{13}}{2} 2 5 + 13 [6 ] gives rise to three Morita equivalent fusion categories H 1 \mathcal{H}_1 H 1 H 2 \mathcal{H}_2 H 2 H 3 \mathcal{H}_3 H 3 [43 ] . The category H 3 \mathcal{H}_3 H 3 1 1 1 a a a a 2 a^2 a 2 ρ \rho ρ a ρ a\rho a ρ a 2 ρ a^2\rho a 2 ρ
a 3 ≅ 1 , a ρ ≅ ρ a 2 , ρ 2 ≅ 1 ⊕ ρ ⊕ a ρ ⊕ a 2 ρ . (1) a^3 \cong1,\qquad a\rho\cong\rho a^2,\qquad\rho^2 \cong1 \oplus\rho\oplus a\rho\oplus a^2\rho. \tag*{(1)} a 3 ≅ 1 , a ρ ≅ ρ a 2 , ρ 2 ≅ 1 ⊕ ρ ⊕ a ρ ⊕ a 2 ρ . (1) The Frobenius–Perron dimension of ρ \rho ρ
d = d e f 3 + 13 2 . (2) d \mathrel{\mathop{=}^{\mathrm{def}}} \frac{3+\sqrt{13}}{2}. \tag*{(2)} d = def 2 3 + 13 . (2) Numerical studies of Haagerup symmetry in conformal field theory include [46 Y+22] . At central charge eight, Evans and Gannon [33 ] showed that a positive vertex operator algebra whose representation category has the modular data of the Drinfeld center Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) γ ∈ { 0 , 1 } \gamma\in\{0,1\} γ ∈ { 0 , 1 } 6 + 13 γ 6+13\gamma 6 + 13 γ V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 [33 Section 2.4] , and Gannon [40 ] began the analysis of its currents. Albert, Honda, Kaidi and Zheng [AHKZ26] suggested that the chiral ( e 8 ) 1 (\mathfrak{e}_8)_1 ( e 8 ) 1 [63 ] , and Gannon and Rayhaun proposed [41 ] , a correspondence between finite non-invertible symmetries and conformal subalgebras of finite index. For the Haagerup categories, Gannon and Rayhaun predicted that a chiral algebra of central charge eight with representation category Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 [41 Section 4.3] . Recently, Jiao [52 ] gave an explicit vertex operator algebra of central charge eight inside V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 s l 2 ⊕ s l 2 \mathfrak{sl}_2 \oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2 ( 3 , 13 ) (3,13) ( 3 , 13 ) Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [66 ] and Huang [50 ] subsequently each announced such an algebra, with currents of levels ( 1 , 39 ) (1,39) ( 1 , 39 ) [35 ] have since announced realizations of the centers of the untwisted Haagerup–Izumi categories of every odd order by non-unitary algebras of negative central charge.
We classify the possible Haagerup algebras and construct the twin of any such algebra by exchanging its associated lattice. The classification starts from either of the two explicit candidates once it is known to be a Haagerup algebra. That the explicit algebra S S S Theorem 1.3 is a Haagerup algebra is proved in [5 Theorem 6.1] . We call Haagerup algebra a positive strongly rational vertex operator algebra U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U) \cong Z(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) U U U q − c 24 ( 1 + O ( q ) ) q^{-\frac{c}{24}}(1+O(q)) q − 24 c ( 1 + O ( q )) Definition 2.2 ). The characters of a Haagerup algebra form such a vector by [72 , 49 ] . Both vectors of [33 ] pass the tests of character theory known to us (their coefficients are nonnegative integers up to q 60 q^{60} q 60
Theorem 1.1 Let U U U U U U γ = 0 \gamma= 0 γ = 0 U U U
The proof has three steps. First, U U U V E 8 V_{E_8} V E 8 Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) Vec Z 3 ⊂ H 3 \operatorname{Vec}_{\mathbb{Z}_3} \subset\mathcal{H}_3 Vec Z 3 ⊂ H 3 E 8 E_8 E 8 V ′ = V E 6 ⊕ A 2 V' = V_{E_6\oplus A_2} V ′ = V E 6 ⊕ A 2 Proposition 4.1 ). Second, the restrictions of the V ′ V' V ′ U U U 3 3 3 s l 3 \mathfrak{sl}_3 sl 3 27 27 27 e 6 \mathfrak{e}_6 e 6 U 1 U_1 U 1 Proposition 4.4 ). Third, for h h h U 1 U_1 U 1 h [ − 1 ] h h[-1]h h [ − 1 ] h h ( 0 ) 2 h^2_{(0)} h ( 0 ) 2 [40 Section 2.4] . We determine the space of such forms exactly: the relations between its coefficients are produced by pairing with weakly holomorphic forms of weight zero for the dual representation and applying the residue theorem, in the spirit of the Serre duality of Borcherds [9 Theorem 3.1] , and their completeness is proved by exhibiting exact elements (Proposition 5.4). For γ = 1 \gamma= 1 γ = 1 d 3 d_3 d 3 U 1 U_1 U 1 dim U 1 = 19 \dim U_1 = 19 dim U 1 = 19 § 7 ). For γ = 0 \gamma= 0 γ = 0 s l 2 \mathfrak{sl}_2 sl 2 e 8 \mathfrak{e}_8 e 8
Theorem 1.2 Let U U U U 1 ≅ s l 2 ⊕ s l 2 U_1 \cong\mathfrak{sl}_2 \oplus\mathfrak{sl}_2 U 1 ≅ sl 2 ⊕ sl 2 U 1 ⊂ e 6 ⊕ s l 3 U_1 \subset\mathfrak{e}_6 \oplus\mathfrak{sl}_3 U 1 ⊂ e 6 ⊕ sl 3 Proposition 4.1 is, up to conjugation, one of two explicit embeddings, with levels ( k ′ , k ′ ′ ) = ( 1 , 39 ) (k', k'') = (1,39) ( k ′ , k ′′ ) = ( 1 , 39 ) ( 3 , 13 ) (3,13) ( 3 , 13 ) U 2 U_2 U 2 U 1 U_1 U 1
The precise statement is Theorem 8.5 with Table 2 . In case (I) the table reproduces the data listed in the working manuscript [67 ] of Riesen, which are therefore forced by the representation category once the levels are ( 1 , 39 ) (1,39) ( 1 , 39 ) [40 Section 3.3] began the analysis of the six-dimensional case and treated the level one factor. The levels 13 and 39 come from diagonal embeddings into e 6 ⊕ s l 3 \mathfrak{e}_6 \oplus\mathfrak{sl}_3 e 6 ⊕ sl 3 V ′ V' V ′ K = Q ( 13 , − 3 ) K = \mathbb{Q}(\sqrt{13},\sqrt{-3}) K = Q ( 13 , − 3 ) C = Com U ( A ) C = \operatorname{Com}_{U}(A) C = Com U ( A ) A A A U 1 U_1 U 1 § 10 ).
Table 2. Lowest weight spaces and U 2 U_2 U 2 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ ( a , b ) (a,b) ( a , b ) V ( a , b ) V(a,b) V ( a , b ) a a a g ′ \mathfrak{g}' g ′ b b b g ′ ′ \mathfrak{g}'' g ′′ R z R_z R z Lemma 8.1 ). Those of the row U 2 U_2 U 2 U 2 U_2 U 2 V ( 4 , 0 ) ⊂ S 2 ( g ′ ) V(4,0) \subset S^2(\mathfrak{g}') V ( 4 , 0 ) ⊂ S 2 ( g ′ ) L s l 2 ( 3 , 0 ) L_{\mathfrak{sl}_2}(3,0) L sl 2 ( 3 , 0 )
Theorem 1.3 Let U U U V ′ V' V ′ C 2 C_2 C 2 182 95 \frac{182}{95} 95 182 32 19 \frac{32}{19} 19 32 U 2 U_2 U 2 V 2 ′ V'_2 V 2 ′ K K K Gal ( K / Q ) \operatorname{Gal}(K/\mathbb{Q}) Gal ( K / Q ) S S S U 1 ⊕ U 2 U_1 \oplus U_2 U 1 ⊕ U 2 C 2 C_2 C 2 q 11 / 3 q^{11/3} q 11/3 U n = S n U_n = S_n U n = S n n ≤ 4 n \leq4 n ≤ 4 S S S [52 ] .
These are Theorems 10.5 , 10.9 , 10.10 , 10.21 and 10.24 , and Propositions 10.12 and 10.27 . The finite weight-four condition (F4) used for strong generation is stated in Definition 10.16 and established by the exact certificate of Proposition 10.17 . In the setting of Theorem 1.3 , the inclusion S ⊂ U S \subset U S ⊂ U Corollary 10.22 ). The proof of identification with Jiao uses only weight-two membership and generation, and is independent of (F4). In case (I), we exclude the last spurious candidate for C 2 C_2 C 2 Proposition 10.35 ), identify the three weight-two generators e e e f f f v v v U R = ⟨ A , e , f , v ⟩ U^{R} = \langle A,e,f,v\rangle U R = ⟨ A , e , f , v ⟩ [66 ] with our data (Proposition 10.36 ), and exchange the remaining choices by automorphisms of V ′ V' V ′ Proposition 10.37 ). Hence every Haagerup algebra contains S S S ψ R ( U R ) \psi_R(U^{R}) ψ R ( U R ) V ′ V' V ′ ψ R \psi_R ψ R [66 ] with V ′ V' V ′ Theorems 10.38 and 10.29 prove equality when the corresponding candidate is a Haagerup algebra.
Our main new result explains the two cases. A strongly rational vertex operator algebra decomposes over the dual pair formed by its Heisenberg commutant Ω U = Com U ( M ( h ) ) \Omega_U = \operatorname{Com}_{U}(M(\mathfrak{h})) Ω U = Com U ( M ( h )) h \mathfrak{h} h U 1 U_1 U 1 V L V_L V L L L L [57 , 62 ] . This decomposition was used by Höhn [48 ] to study the genus of [Hoe03], and it leads to the hyperbolic genus of Moriwaki [61 ] and Möller–Rayhaun [62 ] .
Theorem 1.4 Let U U U ( k ′ , k ′ ′ ) (k',k'') ( k ′ , k ′′ )
The associated lattice of U U U ( 2 1 1 20 ) \begin{pmatrix}2&1\\1&20\end{pmatrix} ( 2 1 1 20 ) ( 6 3 3 8 ) \begin{pmatrix}6&3\\3&8\end{pmatrix} ( 6 3 3 8 ) x 2 + x y + 10 y 2 x^2+xy+10y^2 x 2 + x y + 10 y 2 3 x 2 + 3 x y + 4 y 2 3x^2+3xy+4y^2 3 x 2 + 3 x y + 4 y 2 − 39 -39 − 39 Ω U \Omega_U Ω U Rep ( Ω U ) ≅ H ( 3 ) × C ( L ′ / L ) ‾ \operatorname{Rep}(\Omega_U) \cong\mathcal{H}(3) \times\overline{\mathcal{C}(L'/L)} Rep ( Ω U ) ≅ H ( 3 ) × C ( L ′ / L ) U U U L s l 2 ( k ′ , k ′ ) ⊗ L s l 2 ( k ′ ′ , k ′ ′ ) L_{\mathfrak{sl}_2}(k',k') \otimes L_{\mathfrak{sl}_2}(k'',k'') L sl 2 ( k ′ , k ′ ) ⊗ L sl 2 ( k ′′ , k ′′ )
Replacing L L L U U U Ω U ⊗ V L \Omega_U \otimes V_L Ω U ⊗ V L U ♯ U^{\sharp} U ♯ Ω U \Omega_U Ω U U ↦ U ♯ U \mapsto U^{\sharp} U ↦ U ♯ U U U U ♯ U^{\sharp} U ♯ U U U
The precise statements are Proposition 12.2 and Theorem 12.3. Part 1 combines [62 Proposition 2.14] and the proof of [62 Corollary 4.10] with the weight decomposition in Table 2 and a growth estimate for characters. In part 2, U ♭ U^{\flat} U ♭ U U U Rep ( Ω U ) ⊠ C ( L ′ / L ) → Rep ( Ω U ) ⊠ C ( L ♭ ′ / L ♭ ) \operatorname{Rep}(\Omega_U) \boxtimes C(L^{\prime}/L) \to\operatorname{Rep}(\Omega_U) \boxtimes C(L^{\flat\prime}/L^{\flat}) Rep ( Ω U ) ⊠ C ( L ′ / L ) → Rep ( Ω U ) ⊠ C ( L ♭′ / L ♭ ) U ♭ U^{\flat} U ♭ U U U ± 1 \pm1 ± 1 [62 Proposition 5.3] for holomorphic algebras. What is specific here is its computation for Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 )
Classification and explicit realizations. We write ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) V ≅ S V \cong S V ≅ S ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U R U^{\mathrm{R}} U R Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [5 Theorem 6.1] and Theorem 10.24, ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) [5 ] uses Theorems 10.9, 10.10 and 10.21 and the results of § 11 , and Theorem 10.24 transfers it to Jiao’s algebra. None of these results depends on ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) [5 Theorem 7.13] , which uses the identifications of [20 Proposition 4.1 and Theorem 7.3] , ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R )
Theorem 1.5
If ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U ↦ U ♭ U \mapsto U^{\flat} U ↦ U ♭
The explicit algebra S S S [5 Theorem 6.1] . Consequently the two isomorphism classes are represented by S S S S ♭ S^{\flat} S ♭
The twin S ♭ S^{\flat} S ♭ S ♭ ≅ U H ≅ U R S^{\flat} \cong U^{\mathrm{H}} \cong U^{\mathrm{R}} S ♭ ≅ U H ≅ U R U H U^{\mathrm{H}} U H U R U^{\mathrm{R}} U R [5 Theorem 7.9 and Corollary 7.10] , using the common fixed-point algebra and the Huang–Riesen identification of [20 Theorem 7.3 and Proposition 4.1] .
Theorems 12.3 and 12.4 give part 1 and the construction of S ♭ S^{\flat} S ♭ S ♭ S^{\flat} S ♭ V ′ V^{\prime} V ′ ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) [5 Section 7] . Table 1 records the sources of these results.
Table 1. Dependencies and sources of the main results.
We also prove that the explicit algebras S S S ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) U 1 U_1 U 1 Aut ( V ′ ) \operatorname{Aut}(V^{\prime}) Aut ( V ′ ) C C C § 11 reduces rationality to the semisimplicity of the Zhu algebra. The inclusion of the affine simple-current extension in S S S Table 2 (Theorem 11.6). The exact descent and the branching are proved in [5 Appendix A] , see § 11.5 .
Computations. Many proofs rest on finite computations: exact rational arithmetic for the vector-valued modular forms and the relations between traces, exact arithmetic over K \mathbb{K} K V ♮ V^\natural V ♮ Q \mathbb{Q} Q Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
Notation. Throughout, V ♮ = V E 8 ⊕ A 2 V^\natural= V_{E_8 \oplus A_2} V ♮ = V E 8 ⊕ A 2 U ⊂ V ♮ U \subset V^\natural U ⊂ V ♮ U 1 = g ′ ⊕ g ′ ′ U_1 = \mathfrak{g}' \oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ ( k ′ , k ′ ′ ) (k', k'') ( k ′ , k ′′ ) A = L g ′ ( k ′ , 0 ) ⊗ L g ′ ′ ( k ′ ′ , 0 ) A = L_{\mathfrak{g}'}(k',0) \otimes L_{\mathfrak{g}''}(k'',0) A = L g ′ ( k ′ , 0 ) ⊗ L g ′′ ( k ′′ , 0 ) C = Com U ( A ) C = \operatorname{Com}_U(A) C = Com U ( A ) K = Q ( 13 , − 3 ) \mathbb{K} = \mathbb{Q}(\sqrt{13}, \sqrt{-3}) K = Q ( 13 , − 3 ) S S S Theorem 1.3 , generated by the explicit currents and one of four explicit subspaces of V 2 ♮ V_2^\natural V 2 ♮ § 10.5 ), and U R = ⟨ A , e , f , v ⟩ U^R = \langle A,e,f,v\rangle U R = ⟨ A , e , f , v ⟩ [66 ] . In § 12 , h h h U 1 U_1 U 1 Ω U \Omega_U Ω U L L L L ♯ L^\sharp L ♯
Structure of the paper. § 2 fixes the modular data of Z ( H 3 ) \mathcal{Z}(\mathcal{H}_3) Z ( H 3 ) § 3 treats the character vectors at central charge eight, and § 4 establishes the chain U ⊂ V E 8 ⊕ A 2 ⊂ V E 8 U \subset V_{E_8 \oplus A_2} \subset V_{E_8} U ⊂ V E 8 ⊕ A 2 ⊂ V E 8 § 5 derives the weight-two relations in exact arithmetic and § 6 the relations of higher weight. § 7 proves Theorem 1.1 and § 8 proves Theorem 1.2 . § 9 gives the constraints on the weight-two space. Section 10 determines the commutant of the currents and U 2 U_2 U 2 [52 ] and [66 ] , and proves the uniqueness theorems and the rational forms. Section 11 gives the Zhu-algebra criterion, the modular computations and the references for exact descent and branching. Section 12 determines the associated lattices, constructs the twins and applies the classification to S S S
Preliminaries
The modular data of the Drinfeld center
We use the modular data of Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [33 equations (2.5a) and (2.5b) p. 20] . The simple objects are labelled
0 , b , a , c 0 , c 1 , c 2 , d 1 , … , d 6 , 0,\ b,\ a,\ c_0,\ c_1,\ c_2,\ d_1,\ldots,d_6, 0 , b , a , c 0 , c 1 , c 2 , d 1 , … , d 6 , with 0 0 0
1 , 3 d + 1 , 3 d + 2 , 3 d + 2 , 3 d + 2 , 3 d + 2 , 3 d , … , 3 d , (3) 1,\quad3d+1,\quad3d+2,\quad3d+2,\quad3d+2,\quad3d+2,\quad3d,\ldots,3d, \tag*{(3)} 1 , 3 d + 1 , 3 d + 2 , 3 d + 2 , 3 d + 2 , 3 d + 2 , 3 d , … , 3 d , (3) and their categorical twists are
1 , 1 , 1 , 1 , e 2 π i 3 , e − 2 π i 3 , e 2 π i l 2 13 ( 1 ≤ l ≤ 6 ) . (4) 1,\quad1,\quad1,\quad1,\quad\mathrm{e}^{\frac{2\pi i}{3}},\quad\mathrm{e}^{-\frac{2\pi i}{3}},\quad\mathrm{e}^{\frac{2\pi i l^2}{13}}\quad(1\le l\le6). \tag*{(4)} 1 , 1 , 1 , 1 , e 3 2 π i , e − 3 2 π i , e 13 2 π i l 2 ( 1 ≤ l ≤ 6 ) . (4) Put x = 13 − 3 13 26 x=\frac{13-3\sqrt{13}}{26} x = 26 13 − 3 13 y = 3 13 y=\frac{3}{\sqrt{13}} y = 13 3 c ( j ) = − 2 y cos ( 2 π j 13 ) c(j)=-2y\cos\left(\frac{2\pi j}{13}\right) c ( j ) = − 2 y cos ( 13 2 π j ) S S S 1 3 M \frac{1}{3}M 3 1 M M M M ( x , 1 − x , 1 , 1 , 1 , 1 , y , … , y ) (x,1-x,1,1,1,1,y,\ldots,y) ( x , 1 − x , 1 , 1 , 1 , 1 , y , … , y ) ( 1 − x , x , 1 , 1 , 1 , 1 , − y , … , − y ) (1-x,x,1,1,1,1,-y,\ldots,-y) ( 1 − x , x , 1 , 1 , 1 , 1 , − y , … , − y ) ( a , c 0 , c 1 , c 2 ) (a,c_0,c_1,c_2) ( a , c 0 , c 1 , c 2 )
( 2 − 1 − 1 − 1 − 1 2 − 1 − 1 − 1 − 1 − 1 2 − 1 − 1 2 − 1 ) , \begin{pmatrix}
2 & -1 & -1 & -1\\
-1 & 2 & -1 & -1\\
-1 & -1 & -1 & 2\\
-1 & -1 & 2 & -1
\end{pmatrix}, 2 − 1 − 1 − 1 − 1 2 − 1 − 1 − 1 − 1 − 1 2 − 1 − 1 2 − 1 , the entries M a , 0 , M a , b , … , M c 2 , b M_{a,0},M_{a,b},\ldots,M_{c_2,b} M a , 0 , M a , b , … , M c 2 , b 1 1 1 M d l , 0 = − M d l , b M_{d_l,0}=-M_{d_l,b} M d l , 0 = − M d l , b y y y { a , c 0 , c 1 , c 2 } \{a,c_0,c_1,c_2\} { a , c 0 , c 1 , c 2 } { d 1 , … , d 6 } \{d_1,\ldots,d_6\} { d 1 , … , d 6 } M d l , d l ′ = c ( l l ′ ) M_{d_l,d_{l'}}=c(ll') M d l , d l ′ = c ( l l ′ ) D 2 = 9 ( 3 d + 2 ) 2 D^2=9(3d+2)^2 D 2 = 9 ( 3 d + 2 ) 2 S 0 , 0 = 3 3 ( 3 d + 2 ) S_{0,0}=\frac{3}{\sqrt{3(3d+2)}} S 0 , 0 = 3 ( 3 d + 2 ) 3
The modular data of Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [51 ] . The form above is that of [33 ] , and its identification with the Drinfeld center of the Haagerup category is established in [44 ] . The charge conjugation is trivial by an exact calculation. Indeed, for 1 ≤ a , b ≤ 6 1\le a,b\le6 1 ≤ a , b ≤ 6
∑ l = 1 6 cos ( 2 π a l 13 ) = − 1 2 , ∑ l = 1 6 cos ( 2 π a l 13 ) cos ( 2 π b l 13 ) = 13 δ a , b − 2 4 . \sum_{l=1}^{6}\cos\left(\frac{2\pi al}{13}\right)=-\frac{1}{2},\qquad\sum_{l=1}^{6}\cos\left(\frac{2\pi al}{13}\right)\cos\left(\frac{2\pi bl}{13}\right)=\frac{13\delta_{a,b}-2}{4}. l = 1 ∑ 6 cos ( 13 2 π a l ) = − 2 1 , l = 1 ∑ 6 cos ( 13 2 π a l ) cos ( 13 2 π b l ) = 4 13 δ a , b − 2 . Together with 2 x − 1 = − y 2x-1=-y 2 x − 1 = − y x ( 1 − x ) = 1 / 13 x(1-x)=1/13 x ( 1 − x ) = 1/13 y 2 = 9 / 13 y^2=9/13 y 2 = 9/13 M 2 = 9 Id M^2=9\operatorname{Id} M 2 = 9 Id S 2 = Id S^2=\operatorname{Id} S 2 = Id ( S T ) 3 = S 2 (ST)^3=S^2 ( S T ) 3 = S 2 0 0 0 (3) .
Remark 2.1 We use the labels of [33 ] , whose six-object even part is H 2 \mathcal{H}_2 H 2 [43 ] . The inductions of the units from H 2 \mathcal{H}_2 H 2 H 3 \mathcal{H}_3 H 3 I H 2 ( 1 ) ≅ 0 ⊕ b ⊕ 2 a I\!H_{2}(1) \cong\mathbf{0} \oplus\mathbf{b} \oplus2\mathbf{a} I H 2 ( 1 ) ≅ 0 ⊕ b ⊕ 2 a I H 3 ( 1 ) ≅ 0 ⊕ b ⊕ 2 c 0 I\!H_{3}(1) \cong\mathbf{0} \oplus\mathbf{b} \oplus2\mathbf{c}_{0} I H 3 ( 1 ) ≅ 0 ⊕ b ⊕ 2 c 0 [33 Section 2.2] , [4 Corollary 3.8 and Remark 3.7] . Thus 0 , b , a , c 0 \mathbf{0},\mathbf{b},\mathbf{a},\mathbf{c}_{0} 0 , b , a , c 0 1 , π 1 , σ 1 , π 2 1,\pi_{1},\sigma_{1},\pi_{2} 1 , π 1 , σ 1 , π 2 H 3 \mathcal{H}_{3} H 3 ρ \rho ρ 0 \mathbf{0} 0 b \mathbf{b} b d d d d − ≔ 3 − 13 2 d_{-} \coloneqq\frac{3-\sqrt{13}}{2} d − : = 2 3 − 13 c 0 \mathbf{c}_{0} c 0
Vertex operator algebras and character vectors
A vertex operator algebra V V V C 2 C_{2} C 2 V V V M ≇ V M \not\cong V M ≅ V h M > 0 h_{M} > 0 h M > 0 V V V [49 ] , Rep ( V ) \operatorname{Rep}(V) Rep ( V ) C ( K ) \mathcal{C}(K) C ( K ) K K K Rep ( V K ) \operatorname{Rep}(V_{K}) Rep ( V K )
In the case considered here, a braided equivalence Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U) \cong\mathcal{Z}(\mathcal{H}_{3}) Rep ( U ) ≅ Z ( H 3 ) [42 Theorem 6.3] . Pivotal structures differ by a tensor automorphism of the identity functor [31 ] , and such automorphisms are given by the characters of the universal grading group. Hence the pivotal structure is unique. We use this structure and the normalization S 0 , 0 > 0 S_{0,0} > 0 S 0 , 0 > 0
Let ( S , T ) (S,T) ( S , T ) Irr = { 0 = i 0 , i 1 , … } \operatorname{Irr}=\{0=i_{0},i_{1},\ldots\} Irr = { 0 = i 0 , i 1 , … } h i ∈ [ 0 , 1 ) h_{i}\in[0,1) h i ∈ [ 0 , 1 ) c ∈ Q c\in\mathbb{Q} c ∈ Q T c ≔ e − 2 π i c 24 T T_{c}\coloneqq e^{-\frac{2\pi i c}{24}}T T c : = e − 24 2 π i c T q = e 2 π i τ q=e^{2\pi i\tau} q = e 2 π i τ τ \tau τ H \mathbb{H} H
Definition 2.2 An admissible character vector at central charge c c c ( S , T ) (S,T) ( S , T ) X = ( X i ) i ∈ Irr X=(X_{i})_{i\in\operatorname{Irr}} X = ( X i ) i ∈ Irr H \mathbb{H} H
X ( − 1 τ ) = S X ( τ ) X\left(-\frac{1}{\tau}\right)=SX(\tau) X ( − τ 1 ) = S X ( τ ) X ( τ + 1 ) = T c X ( τ ) X(\tau+1)=T_{c}X(\tau) X ( τ + 1 ) = T c X ( τ ) τ ∈ H \tau\in\mathbb{H} τ ∈ H
for any i ∈ Irr i\in\operatorname{Irr} i ∈ Irr X i ( τ ) = ∑ n ≥ 0 a i , n q h i + n i + n − c 24 X_{i}(\tau)=\sum_{n\geq0}a_{i,n}q^{h_{i}+n_{i}+n-\frac{c}{24}} X i ( τ ) = ∑ n ≥ 0 a i , n q h i + n i + n − 24 c a i , n ∈ Z ≥ 0 a_{i,n}\in\mathbb{Z}_{\geq0} a i , n ∈ Z ≥ 0 n i ≥ 0 n_{i}\geq0 n i ≥ 0 H \mathbb{H} H
X 0 ( τ ) = q − c 24 ( 1 + O ( q ) ) X_{0}(\tau)=q^{-\frac{c}{24}}(1+O(q)) X 0 ( τ ) = q − 24 c ( 1 + O ( q ))
for any i ≠ 0 i\ne0 i = 0 X i X_{i} X i − c 24 -\frac{c}{24} − 24 c
Condition 4 is the positivity of the conformal weights of the nonvacuum components. By Zhu’s modular invariance theorem [72 ] and the identification of the modular transformation of the characters with the categorical S S S [49 ] , the following holds.
Lemma 2.3 Let V V V c c c ( Tr M q L 0 − c 24 ) M ∈ Irr ( Rep ( V ) ) \left(\operatorname{Tr}_{M}q^{L_{0}-\frac{c}{24}}\right)_{M\in\operatorname{Irr}(\operatorname{Rep}(V))} ( Tr M q L 0 − 24 c ) M ∈ Irr ( Rep ( V )) c c c Rep ( V ) \operatorname{Rep}(V) Rep ( V )
Proof. The convergence and the transformation law are [72 ] , with the S S S [49 ] . The coefficients are dimensions of graded pieces. The vacuum component starts with q − c 24 q^{-\frac{c}{24}} q − 24 c V V V
Lattice character vectors
Let L L L n n n L ∗ L^* L ∗ q L : L ∗ / L → Q / Z q_L : L^*/L \to\mathbb{Q}/\mathbb{Z} q L : L ∗ / L → Q / Z q L ( λ ) = ⟨ λ , λ ⟩ 2 q_L(\lambda) = \frac{\langle\lambda,\lambda\rangle}{2} q L ( λ ) = 2 ⟨ λ , λ ⟩
Θ L , λ ( τ ) : = 1 η ( τ ) n ∑ v ∈ L + λ q ⟨ v , v ⟩ 2 , λ ∈ L ∗ / L , (5) \Theta_{L,\lambda}(\tau) \mathrel{\mathop:}= \frac{1}{\eta(\tau)^n}\sum_{v\in L+\lambda}q^{\frac{\langle v,v\rangle}{2}},\qquad\lambda\in L^*/L, \tag*{(5)} Θ L , λ ( τ ) : = η ( τ ) n 1 v ∈ L + λ ∑ q 2 ⟨ v , v ⟩ , λ ∈ L ∗ / L , (5) are the characters of the irreducible V L V_L V L Θ L ( − 1 τ ) = S L Θ L ( τ ) \Theta_L\left(-\frac{1}{\tau}\right)=S^L\Theta_L(\tau) Θ L ( − τ 1 ) = S L Θ L ( τ ) S λ , μ L = ∣ L ∗ / L ∣ − 1 2 e − 2 π i ⟨ λ , μ ⟩ S^L_{\lambda,\mu}=|L^*/L|^{-\frac{1}{2}}e^{-2\pi i\langle\lambda,\mu\rangle} S λ , μ L = ∣ L ∗ / L ∣ − 2 1 e − 2 π i ⟨ λ , μ ⟩
Θ L , 0 ( τ ) = q − n 24 ( 1 + ( n + ∣ R ( L ) ∣ ) q + O ( q 2 ) ) , (6) \Theta_{L,0}(\tau)=q^{-\frac{n}{24}}\left(1+(n+|R(L)|)q+O(q^2)\right), \tag*{(6)} Θ L , 0 ( τ ) = q − 24 n ( 1 + ( n + ∣ R ( L ) ∣ ) q + O ( q 2 ) ) , (6) where R ( L ) R(L) R ( L ) L L L q q q n n n h ( − 1 ) 1 h(-1)1 h ( − 1 ) 1 e v e^v e v ⟨ v , v ⟩ = 2 \langle v,v\rangle=2 ⟨ v , v ⟩ = 2
Let K K K K ′ K' K ′ r r r 8 − r 8-r 8 − r φ : K ∗ / K → K ′ ∗ / K ′ \varphi:K^*/K\to K'^*/K' φ : K ∗ / K → K ′ ∗ / K ′ q K ′ ∘ φ = − q K q_{K'}\circ\varphi=-q_K q K ′ ∘ φ = − q K φ \varphi φ K ⊕ K ′ K\oplus K' K ⊕ K ′ E 8 E_8 E 8 E 8 E_8 E 8 K K K K ′ K' K ′ K ⊥ = K ′ K^\perp=K' K ⊥ = K ′ [65 Propositions 1.4.1 and 1.6.1] . Conversely, a primitive embedding K ⊂ E 8 K\subset E_8 K ⊂ E 8 K ′ = K ⊥ K'=K^\perp K ′ = K ⊥
S φ ( λ ) , φ ( μ ) K ′ = S λ , μ K , T φ ( λ ) K ′ = T λ K , (7) S^{K'}_{\varphi(\lambda),\varphi(\mu)}=S^K_{\lambda,\mu},\qquad T^{K'}_{\varphi(\lambda)}=T^K_\lambda, \tag*{(7)} S φ ( λ ) , φ ( μ ) K ′ = S λ , μ K , T φ ( λ ) K ′ = T λ K , (7) since the bilinear form of K ′ ∗ / K ′ K'^*/K' K ′ ∗ / K ′ K ∗ / K K^*/K K ∗ / K φ \varphi φ
Character vectors of the Haagerup double at central charge eight
Let ϱ \varrho ϱ S L 2 ( Z ) \mathrm{SL}_2(\mathbb{Z}) SL 2 ( Z ) C 12 \mathbb{C}^{12} C 12 ϱ ( ( 0 − 1 1 0 ) ) = S \varrho\left(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\right)=S ϱ ( ( 0 1 − 1 0 ) ) = S ϱ ( ( 1 1 0 1 ) ) = T 8 \varrho\left(\begin{pmatrix}1&1\\0&1\end{pmatrix}\right)=T_8 ϱ ( ( 1 0 1 1 ) ) = T 8 ( S , T ) (S,T) ( S , T ) Z ( H 3 ) \mathcal{Z}(\mathcal{H}_3) Z ( H 3 ) M ( ϱ ) M(\varrho) M ( ϱ ) X : H → C 12 X:\mathbb{H}\to\mathbb{C}^{12} X : H → C 12 X ( γ τ ) = ϱ ( γ ) X ( τ ) X(\gamma\tau)=\varrho(\gamma)X(\tau) X ( γ τ ) = ϱ ( γ ) X ( τ ) γ ∈ S L 2 ( Z ) \gamma\in\mathrm{SL}_2(\mathbb{Z}) γ ∈ SL 2 ( Z ) C [ J ] \mathbb{C}[J] C [ J ] J ( τ ) = q − 1 + 196884 q + ⋯ J(\tau)=q^{-1}+196884q+\cdots J ( τ ) = q − 1 + 196884 q + ⋯
Following [33 equation (2.7) pp. 21–22] , put
Λ : = diag ( 2 3 , 2 3 , 2 3 , 2 3 , 0 , 1 3 , 5 39 , 20 39 , 32 39 , 2 39 , 8 39 , 11 39 ) , (8) \Lambda\mathrel{\mathop:}=\operatorname{diag}\left(\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},0,\frac{1}{3},\frac{5}{39},\frac{20}{39},\frac{32}{39},\frac{2}{39},\frac{8}{39},\frac{11}{39}\right), \tag*{(8)} Λ : = diag ( 3 2 , 3 2 , 3 2 , 3 2 , 0 , 3 1 , 39 5 , 39 20 , 39 32 , 39 2 , 39 8 , 39 11 ) , (8) in the order of the labels, and let χ \chi χ 12 × 12 12\times12 12 × 12 [33 equation (2.7)] . Note that e 2 π i Λ = T 8 e^{2\pi i\Lambda}=T_8 e 2 π i Λ = T 8 F = ( J − 240 ) E 10 Δ = ∑ k ≥ 0 f k q k F=(J-240)\frac{E_{10}}{\Delta}=\sum_{k\geq0}f_kq^k F = ( J − 240 ) Δ E 10 = ∑ k ≥ 0 f k q k G = E 10 Δ = ∑ k ≥ 1 g k q k G=\frac{E_{10}}{\Delta}=\sum_{k\geq1}g_kq^k G = Δ E 10 = ∑ k ≥ 1 g k q k Δ \Delta Δ E 10 = E 4 E 6 E_{10}=E_4E_6 E 10 = E 4 E 6 f 0 = 1 f_0=1 f 0 = 1 f 1 = 0 f_1=0 f 1 = 0 g 1 = 1 g_1=1 g 1 = 1 Ξ n \Xi_n Ξ n n ≥ − 1 n\geq-1 n ≥ − 1 Ξ − 1 = Id \Xi_{-1}=\operatorname{Id} Ξ − 1 = Id Ξ 0 = χ \Xi_0=\chi Ξ 0 = χ n ≥ 1 n\geq1 n ≥ 1
( Λ i i − Λ j j + n + 1 ) ( Ξ n ) i j = ( ∑ l = − 1 n − 1 Ξ l ( f n − l ( Λ − Id ) + g n − l ( χ + [ Λ , χ ] ) ) ) i j . (9) (\Lambda_{ii}-\Lambda_{jj}+n+1)(\Xi_n)_{ij}=\left(\sum_{l=-1}^{n-1}\Xi_l\left(f_{n-l}(\Lambda-\operatorname{Id})+g_{n-l}(\chi+[\Lambda,\chi])\right)\right)_{ij}. \tag*{(9)} ( Λ ii − Λ j j + n + 1 ) ( Ξ n ) ij = ( l = − 1 ∑ n − 1 Ξ l ( f n − l ( Λ − Id ) + g n − l ( χ + [ Λ , χ ]) ) ) ij . (9) The coefficient on the left is nonzero since ∣ Λ i i − Λ j j ∣ < 1 |\Lambda_{ii}-\Lambda_{jj}|<1 ∣ Λ ii − Λ j j ∣ < 1
Ξ ( τ ) : = q Λ ( q − 1 Id + ∑ n ≥ 0 Ξ n q n ) . (10) \Xi(\tau)\mathrel{\mathop:}=q^\Lambda\left(q^{-1}\operatorname{Id}+\sum_{n\geq0}\Xi_nq^n\right). \tag*{(10)} Ξ ( τ ) : = q Λ ( q − 1 Id + n ≥ 0 ∑ Ξ n q n ) . (10) The recursion (9) expresses the differential equation q d Ξ d q = Ξ ( ( Λ − Id ) F + ( χ + [ Λ , χ ] ) G ) q\frac{\mathrm{d}\Xi}{\mathrm{d}q}=\Xi\bigl((\Lambda-\operatorname{Id})F+(\chi+[\Lambda,\chi])G\bigr) q d q d Ξ = Ξ ( ( Λ − Id ) F + ( χ + [ Λ , χ ]) G ) q Λ + n q^{\Lambda+n} q Λ + n ( Λ + n ) Ξ n (\Lambda+n)\Xi_n ( Λ + n ) Ξ n l = n l=n l = n Ξ n ( Λ − Id ) \Xi_n(\Lambda-\operatorname{Id}) Ξ n ( Λ − Id ) f 0 = 1 f_0=1 f 0 = 1 [7 Section 3] . The conditions f 1 = 0 f_1=0 f 1 = 0 g 1 = 1 g_1=1 g 1 = 1 n = 0 n=0 n = 0
Theorem 3.1. The following hold.
The columns of Ξ \Xi Ξ M ( ϱ ) \mathcal{M}(\varrho) M ( ϱ ) M ( ϱ ) \mathcal{M}(\varrho) M ( ϱ ) C [ J ] \mathbb{C}[J] C [ J ]
Every admissible character vector at central charge eight for the modular data of Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 )
X ( γ ) = Ξ ∙ , 0 + γ Ξ ∙ , d 3 , γ ∈ { 0 , 1 } , (11) X^{(\gamma)}=\Xi_{\bullet,0}+\gamma\Xi_{\bullet,d_3},\qquad\gamma\in\{0,1\}, \tag*{(11)} X ( γ ) = Ξ ∙ , 0 + γ Ξ ∙ , d 3 , γ ∈ { 0 , 1 } , (11) where Ξ ∙ , j \Xi_{\bullet,j} Ξ ∙ , j j j j
X 0 ( τ ) = q − 1 3 ( 1 + ( 6 + 13 γ ) q + ( 120 + 78 γ ) q 2 + ( 956 + 351 γ ) q 3 + O ( q 4 ) ) , γ ∈ { 0 , 1 } . (12) X_0(\tau)=q^{-\frac{1}{3}}\left(1+(6+13\gamma)q+(120+78\gamma)q^2+(956+351\gamma)q^3+O(q^4)\right),\qquad\gamma\in\{0,1\}. \tag*{(12)} X 0 ( τ ) = q − 3 1 ( 1 + ( 6 + 13 γ ) q + ( 120 + 78 γ ) q 2 + ( 956 + 351 γ ) q 3 + O ( q 4 ) ) , γ ∈ { 0 , 1 } . (12) The second assertion is contained in [33 Section 2.4 p. 22] . We give the argument in the form needed below, together with an independent verification of the data. We do not use, and do not prove, the converse: the coefficients of X ( 0 ) X^{(0)} X ( 0 ) X ( 1 ) X^{(1)} X ( 1 ) q 60 q^{60} q 60
Proof: 1. Assume that the columns of Ξ \Xi Ξ Ξ − 1 = Id \Xi_{-1}=\operatorname{Id} Ξ − 1 = Id C ( J ) \mathbb{C}(J) C ( J ) [7 equation (2.22) p. 6] , the C [ J ] \mathbb{C}[J] C [ J ] M ( ϱ ) \mathcal{M}(\varrho) M ( ϱ )
Tr ( Λ ) = 5 ⋅ 12 12 + 1 4 Tr ( S ) + 2 3 3 Re ( e − π i 6 Tr ( S T 8 − 1 ) ) . (13) \operatorname{Tr}(\Lambda)=\frac{5\cdot12}{12}+\frac{1}{4}\operatorname{Tr}(S)+\frac{2}{3\sqrt{3}}\operatorname{Re}\left(e^{-\frac{\pi i}{6}}\operatorname{Tr}(ST_8^{-1})\right). \tag*{(13)} Tr ( Λ ) = 12 5 ⋅ 12 + 4 1 Tr ( S ) + 3 3 2 Re ( e − 6 π i Tr ( S T 8 − 1 ) ) . (13) The left-hand side is 8 3 + 1 3 + 78 39 = 5 \frac{8}{3}+\frac{1}{3}+\frac{78}{39}=5 3 8 + 3 1 + 39 78 = 5 Tr ( S ) = 0 \operatorname{Tr}(S)=0 Tr ( S ) = 0 Tr ( S T 8 − 1 ) = − 3 2 + 3 3 2 i \operatorname{Tr}(ST_8^{-1})=-\frac{3}{2}+\frac{3\sqrt{3}}{2}i Tr ( S T 8 − 1 ) = − 2 3 + 2 3 3 i Re ( e − π i 6 Tr ( S T 8 − 1 ) ) = − 3 3 4 + 3 3 4 = 0 \operatorname{Re}\left(e^{-\frac{\pi i}{6}}\operatorname{Tr}(ST_8^{-1})\right)=-\frac{3\sqrt{3}}{4}+\frac{3\sqrt{3}}{4}=0 Re ( e − 6 π i Tr ( S T 8 − 1 ) ) = − 4 3 3 + 4 3 3 = 0 Tr ( S ) \operatorname{Tr}(S) Tr ( S ) 1 3 ( 2 x + 2 + ∑ l = 1 6 c ( l 2 ) ) \frac{1}{3}(2x+2+\sum_{l=1}^{6}c(l^2)) 3 1 ( 2 x + 2 + ∑ l = 1 6 c ( l 2 )) ∑ l = 0 12 e 2 π i l 2 13 = 13 \sum_{l=0}^{12}e^{\frac{2\pi i l^2}{13}}=\sqrt{13} ∑ l = 0 12 e 13 2 π i l 2 = 13 ∑ l = 1 6 c ( l 2 ) = − 3 + 3 13 = − ( 2 x + 2 ) \sum_{l=1}^{6}c(l^2)=-3+\frac{3}{\sqrt{13}}=-(2x+2) ∑ l = 1 6 c ( l 2 ) = − 3 + 13 3 = − ( 2 x + 2 ) (13) equals 5.
The modularity of the columns is the content of [33 Section 2.4] . Together with (13) , it proves the assertion.
Let X X X X = ∑ j Ξ ∙ , j P j ( J ) X=\sum_j\Xi_{\bullet,j}P_j(J) X = ∑ j Ξ ∙ , j P j ( J ) P j P_j P j m m m P j P_j P j m ≥ 1 m\geq1 m ≥ 1 j j j deg P j = m \deg P_j=m deg P j = m X j X_j X j λ m q Λ j j − 1 − m \lambda_m q^{\Lambda_{jj}-1-m} λ m q Λ j j − 1 − m λ m ≠ 0 \lambda_m\ne0 λ m = 0 P j P_j P j X X X j j j Λ j j − m \Lambda_{jj}-m Λ j j − m Λ j j ≤ 32 39 \Lambda_{jj}\leq\frac{32}{39} Λ j j ≤ 39 32 Λ j j − 1 − m ≤ 32 39 − 2 < − 1 3 \Lambda_{jj}-1-m\leq\frac{32}{39}-2<-\frac{1}{3} Λ j j − 1 − m ≤ 39 32 − 2 < − 3 1 P j P_j P j v j v_j v j
Now, the column j j j Λ j j − 1 \Lambda_{jj}-1 Λ j j − 1 j j j Λ j j \Lambda_{jj} Λ j j v 0 = 1 v_0=1 v 0 = 1 Λ 00 − 1 = − 1 3 \Lambda_{00}-1=-\frac{1}{3} Λ 00 − 1 = − 3 1 j ≠ 0 j\ne0 j = 0 v j = 0 v_j=0 v j = 0 Λ j j − 1 ≤ − 1 3 \Lambda_{jj}-1\leq-\frac{1}{3} Λ j j − 1 ≤ − 3 1 Λ j j ≤ 2 3 \Lambda_{jj}\leq\frac{2}{3} Λ j j ≤ 3 2 j ≠ 0 j\ne0 j = 0 j = d 3 j=d_3 j = d 3 Λ j j > 2 3 \Lambda_{jj}>\frac{2}{3} Λ j j > 3 2 X = Ξ ∙ , 0 + γ Ξ ∙ , d 3 X=\Xi_{\bullet,0}+\gamma\Xi_{\bullet,d_3} X = Ξ ∙ , 0 + γ Ξ ∙ , d 3 γ = v d 3 \gamma=v_{d_3} γ = v d 3 X d 3 X_{d_3} X d 3 γ \gamma γ γ ∈ Z ≥ 0 \gamma\in\mathbb{Z}_{\geq0} γ ∈ Z ≥ 0 X d 4 X_{d_4} X d 4
( Ξ 0 ) d 4 , 0 + γ ( Ξ 0 ) d 4 , d 3 = X d 4 , 0 + γ X d 4 , d 3 = 5 − 3 γ , (\Xi_0)_{d_4,0}+\gamma(\Xi_0)_{d_4,d_3}=X_{d_4,0}+\gamma X_{d_4,d_3}=5-3\gamma, ( Ξ 0 ) d 4 , 0 + γ ( Ξ 0 ) d 4 , d 3 = X d 4 , 0 + γ X d 4 , d 3 = 5 − 3 γ , which is nonnegative only for γ ≤ 1 \gamma\leq1 γ ≤ 1 (12) is read from the matrices Ξ 0 = χ \Xi_0=\chi Ξ 0 = χ Ξ 1 \Xi_1 Ξ 1 Ξ 2 \Xi_2 Ξ 2
Remark 3.2 The first coefficients of all components produced by our recursion coincide with those printed in [33 p. 22] . For instance, the components d 1 d_1 d 1 d 4 d_4 d 4 d 5 d_5 d 5 d 6 d_6 d 6 7 − γ 7-\gamma 7 − γ 5 − 3 γ 5-3\gamma 5 − 3 γ 13 − 5 γ 13-5\gamma 13 − 5 γ 14 + 7 γ 14+7\gamma 14 + 7 γ γ = 0 \gamma=0 γ = 0 dim U 1 = 6 \dim U_1=6 dim U 1 = 6 dim U 2 = 120 \dim U_2=120 dim U 2 = 120 dim U 3 = 956 \dim U_3=956 dim U 3 = 956 dim U 4 = 6010 \dim U_4=6010 dim U 4 = 6010 U U U Ξ 3 \Xi_3 Ξ 3 § 10 . These numbers agree with the algebra announced in [66 , 50 ] . The vector γ = 1 \gamma=1 γ = 1 Theorem 7.1 , it is not the character vector of a positive strongly rational vertex operator algebra U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U)\cong\mathcal{Z}(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 )
Remark 3.3 The modularity of Ξ \Xi Ξ [33 Section 2.4] . As independent controls, we compute its first 60 coefficients in rational arithmetic, check their integrality and test Ξ ( − 1 / τ ) = S Ξ ( τ ) \Xi(-1/\tau)=S\Xi(\tau) Ξ ( − 1/ τ ) = S Ξ ( τ ) 1.4 × 10 − 50 1.4\times10^{-50} 1.4 × 1 0 − 50 (13) . The floating-point tests do not enter the proof.
The chain U ⊂ V E 6 ⊕ A 2 ⊂ V E 8 U\subset V_{E_6\oplus A_2}\subset V_{E_8} U ⊂ V E 6 ⊕ A 2 ⊂ V E 8
Throughout this section, U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U)\cong\mathcal{Z}(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) z ∈ Irr ( Z ( H 3 ) ) z\in\operatorname{Irr}(\mathcal{Z}(\mathcal{H}_3)) z ∈ Irr ( Z ( H 3 )) M z M_z M z U U U R z R_z R z Lemma 2.3 and Theorem 3.1 , the character vector of U U U X ( γ ) X(\gamma) X ( γ ) γ ∈ { 0 , 1 } \gamma\in\{0,1\} γ ∈ { 0 , 1 } Theorem 3.1 , the lowest conformal weights and the dimensions of the lowest weight spaces are
z 0 b a c 0 c 1 c 2 h z 0 1 1 1 1 3 2 3 dim R z 1 80 − 13 γ 81 81 3 27 z d 1 d 2 d 3 d 4 d 5 d 6 h z 6 13 11 13 2 13 + ε 5 13 7 13 8 13 dim R z 7 − γ 42 + 16 γ γ + 119 ( 1 − γ ) 5 − 3 γ 13 − 5 γ 14 + 7 γ (14) \begin{array}{c|cccccc}
z & 0 & b & a & c_0 & c_1 & c_2 \\
h_z & 0 & 1 & 1 & 1 & \frac{1}{3} & \frac{2}{3} \\
\dim R_z & 1 & 80-13\gamma& 81 & 81 & 3 & 27
\end{array}
\qquad
\begin{array}{c|cccccc}
z & d_1 & d_2 & d_3 & d_4 & d_5 & d_6 \\
h_z & \frac{6}{13} & \frac{11}{13} & \frac{2}{13}+\varepsilon& \frac{5}{13} & \frac{7}{13} & \frac{8}{13} \\
\dim R_z & 7-\gamma& 42+16\gamma& \gamma+119(1-\gamma) & 5-3\gamma& 13-5\gamma& 14+7\gamma \tag*{(14)}
\end{array} z h z dim R z 0 0 1 b 1 80 − 13 γ a 1 81 c 0 1 81 c 1 3 1 3 c 2 3 2 27 z h z dim R z d 1 13 6 7 − γ d 2 13 11 42 + 16 γ d 3 13 2 + ε γ + 119 ( 1 − γ ) d 4 13 5 5 − 3 γ d 5 13 7 13 − 5 γ d 6 13 8 14 + 7 γ (14) where ε = 0 \varepsilon=0 ε = 0 γ = 1 \gamma=1 γ = 1 ε = 1 \varepsilon=1 ε = 1 γ = 0 \gamma=0 γ = 0 dim U 1 = 6 + 13 γ \dim U_1=6+13\gamma dim U 1 = 6 + 13 γ dim U 2 = 120 + 78 γ \dim U_2=120+78\gamma dim U 2 = 120 + 78 γ dim U 3 = 956 + 351 γ \dim U_3=956+351\gamma dim U 3 = 956 + 351 γ
The holomorphic extension and the Z 3 \mathbb{Z}_3 Z 3
Proposition 4.1 Let U U U
The Lagrangian algebra I H 2 ( 1 ) ≅ 0 ⊕ b ⊕ 2 a \mathcal{I}\mathcal{H}_2(1)\cong0\oplus b\oplus2a I H 2 ( 1 ) ≅ 0 ⊕ b ⊕ 2 a H = U ⊕ M b ⊕ M a ⊕ M a H=U\oplus M_b\oplus M_a\oplus M_a H = U ⊕ M b ⊕ M a ⊕ M a U U U H ≅ V E 8 H\cong V_{E_8} H ≅ V E 8
The subspace V ′ ≔ U ⊕ M b V^\prime\coloneqq U\oplus M_b V ′ : = U ⊕ M b H H H g g g H H H H g = V ′ H^g=V^\prime H g = V ′ g g g M a M_a M a U U U
There is an isomorphism V ′ ≅ V E 6 ⊕ A 2 V^\prime\cong V_{E_6\oplus A_2} V ′ ≅ V E 6 ⊕ A 2 V 1 ′ V^\prime_1 V 1 ′ e 6 ⊕ s l 3 \mathfrak{e}_6\oplus\mathfrak{sl}_3 e 6 ⊕ sl 3
Proof. 1. The Lagrangian algebra I H 2 ( 1 ) \mathcal{I}\mathcal{H}_2(1) I H 2 ( 1 ) Rep ( U ) \operatorname{Rep}(U) Rep ( U ) [45 , 15 ] , it defines a simple extension H H H U U U U U U H H H (14) , so H H H L ( 1 ) L(1) L ( 1 ) U 1 U_1 U 1 U U U L ( 1 ) H 1 = 0 L(1)H_1=0 L ( 1 ) H 1 = 0 [55 ] . The arguments for C 2 C_2 C 2 Theorem 12.3 , with U U U Ω U ⊗ V L ♮ \Omega_U \otimes V_{L^\natural} Ω U ⊗ V L ♮ H H H Rep ( H ) ≅ Vec \operatorname{Rep}(H) \cong\operatorname{Vec} Rep ( H ) ≅ Vec V E 8 V_{E_8} V E 8 [26 Theorem 1 p. 2] .
By [23 Theorem 4.10 p. 18] , the subalgebras of I H 2 ( 1 ) I\mathcal{H}_2(1) I H 2 ( 1 ) H 2 \mathcal{H}_2 H 2 Vec Z 3 \operatorname{Vec}_{\mathbb{Z}_3} Vec Z 3 H 2 \mathcal{H}_2 H 2 A A A 9 d + 6 3 = 3 d + 2 \frac{9d+6}{3}=3d+2 3 9 d + 6 = 3 d + 2 0 ⊕ b ⊕ 2 a 0\oplus b\oplus2a 0 ⊕ b ⊕ 2 a 0 0 0 0 ⊕ b 0\oplus b 0 ⊕ b (3) . By [45 ] and the arguments recalled in part 1, V ′ = U ⊕ M b V'=U\oplus M_b V ′ = U ⊕ M b H H H Rep ( V ′ ) ≅ Z ( H 3 ) A \operatorname{Rep}(V')\cong\mathcal{Z}(\mathcal{H}_3)_A Rep ( V ′ ) ≅ Z ( H 3 ) A 9 ( 3 d + 2 ) 2 ( 3 d + 2 ) 2 = 9 \frac{9(3d+2)^2}{(3d+2)^2}=9 ( 3 d + 2 ) 2 9 ( 3 d + 2 ) 2 = 9 [23 Corollary 3.32 p. 15] . It is Witt equivalent to Z ( H 3 ) \mathcal{Z}(\mathcal{H}_3) Z ( H 3 ) [23 Proposition 5.4 p. 22] , hence of the form Z ( D ) \mathcal{Z}(D) Z ( D ) D D D [23 Proposition 5.8 p. 23] , with FPdim ( D ) = 3 \operatorname{FPdim}(D)=3 FPdim ( D ) = 3 [31 Corollary 8.30] . Here D ≅ Vec Z 3 ω D\cong\operatorname{Vec}_{\mathbb{Z}_3}^{\omega} D ≅ Vec Z 3 ω ω \omega ω Z 3 \mathbb{Z}_3 Z 3 H 2 ( Z 3 , C × ) = 0 H^2(\mathbb{Z}_3,\mathbb{C}^{\times})=0 H 2 ( Z 3 , C × ) = 0 Rep ( V ′ ) \operatorname{Rep}(V') Rep ( V ′ )
For every nonvacuum simple V ′ V' V ′ N N N R ( N ) R(N) R ( N ) 3 d + 2 3d+2 3 d + 2
Hom U ( U , R ( N ) ) ≅ Hom V ′ ( V ′ , N ) = 0. \operatorname{Hom}_U(U,R(N))\cong\operatorname{Hom}_{V'}(V',N)=0. Hom U ( U , R ( N )) ≅ Hom V ′ ( V ′ , N ) = 0. The list (3) therefore forces R ( N ) R(N) R ( N ) U U U V ′ V' V ′ H H H M a ⊕ M a M_a\oplus M_a M a ⊕ M a V ′ V' V ′ H = V ′ ⊕ N 1 ⊕ N 2 H=V'\oplus N_1\oplus N_2 H = V ′ ⊕ N 1 ⊕ N 2 N 1 , N 2 N_1,N_2 N 1 , N 2 V ′ V' V ′ M a M_a M a U U U H H H Rep ( V ′ ) \operatorname{Rep}(V') Rep ( V ′ ) End H ( H ) ≅ Hom ( 1 , H ) = C \operatorname{End}_H(H)\cong\operatorname{Hom}(1,H)=\mathbb{C} End H ( H ) ≅ Hom ( 1 , H ) = C J J J J ⊗ H J\otimes H J ⊗ H Hom ( J , H ) ≅ Hom H ( J ⊗ H , H ) \operatorname{Hom}(J,H)\cong\operatorname{Hom}_H(J\otimes H,H) Hom ( J , H ) ≅ Hom H ( J ⊗ H , H ) J J J H H H H H H J ↦ J ⊗ H J\mapsto J\otimes H J ↦ J ⊗ H FPdim Rep ( V ′ ) ( H ) = 3 \operatorname{FPdim}_{\operatorname{Rep}(V')}(H)=3 FPdim Rep ( V ′ ) ( H ) = 3 e 2 π i k / 3 e^{2\pi i k/3} e 2 π ik /3 N k N_k N k g g g H g = V ′ H^g=V' H g = V ′
The automorphism group of V E 8 V_{E_8} V E 8 E 8 E_8 E 8 [28 Theorem 2.1] . An element of order three is conjugate to e 2 π i β ( 0 ) e^{2\pi i\beta(0)} e 2 π i β ( 0 ) β \beta β [54 Chapter 8] . There are four classes of elements of order three, whose centralizers in e 8 \mathfrak{e}_8 e 8 A 8 A_8 A 8 E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2 D 7 ⊕ C D_7\oplus\mathbb{C} D 7 ⊕ C E 7 ⊕ C E_7\oplus\mathbb{C} E 7 ⊕ C dim V 1 ′ = ( 6 + 13 γ ) + ( 80 − 13 γ ) = 86 \dim V'_1=(6+13\gamma)+(80-13\gamma)=86 dim V 1 ′ = ( 6 + 13 γ ) + ( 80 − 13 γ ) = 86 (14) , so that g g g E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2 g = e 2 π i β ( 0 ) g=e^{2\pi i\beta(0)} g = e 2 π i β ( 0 ) { α ∈ E 8 : ⟨ α , β ⟩ ∈ Z } \{\alpha\in E_8:\langle\alpha,\beta\rangle\in\mathbb{Z}\} { α ∈ E 8 : ⟨ α , β ⟩ ∈ Z } E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2 E 8 E_8 E 8 E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2
Remark 4.2 The realization of V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 Z 3 \mathbb{Z}_3 Z 3 V E 8 V_{E_8} V E 8 [33 Section 2.4 p. 22] . The physical picture of a hypergroup acting on the Z 3 \mathbb{Z}_3 Z 3 [41 Section 4.3 pp. 83–88] , and the announced algebra of [66 , 50 ] is constructed inside V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 γ \gamma γ
Restriction of the lattice modules
The discriminant group of E 6 ⊕ A 2 E_6 \oplus A_2 E 6 ⊕ A 2 Z 3 × Z 3 \mathbb{Z}_3 \times\mathbb{Z}_3 Z 3 × Z 3 q ( i , j ) = 2 i 2 + j 2 3 m o d 1 q(i,j)=\frac{2i^2+j^2}{3}\bmod1 q ( i , j ) = 3 2 i 2 + j 2 mod 1 V ( i , j ) V_{(i,j)} V ( i , j ) V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 { ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 ) } \{(0,0),(1,1),(2,2)\} {( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 )} { ( 0 , 0 ) , ( 1 , 2 ) , ( 2 , 1 ) } \{(0,0),(1,2),(2,1)\} {( 0 , 0 ) , ( 1 , 2 ) , ( 2 , 1 )} H H H H H H { ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 ) } \{(0,0),(1,1),(2,2)\} {( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 )} A 2 A_2 A 2 H H H S S S V ( i , j ) V_{(i,j)} V ( i , j ) e 6 ⊕ s l 3 \mathfrak{e}_6\oplus\mathfrak{sl}_3 e 6 ⊕ sl 3 ( 27 i , 3 j ) (27^i,3^j) ( 2 7 i , 3 j ) 27 0 = 3 0 27^0=3^0 2 7 0 = 3 0 27 1 = 27 27^1=27 2 7 1 = 27 27 2 = 27 ‾ 27^2=\overline{27} 2 7 2 = 27 3 3 3
Lemma 4.3 Use the coordinates adapted to H H H ( i , j ) ≠ ( 0 , 0 ) (i,j)\ne(0,0) ( i , j ) = ( 0 , 0 ) V ( i , j ) V_{(i,j)} V ( i , j ) U U U M c 1 M_{c_1} M c 1 i = 0 i=0 i = 0 M c 2 M_{c_2} M c 2 j = 0 j=0 j = 0 M a M_a M a ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) E 8 / ( E 6 ⊕ A 2 ) E_8/(E_6\oplus A_2) E 8 / ( E 6 ⊕ A 2 ) M c 0 M_{c_0} M c 0 ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 2 , 1 ) (2,1) ( 2 , 1 )
Proof: The Frobenius–Perron dimension and free-module adjunction argument in the proof of Proposition 4.1, part 2, shows that the restriction R ( w ) R(w) R ( w ) V ′ V' V ′ a a a c 0 c_0 c 0 c 1 c_1 c 1 c 2 c_2 c 2 R ( w ) R(w) R ( w ) e 2 π i q ( i , j ) e^{2\pi i q(i,j)} e 2 π i q ( i , j ) a a a c 0 c_0 c 0 c 1 c_1 c 1 c 2 c_2 c 2 e 2 π i 3 e^{\frac{2\pi i}{3}} e 3 2 π i e − 2 π i 3 e^{-\frac{2\pi i}{3}} e − 3 2 π i (4) , while q ( 0 , ± 1 ) = 1 3 q(0,\pm1)=\frac{1}{3} q ( 0 , ± 1 ) = 3 1 q ( ± 1 , 0 ) = 2 3 q(\pm1,0)=\frac{2}{3} q ( ± 1 , 0 ) = 3 2 q ( ± 1 , ± 1 ) = 0 q(\pm1,\pm1)=0 q ( ± 1 , ± 1 ) = 0 E 8 E_8 E 8 M a M_a M a ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) M c 0 M_{c_0} M c 0 z z z Z ( H 3 ) Z(H_3) Z ( H 3 ) w ∈ Z 3 × Z 3 w\in\mathbb{Z}_3\times\mathbb{Z}_3 w ∈ Z 3 × Z 3 n z , w ∈ { 0 , 1 } n_{z,w}\in\{0,1\} n z , w ∈ { 0 , 1 } M z M_z M z V w V_w V w v ∈ U v\in U v ∈ U Z V w ( v , τ ) = ∑ z n z , w Z M z ( v , τ ) Z_{V_w}(v,\tau)=\sum_z n_{z,w}Z_{M_z}(v,\tau) Z V w ( v , τ ) = ∑ z n z , w Z M z ( v , τ ) τ ↦ − 1 τ \tau\mapsto-\frac{1}{\tau} τ ↦ − τ 1 S S S [72 Section 5] , [49 ] . Applying this to U U U V ′ V' V ′ S n = n S L Sn=nS^L S n = n S L S L S^L S L S S S V ′ = V L V'=V_L V ′ = V L L = E 6 ⊕ A 2 L=E_6\oplus A_2 L = E 6 ⊕ A 2 (5) . All the entries S w , ( 0 , 0 ) L S^L_{w,(0,0)} S w , ( 0 , 0 ) L 1 3 \frac{1}{3} 3 1 V ( 0 , 0 ) = V ′ V_{(0,0)}=V' V ( 0 , 0 ) = V ′ U ⊕ M b U\oplus M_b U ⊕ M b ( c 0 , ( 0 , 0 ) ) (c_0,(0,0)) ( c 0 , ( 0 , 0 )) S c 0 , 0 + S c 0 , b = 1 3 # { w : n c 0 , w = 1 } S_{c_0,0}+S_{c_0,b}=\frac{1}{3}\#\{w:n_{c_0,w}=1\} S c 0 , 0 + S c 0 , b = 3 1 # { w : n c 0 , w = 1 } 1 3 + 1 3 = 2 3 \frac{1}{3}+\frac{1}{3}=\frac{2}{3} 3 1 + 3 1 = 3 2 S S S M c 0 M_{c_0} M c 0 ( 1 , 2 ) (1,2) ( 1 , 2 ) ( 2 , 1 ) (2,1) ( 2 , 1 )
We have checked Lemma 4.3 on characters: for both values of γ \gamma γ Θ ( 0 , 0 ) = X 0 + X b \Theta_{(0,0)}=X_0+X_b Θ ( 0 , 0 ) = X 0 + X b Θ ( ± 1 , ± 1 ) = X a = X c 0 \Theta_{(\pm1,\pm1)}=X_a=X_{c_0} Θ ( ± 1 , ± 1 ) = X a = X c 0 Θ ( 0 , ± 1 ) = X c 1 \Theta_{(0,\pm1)}=X_{c_1} Θ ( 0 , ± 1 ) = X c 1 Θ ( ± 1 , 0 ) = X c 2 \Theta_{(\pm1,0)}=X_{c_2} Θ ( ± 1 , 0 ) = X c 2 q 4 q^4 q 4 a a a c 0 c_0 c 0
Self-duality constraints on the currents
Let s : = U 1 ⊂ e 6 ⊕ s l 3 \mathfrak{s}\mathrel{\mathop:}=U_1\subset\mathfrak{e}_6\oplus\mathfrak{sl}_3 s : = U 1 ⊂ e 6 ⊕ sl 3 p 1 p_1 p 1 p 2 p_2 p 2 e 6 \mathfrak{e}_6 e 6 s l 3 \mathfrak{sl}_3 sl 3 s \mathfrak{s} s [25 Theorem 1, p. 2] . For a simple ideal g \mathfrak{g} g s \mathfrak{s} s i 1 ( g ) i_1(\mathfrak{g}) i 1 ( g ) i 2 ( g ) i_2(\mathfrak{g}) i 2 ( g ) p 1 ∣ g p_1|\mathfrak{g} p 1 ∣ g p 2 ∣ g p_2|\mathfrak{g} p 2 ∣ g e 6 \mathfrak{e}_6 e 6 s l 3 \mathfrak{sl}_3 sl 3 e 8 \mathfrak{e}_8 e 8 g \mathfrak{g} g U U U k ( g ) = i 1 ( g ) + i 2 ( g ) k(\mathfrak{g})=i_1(\mathfrak{g})+i_2(\mathfrak{g}) k ( g ) = i 1 ( g ) + i 2 ( g )
Proposition 4.4
The representations 3 3 3 3 ‾ \overline{3} 3 p 2 ( s ) p_2(\mathfrak{s}) p 2 ( s ) 27 27 27 27 ‾ \overline{27} 27 p 1 ( s ) p_1(\mathfrak{s}) p 1 ( s )
The Lie algebra p 2 ( s ) p_2(\mathfrak{s}) p 2 ( s ) ( t , − t , 0 ) (t,-t,0) ( t , − t , 0 ) 3 3 3 s l 2 \mathfrak{sl}_2 sl 2 s o 3 \mathfrak{so}_3 so 3 g \mathfrak{g} g s \mathfrak{s} s p 2 ( g ) ≠ 0 p_2(\mathfrak{g})\ne0 p 2 ( g ) = 0 s l 2 \mathfrak{sl}_2 sl 2 i 2 ( g ) ∈ { 1 , 4 } i_2(\mathfrak{g})\in\{1,4\} i 2 ( g ) ∈ { 1 , 4 }
The rank of p 1 ( s ) p_1(\mathfrak{s}) p 1 ( s )
Proof. 1. By Lemma 4.3 , V ( 0 , 1 ) V_{(0,1)} V ( 0 , 1 ) V ( 0 , 2 ) V_{(0,2)} V ( 0 , 2 ) U U U x ( 0 ) x(0) x ( 0 ) x ∈ U 1 x \in U_1 x ∈ U 1 ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 1 , 3 ‾ ) (1,\overline{3}) ( 1 , 3 ) x ∈ s x \in\mathfrak{s} x ∈ s p 2 ( x ) p_2(x) p 2 ( x ) V ( 1 , 0 ) V_{(1,0)} V ( 1 , 0 ) V ( 2 , 0 ) V_{(2,0)} V ( 2 , 0 )
The action of s \mathfrak{s} s U U U [25 Theorem 1 p. 2] . By Lemma 4.3 , this applies in particular to 3 \mathbf{3} 3 27 \mathbf{27} 27 p 2 ( s ) p_2(\mathfrak{s}) p 2 ( s ) p 1 ( s ) p_1(\mathfrak{s}) p 1 ( s ) s l 3 \mathfrak{sl}_3 sl 3 0 0 0 s l 2 \mathfrak{sl}_2 sl 2 3 = 2 ⊕ 1 \mathbf{3}=\mathbf{2}\oplus\mathbf{1} 3 = 2 ⊕ 1 g l 2 \mathfrak{gl}_2 gl 2 s o 3 \mathfrak{so}_3 so 3 3 \mathbf{3} 3 s l 3 \mathfrak{sl}_3 sl 3 3 \mathbf{3} 3 g l 2 \mathfrak{gl}_2 gl 2 s l 3 \mathfrak{sl}_3 sl 3 3 \mathbf{3} 3 ( t , − t , 0 ) (t,-t,0) ( t , − t , 0 ) s l 2 \mathfrak{sl}_2 sl 2 s o 3 \mathfrak{so}_3 so 3
Choose a Cartan subalgebra h \mathfrak{h} h e 6 \mathfrak{e}_6 e 6 p 1 ( s ) p_1(\mathfrak{s}) p 1 ( s ) Ω \Omega Ω 27 \mathbf{27} 27 ω 1 \omega_1 ω 1 t \mathfrak{t} t p 1 ( s ) p_1(\mathfrak{s}) p 1 ( s ) Ω \Omega Ω t \mathfrak{t} t λ ∈ Ω \lambda\in\Omega λ ∈ Ω t \mathfrak{t} t Ω \Omega Ω t \mathfrak{t} t h ∗ \mathfrak{h}^{*} h ∗ C λ \mathbb{C}\lambda C λ λ = ω 1 \lambda=\omega_1 λ = ω 1
Let P P P Q Q Q E 6 E_6 E 6
ω 1 = 4 α 1 + 3 α 2 + 5 α 3 + 6 α 4 + 4 α 5 + 2 α 6 3 , ⟨ ω 1 , ω 1 ⟩ = 4 3 , ⟨ ω 1 , α 1 ⟩ = 1. \omega_1=\frac{4\alpha_1+3\alpha_2+5\alpha_3+6\alpha_4+4\alpha_5+2\alpha_6}{3},\qquad\langle\omega_1,\omega_1\rangle=\frac{4}{3},\qquad\langle\omega_1,\alpha_1\rangle=1. ω 1 = 3 4 α 1 + 3 α 2 + 5 α 3 + 6 α 4 + 4 α 5 + 2 α 6 , ⟨ ω 1 , ω 1 ⟩ = 3 4 , ⟨ ω 1 , α 1 ⟩ = 1. The weight μ = s α 1 ( ω 1 ) = ω 1 − α 1 \mu=s_{\alpha_1}(\omega_1)=\omega_1-\alpha_1 μ = s α 1 ( ω 1 ) = ω 1 − α 1 Ω \Omega Ω ν ∈ Ω \nu\in\Omega ν ∈ Ω c ∈ C c\in\mathbb{C} c ∈ C μ + ν = c ω 1 \mu+\nu=c\omega_1 μ + ν = c ω 1 Ω \Omega Ω 4 / 3 4/3 4/3 ⟨ μ , ω 1 ⟩ = 1 / 3 \langle\mu,\omega_1\rangle=1/3 ⟨ μ , ω 1 ⟩ = 1/3
0 = ⟨ c ω 1 − μ , c ω 1 − μ ⟩ − ⟨ μ , μ ⟩ = 4 3 c 2 − 2 3 c , 0=\langle c\omega_1-\mu,c\omega_1-\mu\rangle-\langle\mu,\mu\rangle=\frac{4}{3}c^2-\frac{2}{3}c, 0 = ⟨ c ω 1 − μ , c ω 1 − μ ⟩ − ⟨ μ , μ ⟩ = 3 4 c 2 − 3 2 c , so c = 0 c=0 c = 0 c = 1 / 2 c=1/2 c = 1/2 Ω \Omega Ω ω 1 + Q \omega_1+Q ω 1 + Q 2 ω 1 ∉ Q 2\omega_1\notin Q 2 ω 1 ∈ / Q μ + ν ∈ P \mu+\nu\in P μ + ν ∈ P ω 1 / 2 ∉ P \omega_1/2\notin P ω 1 /2 ∈ / P
For comparison, the finite calculation can be retained independently. If two multisets of ambient weights coincide modulo a line while differing before restriction, that line is spanned by a nonzero sum w + w ′ w+w' w + w ′ Ω \Omega Ω v v v Q v \mathbb{Q}v Q v
Remark 4.5 The announced algebra of [66 , 50 ] has U 1 ≅ s l 2 ⊕ s l 2 U_1\cong\mathfrak{sl}_2\oplus\mathfrak{sl}_2 U 1 ≅ sl 2 ⊕ sl 2 Proposition 4.4 : e 6 ⊃ s l 6 ⊕ s l 2 \mathfrak{e}_6\supset\mathfrak{sl}_6\oplus\mathfrak{sl}_2 e 6 ⊃ sl 6 ⊕ sl 2 27 = ( 6 , 2 ) ⊕ ( 15 , 1 ) \mathbf{27}=(\mathbf{6},2)\oplus(\mathbf{15},1) 27 = ( 6 , 2 ) ⊕ ( 15 , 1 ) s l 2 \mathfrak{sl}_2 sl 2 s l 2 \mathfrak{sl}_2 sl 2 s l 6 \mathfrak{sl}_6 sl 6 s o 3 \mathfrak{so}_3 so 3 s l 3 \mathfrak{sl}_3 sl 3 Theorem 8.5 . We do not use this description before § 8 .
Weight-two relations
We now combine the modularity of one-point functions with the explicit lowest weight spaces given by the chain. This is a vector-valued version of the trace identities of Schellekens [69 ] . and Dong–Mason [26 ] . In the context of the Haagerup modular data, the weight-two one-point functions of currents were introduced by Gannon [40 Section 2.4] , who called them the Killing vector and used them, under the assumptions that U U U V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 [40 Section 3.3] . The novelty below is that the relations are obtained in exact arithmetic, for both character vectors, without unitarity.
Lemma 5.1 Let V V V c c c h ∈ V 1 h \in V_1 h ∈ V 1 L ( 1 ) h = 0 L(1)h = 0 L ( 1 ) h = 0 V 1 V_1 V 1 V V V M M M
Z M ( h [ − 1 ] h , τ ) = Tr M h ( 0 ) 2 q L ( 0 ) − c 24 − ⟨ h , h ⟩ 12 E 2 ( τ ) Tr M q L ( 0 ) − c 24 , (15) Z_M(h[-1]h,\tau) = \operatorname{Tr}_M h(0)^2q^{L(0)-\frac{c}{24}}-\frac{\langle h,h\rangle}{12}E_2(\tau)\operatorname{Tr}_M q^{L(0)-\frac{c}{24}}, \tag*{(15)} Z M ( h [ − 1 ] h , τ ) = Tr M h ( 0 ) 2 q L ( 0 ) − 24 c − 12 ⟨ h , h ⟩ E 2 ( τ ) Tr M q L ( 0 ) − 24 c , (15) where h ( 1 ) h = ⟨ h , h ⟩ 1 h(1)h = \langle h,h\rangle1 h ( 1 ) h = ⟨ h , h ⟩ 1 ( Z M z ( h [ − 1 ] h , τ ) ) z (Z_{M_z}(h[-1]h,\tau))_z ( Z M z ( h [ − 1 ] h , τ ) ) z S L 2 ( Z ) \mathrm{SL}_2(\mathbb{Z}) SL 2 ( Z )
Proof : From Y [ h , z ] = Y ( h , e z − 1 ) e z Y[h,z] = Y(h,e^z-1)e^z Y [ h , z ] = Y ( h , e z − 1 ) e z e z e z − 1 = 1 z + 1 2 + O ( z ) \frac{e^z}{e^z-1} = \frac{1}{z}+\frac{1}{2}+O(z) e z − 1 e z = z 1 + 2 1 + O ( z ) e z ( e z − 1 ) 2 = 1 z 2 − 1 12 + O ( z ) \frac{e^z}{(e^z-1)^2} = \frac{1}{z^2}-\frac{1}{12}+O(z) ( e z − 1 ) 2 e z = z 2 1 − 12 1 + O ( z ) h [ − 1 ] h = h ( − 1 ) h + 1 2 h ( 0 ) h − 1 12 h ( 1 ) h = h ( − 1 ) h − ⟨ h , h ⟩ 12 1 h[-1]h = h(-1)h+\frac{1}{2}h(0)h-\frac{1}{12}h(1)h = h(-1)h-\frac{\langle h,h\rangle}{12}1 h [ − 1 ] h = h ( − 1 ) h + 2 1 h ( 0 ) h − 12 1 h ( 1 ) h = h ( − 1 ) h − 12 ⟨ h , h ⟩ 1 h ( 0 ) h = 0 h(0)h = 0 h ( 0 ) h = 0 h ( n ) h = 0 h(n)h = 0 h ( n ) h = 0 n ≥ 2 n \ge2 n ≥ 2 h ( − 1 ) h h(-1)h h ( − 1 ) h h ( 0 ) 2 + 2 ∑ n ≥ 1 h ( − n ) h ( n ) h(0)^2+2\sum_{n\ge1}h(-n)h(n) h ( 0 ) 2 + 2 ∑ n ≥ 1 h ( − n ) h ( n ) κ = ⟨ h , h ⟩ \kappa=\langle h,h\rangle κ = ⟨ h , h ⟩ Q = q L ( 0 ) − c / 24 Q=q^{L(0)-c/24} Q = q L ( 0 ) − c /24 m ≥ 1 m\ge1 m ≥ 1 [ h ( m ) , h ( − m ) ] = m κ Id [h(m),h(-m)]=m\kappa\operatorname{Id} [ h ( m ) , h ( − m )] = mκ Id
A m : = Tr M h ( − m ) h ( m ) Q = q m ( A m + m κ Tr M Q ) . A_m \mathrel{\mathop:}= \operatorname{Tr}_M h(-m)h(m)Q = q^m(A_m+m\kappa\operatorname{Tr}_M Q). A m : = Tr M h ( − m ) h ( m ) Q = q m ( A m + mκ Tr M Q ) . Thus A m = m κ q m 1 − q m Tr M Q A_m=\frac{m\kappa q^m}{1-q^m}\operatorname{Tr}_M Q A m = 1 − q m mκ q m Tr M Q ∑ m ≥ 1 m q m 1 − q m = ( 1 − E 2 ) / 24 \sum_{m\ge1}\frac{mq^m}{1-q^m}=(1-E_2)/24 ∑ m ≥ 1 1 − q m m q m = ( 1 − E 2 ) /24 (15) . This is first an identity of graded series and then of convergent functions. It also covers κ = 0 \kappa=0 κ = 0 h [ − 1 ] h h[-1]h h [ − 1 ] h L [ 0 ] L[0] L [ 0 ] [72 ] together with the identification of the representation [49 ] .
The obstruction principle
We denote by ϱ ∗ : = ( ϱ T ) − 1 \varrho^* \mathrel{\mathop:}= (\varrho^T)^{-1} ϱ ∗ : = ( ϱ T ) − 1 § 2 , we have T t = S T^t=S T t = S S 2 = Id S^2=\operatorname{Id} S 2 = Id T 8 ∗ = T 8 − 1 T_8^* = T_8^{-1} T 8 ∗ = T 8 − 1 T 8 T_8 T 8 ϱ ∗ ( S ) = S \varrho^*(S)=S ϱ ∗ ( S ) = S ϱ ∗ ( T ) = T 8 \varrho^*(T)=T_8 ϱ ∗ ( T ) = T 8 W W W H \mathbb{H} H C 12 \mathbb{C}^{12} C 12 G G G W ⋅ G = ∑ z W z G z W\cdot G=\sum_z W_zG_z W ⋅ G = ∑ z W z G z D = q d d q D=q\frac{d}{dq} D = q d q d ϑ k = D − k 12 E 2 \vartheta_k=D-\frac{k}{12}E_2 ϑ k = D − 12 k E 2 k k k ϑ k \vartheta_k ϑ k k k k ϱ \varrho ϱ k + 2 k+2 k + 2 ϱ \varrho ϱ D J = − E 4 2 E 6 Δ DJ=-\frac{E_4^2E_6}{\Delta} D J = − Δ E 4 2 E 6
Lemma 5.2 Let k ∈ 2 Z k\in2\mathbb{Z} k ∈ 2 Z W W W k k k ϱ \varrho ϱ G G G 2 − k 2-k 2 − k ϱ ∗ \varrho^* ϱ ∗ q q q W ⋅ G W\cdot G W ⋅ G
Proof : For γ = ( a b c d ) ∈ S L 2 ( Z ) \gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb{Z}) γ = ( a c b d ) ∈ SL 2 ( Z )
( W ⋅ G ) ( γ τ ) = ( c τ + d ) k ( c τ + d ) 2 − k ( ϱ ( γ ) W ( τ ) ) t ϱ ( γ ) − t G ( τ ) = ( c τ + d ) 2 ( W ⋅ G ) ( τ ) . (W\cdot G)(\gamma\tau)=(c\tau+d)^k(c\tau+d)^{2-k}(\varrho(\gamma)W(\tau))^t\varrho(\gamma)^{-t}G(\tau)=(c\tau+d)^2(W\cdot G)(\tau). ( W ⋅ G ) ( γ τ ) = ( c τ + d ) k ( c τ + d ) 2 − k ( ϱ ( γ ) W ( τ ) ) t ϱ ( γ ) − t G ( τ ) = ( c τ + d ) 2 ( W ⋅ G ) ( τ ) . Hence f = W ⋅ G f=W\cdot G f = W ⋅ G S L 2 ( Z ) \mathrm{SL}_2(\mathbb{Z}) SL 2 ( Z ) f ( τ ) d τ f(\tau)\,d\tau f ( τ ) d τ f ( τ ) d τ f(\tau)\,d\tau f ( τ ) d τ s s s τ = i \tau=i τ = i τ ↦ − 1 τ \tau\mapsto-\frac{1}{\tau} τ ↦ − τ 1 s ↦ − s s\mapsto-s s ↦ − s F ( s ) d s F(s)\,ds F ( s ) d s F ( − s ) = − F ( s ) F(-s)=-F(s) F ( − s ) = − F ( s ) F ( s ) = s G ( s 2 ) F(s)=sG(s^2) F ( s ) = s G ( s 2 ) G G G F ( s ) d s = G ( t ) d t / 2 F(s)\,ds=G(t)\,dt/2 F ( s ) d s = G ( t ) d t /2 t = s 2 t=s^2 t = s 2 e 2 π i / 3 e^{2\pi i/3} e 2 π i /3 q q q f ( τ ) d τ = 1 2 π i f d q q f(\tau)\,d\tau=\frac{1}{2\pi i}f\frac{dq}{q} f ( τ ) d τ = 2 π i 1 f q d q f f f 2 π i 2\pi i 2 π i [9 Theorem 3.1 p. 5] .
We now produce forms for ϱ ∗ \varrho^* ϱ ∗ ϱ \varrho ϱ π \pi π c 1 c_1 c 1 c 2 c_2 c 2 d l d_l d l d l ′ d_{l'} d l ′ l ′ ≡ ± 5 l ( m o d 13 ) l'\equiv\pm5l\pmod{13} l ′ ≡ ± 5 l ( mod 13 )
π = ( c 1 c 2 ) ( d 1 d 5 ) ( d 2 d 3 ) ( d 4 d 6 ) . (16) \pi=(c_1\ c_2)(d_1\ d_5)(d_2\ d_3)(d_4\ d_6). \tag*{(16)} π = ( c 1 c 2 ) ( d 1 d 5 ) ( d 2 d 3 ) ( d 4 d 6 ) . (16) Lemma 5.3. Let Y Y Y k k k ϱ \varrho ϱ G G G G z = η − 8 Y π ( z ) G_z=\eta^{-8}Y_{\pi(z)} G z = η − 8 Y π ( z ) k − 4 k-4 k − 4 ϱ ∗ \varrho^* ϱ ∗
Proof. We first check that S π ( z ) , π ( w ) = S z , w S_{\pi(z),\pi(w)}=S_{z,w} S π ( z ) , π ( w ) = S z , w z , w z,w z , w ( a , c 0 , c 1 , c 2 ) (a,c_0,c_1,c_2) ( a , c 0 , c 1 , c 2 ) c 1 c_1 c 1 c 2 c_2 c 2 0 0 0 b b b { a , c 0 , c 1 , c 2 } \{a,c_0,c_1,c_2\} { a , c 0 , c 1 , c 2 } { d 1 , … , d 6 } \{d_1,\ldots,d_6\} { d 1 , … , d 6 } c ( l l ′ ) c(ll') c ( l l ′ ) l l ′ ll' l l ′ 25 ≡ − 1 ( m o d 13 ) 25\equiv-1\pmod{13} 25 ≡ − 1 ( mod 13 ) c ( 5 l ⋅ 5 l ′ ) = c ( − l l ′ ) = c ( l l ′ ) c(5l\cdot5l')=c(-ll')=c(ll') c ( 5 l ⋅ 5 l ′ ) = c ( − l l ′ ) = c ( l l ′ ) Λ π ( z ) − 1 3 ≡ − Λ z ( m o d 1 ) \Lambda_{\pi(z)}-\frac{1}{3}\equiv-\Lambda_z\pmod1 Λ π ( z ) − 3 1 ≡ − Λ z ( mod 1 ) z z z z ∈ { 0 , b , a , c 0 } z\in\{0,b,a,c_0\} z ∈ { 0 , b , a , c 0 } 2 3 − 1 3 ≡ − 2 3 \frac{2}{3}-\frac{1}{3}\equiv-\frac{2}{3} 3 2 − 3 1 ≡ − 3 2 z ∈ { c 1 , c 2 } z\in\{c_1,c_2\} z ∈ { c 1 , c 2 } Λ c 2 − 1 3 = 0 = − Λ c 1 \Lambda_{c_2}-\frac{1}{3}=0=-\Lambda_{c_1} Λ c 2 − 3 1 = 0 = − Λ c 1 Λ c 1 − 1 3 = − 1 3 = − Λ c 2 \Lambda_{c_1}-\frac{1}{3}=-\frac{1}{3}=-\Lambda_{c_2} Λ c 1 − 3 1 = − 3 1 = − Λ c 2 z = d l z=d_l z = d l Λ d l ≡ 6 l 2 13 − 1 3 ( m o d 1 ) \Lambda_{d_l}\equiv\frac{6l^2}{13}-\frac{1}{3}\pmod1 Λ d l ≡ 13 6 l 2 − 3 1 ( mod 1 ) Λ π ( d l ) − 1 3 ≡ 150 l 2 13 − 2 3 ≡ − 6 l 2 13 + 1 3 ≡ − Λ d l ( m o d 1 ) \Lambda_{\pi(d_l)}-\frac{1}{3}\equiv\frac{150l^2}{13}-\frac{2}{3}\equiv-\frac{6l^2}{13}+\frac{1}{3}\equiv-\Lambda_{d_l}\pmod1 Λ π ( d l ) − 3 1 ≡ 13 150 l 2 − 3 2 ≡ − 13 6 l 2 + 3 1 ≡ − Λ d l ( mod 1 ) 150 ≡ − 6 ( m o d 13 ) 150\equiv-6\pmod{13} 150 ≡ − 6 ( mod 13 ) η ( τ + 1 ) − 8 = e − 2 π i / 3 η ( τ ) − 8 \eta(\tau+1)^{-8}=e^{-2\pi i/3}\eta(\tau)^{-8} η ( τ + 1 ) − 8 = e − 2 π i /3 η ( τ ) − 8 G z ( τ + 1 ) = e 2 π i ( Λ π ( z ) − 1 / 3 ) G z ( τ ) = e − 2 π i Λ z G z ( τ ) = ( T ∗ G ( τ ) ) z G_z(\tau+1)=e^{2\pi i(\Lambda_{\pi(z)}-1/3)}G_z(\tau)=e^{-2\pi i\Lambda_z}G_z(\tau)=(T^*G(\tau))_z G z ( τ + 1 ) = e 2 π i ( Λ π ( z ) − 1/3 ) G z ( τ ) = e − 2 π i Λ z G z ( τ ) = ( T ∗ G ( τ ) ) z η ( − 1 / τ ) − 8 = τ − 4 η ( τ ) − 8 \eta(-1/\tau)^{-8}=\tau^{-4}\eta(\tau)^{-8} η ( − 1/ τ ) − 8 = τ − 4 η ( τ ) − 8
G z ( − 1 τ ) = τ k − 4 η ( τ ) − 8 ∑ w S π ( z ) , w Y w ( τ ) = τ k − 4 ∑ w S π ( z ) , π ( w ) G w ( τ ) = τ k − 4 ( S G ( τ ) ) z . G_z\left(-\frac{1}{\tau}\right)=\tau^{k-4}\eta(\tau)^{-8}\sum_w S_{\pi(z),w}Y_w(\tau)=\tau^{k-4}\sum_w S_{\pi(z),\pi(w)}G_w(\tau)=\tau^{k-4}(SG(\tau))_z. G z ( − τ 1 ) = τ k − 4 η ( τ ) − 8 w ∑ S π ( z ) , w Y w ( τ ) = τ k − 4 w ∑ S π ( z ) , π ( w ) G w ( τ ) = τ k − 4 ( S G ( τ ) ) z . Since η \eta η H \mathbb{H} H G G G H \mathbb{H} H
The π \pi π S S S
The relations
Let X = X ( γ ) X=X^{(\gamma)} X = X ( γ ) μ z \mu_z μ z X z X_z X z μ z = h z − 1 3 \mu_z=h_z-\frac{1}{3} μ z = h z − 3 1 (14) . For k ∈ 2 Z ≥ 0 k\in2\mathbb{Z}_{\geq0} k ∈ 2 Z ≥ 0 M k ( μ ) M_k(\mu) M k ( μ ) W W W k k k ϱ \varrho ϱ W z ∈ q μ z C [ [ q ] ] W_z\in q^{\mu_z}\mathbb{C}[[q]] W z ∈ q μ z C [[ q ]] z z z W ∈ M k ( μ ) W\in M_k(\mu) W ∈ M k ( μ )
w 0 = [ q − 1 / 3 ] W 0 , w 0 ′ = [ q 2 / 3 ] W 0 , w z = [ q μ z ] W z ( z ≠ 0 ) . (17) w_0=[q^{-1/3}]W_0,\quad w'_0=[q^{2/3}]W_0,\quad w_z=[q^{\mu_z}]W_z\quad(z\ne0). \tag*{(17)} w 0 = [ q − 1/3 ] W 0 , w 0 ′ = [ q 2/3 ] W 0 , w z = [ q μ z ] W z ( z = 0 ) . (17) Proposition 5.4. For γ = 1 \gamma=1 γ = 1 dim M 2 ( μ ) = 5 \dim M_2(\mu)=5 dim M 2 ( μ ) = 5 (17) satisfy exactly eight linearly independent relations, among which
w d 3 − 5 21 w d 4 − 1 21 w d 5 + 1 21 w d 6 = 0 , w c 1 + 1 6 w c 2 − 3 52 w d 2 − 317 364 w d 4 − 55 364 w d 5 + 17 182 w d 6 = 0 , w_{d_3}-\frac{5}{21}w_{d_4}-\frac{1}{21}w_{d_5}+\frac{1}{21}w_{d_6}=0,\quad
w_{c_1}+\frac{1}{6}w_{c_2}-\frac{3}{52}w_{d_2}-\frac{317}{364}w_{d_4}-\frac{55}{364}w_{d_5}+\frac{17}{182}w_{d_6}=0, w d 3 − 21 5 w d 4 − 21 1 w d 5 + 21 1 w d 6 = 0 , w c 1 + 6 1 w c 2 − 52 3 w d 2 − 364 317 w d 4 − 364 55 w d 5 + 182 17 w d 6 = 0 , w d 1 − 1 6 w d 2 − 209 126 w d 4 + 17 126 w d 5 + 44 63 w d 6 = 0 , w 0 + 1 78 w d 2 + 317 1638 w d 4 + 55 1638 w d 5 − 17 819 w d 6 = 0. (18) w_{d_1}-\frac{1}{6}w_{d_2}-\frac{209}{126}w_{d_4}+\frac{17}{126}w_{d_5}+\frac{44}{63}w_{d_6}=0,\quad
w_0+\frac{1}{78}w_{d_2}+\frac{317}{1638}w_{d_4}+\frac{55}{1638}w_{d_5}-\frac{17}{819}w_{d_6}=0. \tag*{(18)} w d 1 − 6 1 w d 2 − 126 209 w d 4 + 126 17 w d 5 + 63 44 w d 6 = 0 , w 0 + 78 1 w d 2 + 1638 317 w d 4 + 1638 55 w d 5 − 819 17 w d 6 = 0. (18) For γ = 0 \gamma=0 γ = 0 dim M 2 ( μ ) = 4 \dim M_2(\mu)=4 dim M 2 ( μ ) = 4
Proof. Let P = { 0 , d 3 } P=\{0,d_{3}\} P = { 0 , d 3 } γ = 1 \gamma=1 γ = 1 P = { 0 } P=\{0\} P = { 0 } γ = 0 \gamma=0 γ = 0 z z z μ z = Λ z − 1 \mu_{z}=\Lambda_{z}-1 μ z = Λ z − 1 Ξ ∙ , j \Xi_{\bullet,j} Ξ ∙ , j Theorem 3.1 .
Step 1: the relations. Consider the 4 ⋅ 12 ⋅ 2 = 96 4\cdot12\cdot2=96 4 ⋅ 12 ⋅ 2 = 96 E 4 Ξ ∙ , j J m E_{4}\Xi_{\bullet,j}J^{m} E 4 Ξ ∙ , j J m ϑ 2 D ( Ξ ∙ , j J m ) \vartheta_{2}D(\Xi_{\bullet,j}J^{m}) ϑ 2 D ( Ξ ∙ , j J m ) D J ⋅ D ( Ξ ∙ , j J m ) DJ\cdot D(\Xi_{\bullet,j}J^{m}) D J ⋅ D ( Ξ ∙ , j J m ) ( D J ) 2 Ξ ∙ , j J m (DJ)^{2}\Xi_{\bullet,j}J^{m} ( D J ) 2 Ξ ∙ , j J m 0 ≤ j ≤ 11 0\leq j\leq11 0 ≤ j ≤ 11 m ∈ { 0 , 1 } m\in\{0,1\} m ∈ { 0 , 1 } Y Y Y G G G ϱ ∗ \varrho^{*} ϱ ∗ Lemma 5.3 . The exponents of G z G_{z} G z − Λ z -\Lambda_{z} − Λ z W ⋅ G W\cdot G W ⋅ G ∑ z ∑ m [ q Λ z + m ] W z ⋅ [ q − Λ z − m ] G z \sum_{z}\sum_{m}[q^{\Lambda_{z}+m}]W_{z}\cdot[q^{-\Lambda_{z}-m}]G_{z} ∑ z ∑ m [ q Λ z + m ] W z ⋅ [ q − Λ z − m ] G z G G G [ q Λ z + m ] W z [q^{\Lambda_{z}+m}]W_{z} [ q Λ z + m ] W z W W W G G G Lemma 5.2 , every solution gives a linear relation, with rational coefficients, between the coefficients (5.3) of every W ∈ M 2 ( μ ) W\in\mathcal{M}_{2}(\mu) W ∈ M 2 ( μ ) q q q Ξ n \Xi_{n} Ξ n 43 × 96 43\times96 43 × 96 γ = 1 \gamma=1 γ = 1 42 × 96 42\times96 42 × 96 γ = 0 \gamma=0 γ = 0 53 53 53 54 54 54 γ = 1 \gamma=1 γ = 1 γ = 0 \gamma=0 γ = 0 m ≤ 2 m\leq2 m ≤ 2
Step 2: exact elements. The forms D ( Ξ ∙ , j J m ) D(\Xi_{\bullet,j}J^{m}) D ( Ξ ∙ , j J m ) D J ⋅ Ξ ∙ , j J m DJ\cdot\Xi_{\bullet,j}J^{m} D J ⋅ Ξ ∙ , j J m m ∈ { 0 , 1 } m\in\{0,1\} m ∈ { 0 , 1 } H \mathbb{H} H W z ∈ q μ z C [ [ q ] ] W_{z}\in q^{\mu_{z}}\mathbb{C}[[q]] W z ∈ q μ z C [[ q ]] M 2 ( μ ) \mathcal{M}_{2}(\mu) M 2 ( μ ) γ = 1 \gamma=1 γ = 1 γ = 0 \gamma=0 γ = 0 13 − 8 = 5 13-8=5 13 − 8 = 5 13 − 9 = 4 13-9=4 13 − 9 = 4 M 2 ( μ ) \mathcal{M}_{2}(\mu) M 2 ( μ )
Step 3: injectivity. Let W ∈ M 2 ( μ ) W\in\mathcal{M}_{2}(\mu) W ∈ M 2 ( μ ) Theorem 3.1 , F = W E 10 Δ F=W\frac{E_{10}}{\Delta} F = W Δ E 10 F = Ξ ( a + J b ) F=\Xi(a+Jb) F = Ξ ( a + J b ) a ∈ C 12 a\in\mathbb{C}^{12} a ∈ C 12 b b b P P P Δ E 10 = q + O ( q 2 ) \frac{\Delta}{E_{10}}=q+O(q^{2}) E 10 Δ = q + O ( q 2 ) [ q Λ z − 1 ] W z [q^{\Lambda_{z}-1}]W_{z} [ q Λ z − 1 ] W z b z b_{z} b z z ∈ P z\in P z ∈ P b = 0 b=0 b = 0 [ q Λ z ] W z = a z [q^{\Lambda_{z}}]W_{z}=a_{z} [ q Λ z ] W z = a z a z = 0 a_{z}=0 a z = 0 z ≠ d 3 z\ne d_{3} z = d 3 z = d 3 z=d_{3} z = d 3 γ = 0 \gamma=0 γ = 0 γ = 1 \gamma=1 γ = 1 W = a d 3 Ξ ∙ , d 3 Δ E 10 W=a_{d_{3}}\Xi_{\bullet,d_{3}}\frac{\Delta}{E_{10}} W = a d 3 Ξ ∙ , d 3 E 10 Δ [ q Λ d 3 ] W d 3 = a d 3 [q^{\Lambda_{d_{3}}}]W_{d_{3}}=a_{d_{3}} [ q Λ d 3 ] W d 3 = a d 3 W 0 ∈ q 1 3 C [ [ q ] ] W_{0}\in q^{\frac{1}{3}}\mathbb{C}[[q]] W 0 ∈ q 3 1 C [[ q ]] W z ∈ q Λ z + 1 C [ [ q ] ] W_{z}\in q^{\Lambda_{z}+1}\mathbb{C}[[q]] W z ∈ q Λ z + 1 C [[ q ]] z ∉ { 0 , d 3 } z\notin\{0,d_{3}\} z ∈ / { 0 , d 3 } G z = η − 8 E 4 Ξ π ( z ) , d 2 G_{z}=\eta^{-8}E_{4}\Xi_{\pi(z),d_{2}} G z = η − 8 E 4 Ξ π ( z ) , d 2 Lemma 5.3 . Put k z = Λ z + Λ π ( z ) − 1 / 3 ∈ { 0 , 1 } k_{z}=\Lambda_{z}+\Lambda_{\pi(z)}-1/3\in\{0,1\} k z = Λ z + Λ π ( z ) − 1/3 ∈ { 0 , 1 } Ξ \Xi Ξ G d 3 = q − Λ d 3 ( 1 + O ( q ) ) G_{d_{3}}=q^{-\Lambda_{d_{3}}}(1+O(q)) G d 3 = q − Λ d 3 ( 1 + O ( q )) G z ∈ q − Λ z + k z C [ [ q ] ] G_{z}\in q^{-\Lambda_{z}+k_{z}}\mathbb{C}[[q]] G z ∈ q − Λ z + k z C [[ q ]] z ≠ d 3 z\ne d_{3} z = d 3 [ q 0 ] ( W ⋅ G ) = a d 3 [q^{0}](W\cdot G)=a_{d_{3}} [ q 0 ] ( W ⋅ G ) = a d 3 Lemma 5.2 . Hence the thirteen coefficients are injective on M 2 ( μ ) \mathcal{M}_{2}(\mu) M 2 ( μ ) dim M 2 ( μ ) \dim\mathcal{M}_{2}(\mu) dim M 2 ( μ )
Remark 5.5 Proposition 5.4 is a computer-assisted result using exact rational linear algebra. The modularity of Ξ \Xi Ξ Theorem 3.1 . The floating-point checks do not enter the proof.
We now insert the explicit data. Let h h h g \mathfrak{g} g h ( 0 ) h(0) h ( 0 ) U U U U U U ⟨ h , h ⟩ = 1 \langle h,h\rangle=1 ⟨ h , h ⟩ = 1 ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ e 8 \mathfrak{e}_{8} e 8 h = ( h 1 , h 2 ) ∈ e 6 ⊕ s l 3 h=(h_{1},h_{2})\in\mathfrak{e}_{6}\oplus\mathfrak{sl}_{3} h = ( h 1 , h 2 ) ∈ e 6 ⊕ sl 3 h 1 h_{1} h 1 27 \mathbf{27} 27 h 2 h_{2} h 2 3 \mathbf{3} 3 Lemma 4.3 . These faithful weight systems span the respective Cartan duals, so h 1 h_{1} h 1 h 2 h_{2} h 2 0 ≤ ⟨ h 1 , h 1 ⟩ ≤ 1 0 \le\langle h_1,h_1\rangle\le1 0 ≤ ⟨ h 1 , h 1 ⟩ ≤ 1
u = d e f ⟨ h 1 , h 1 ⟩ ∈ [ 0 , 1 ] , T = d e f Tr s ( ad h ) 2 , τ j = d e f Tr R d j h ( 0 ) 2 ≥ 0. u \mathrel{\mathop{\mathop{=}\limits^{\mathrm{def}}}} \langle h_1,h_1\rangle\in[0,1],\qquad T \mathrel{\mathop{\mathop{=}\limits^{\mathrm{def}}}} \operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^2,\qquad\tau_j \mathrel{\mathop{\mathop{=}\limits^{\mathrm{def}}}} \operatorname{Tr}_{R_{d_j}} h(0)^2 \ge0. u = def ⟨ h 1 , h 1 ⟩ ∈ [ 0 , 1 ] , T = def Tr s ( ad h ) 2 , τ j = def Tr R d j h ( 0 ) 2 ≥ 0. By Proposition 4.1 and Lemma 4.3, the lowest weight spaces of M b M_b M b M a M_a M a M c 0 M_{c_0} M c 0 M c 1 M_{c_1} M c 1 M c 2 M_{c_2} M c 2 ( e 6 ⊕ s l 3 ) ⊖ s (\mathfrak{e}_6\oplus\mathfrak{sl}_3)\ominus\mathfrak{s} ( e 6 ⊕ sl 3 ) ⊖ s ( 27 , 3 ) (27,3) ( 27 , 3 ) ( 27 , 3 ) (27,3) ( 27 , 3 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 27 , 1 ) (27,1) ( 27 , 1 ) Tr e 6 ( ad h 1 ) 2 = 24 u \operatorname{Tr}_{\mathfrak{e}_6}(\operatorname{ad}h_1)^2=24u Tr e 6 ( ad h 1 ) 2 = 24 u Tr s l 3 ( ad h 2 ) 2 = 6 ( 1 − u ) \operatorname{Tr}_{\mathfrak{sl}_3}(\operatorname{ad}h_2)^2=6(1-u) Tr sl 3 ( ad h 2 ) 2 = 6 ( 1 − u ) Tr 27 h 1 2 = 6 u \operatorname{Tr}_{27}h_1^2=6u Tr 27 h 1 2 = 6 u Tr 3 h 2 2 = 1 − u \operatorname{Tr}_{3}h_2^2=1-u Tr 3 h 2 2 = 1 − u h ( 0 ) 2 h(0)^2 h ( 0 ) 2 6 + 18 u − T 6+18u-T 6 + 18 u − T 27 − 9 u 27-9u 27 − 9 u 27 − 9 u 27-9u 27 − 9 u 1 − u 1-u 1 − u 6 u 6u 6 u U ⊂ V E 8 U\subset V_{E_8} U ⊂ V E 8 L ( 1 ) U 1 = 0 L(1)U_1=0 L ( 1 ) U 1 = 0 W = ( Z M z ( h [ − 1 ] h , τ ) ) z W=(Z_{M_z}(h[-1]h,\tau))_z W = ( Z M z ( h [ − 1 ] h , τ ) ) z M 2 ( μ ) \mathcal{M}_2(\mu) M 2 ( μ ) w 0 = − 1 12 w_0=-\frac{1}{12} w 0 = − 12 1 w 0 ′ = T + 18 − 13 γ 12 w_0'=T+\frac{18-13\gamma}{12} w 0 ′ = T + 12 18 − 13 γ w z = Tr R z h ( 0 ) 2 − dim R z 12 w_z=\operatorname{Tr}_{R_z}h(0)^2-\frac{\dim R_z}{12} w z = Tr R z h ( 0 ) 2 − 12 d i m R z
Corollary 5.6. For γ = 1 \gamma=1 γ = 1
τ 1 = T 15 − 7 τ 5 15 − 3 τ 6 5 − 3 u 5 + 17 5 , τ 2 = 317 T 270 − 209 τ 5 270 − 7 τ 6 5 − 317 u 30 + 2009 90 , \tau_1=\frac{T}{15}-\frac{7\tau_5}{15}-\frac{3\tau_6}{5}-\frac{3u}{5}+\frac{17}{5},\qquad
\tau_2=\frac{317T}{270}-\frac{209\tau_5}{270}-\frac{7\tau_6}{5}-\frac{317u}{30}+\frac{2009}{90}, τ 1 = 15 T − 15 7 τ 5 − 5 3 τ 6 − 5 3 u + 5 17 , τ 2 = 270 317 T − 270 209 τ 5 − 5 7 τ 6 − 30 317 u + 90 2009 , τ 3 = − T 54 + τ 5 54 + u 6 − 1 18 , τ 4 = − 7 T 90 − 11 τ 5 90 + τ 6 5 + 7 u 10 − 19 30 . \tau_3=-\frac{T}{54}+\frac{\tau_5}{54}+\frac{u}{6}-\frac{1}{18},\qquad
\tau_4=-\frac{7T}{90}-\frac{11\tau_5}{90}+\frac{\tau_6}{5}+\frac{7u}{10}-\frac{19}{30}. τ 3 = − 54 T + 54 τ 5 + 6 u − 18 1 , τ 4 = − 90 7 T − 90 11 τ 5 + 5 τ 6 + 10 7 u − 30 19 . For γ = 0 \gamma=0 γ = 0
τ 1 = − 2 T − 3 τ 6 + 18 u − 4 5 , τ 2 = 2 T − 7 τ 6 − 18 u + 74 5 , τ 3 = 3 ( − 3 T + 3 τ 6 + 27 u + 29 ) 5 , \tau_1=\frac{-2T-3\tau_6+18u-4}{5},\qquad
\tau_2=\frac{2T-7\tau_6-18u+74}{5},\qquad
\tau_3=\frac{3(-3T+3\tau_6+27u+29)}{5}, τ 1 = 5 − 2 T − 3 τ 6 + 18 u − 4 , τ 2 = 5 2 T − 7 τ 6 − 18 u + 74 , τ 3 = 5 3 ( − 3 T + 3 τ 6 + 27 u + 29 ) , τ 4 = − T + τ 6 + 9 u − 7 5 , τ 5 = T − 9 u + 9. \tau_4=\frac{-T+\tau_6+9u-7}{5},\qquad\tau_5=T-9u+9. τ 4 = 5 − T + τ 6 + 9 u − 7 , τ 5 = T − 9 u + 9. Proof: This is exact linear algebra on the relations. All relations are satisfied identically after substitution.
Remark 5.7 For γ = 0 \gamma=0 γ = 0 [67 ] lists the lowest weight spaces of the twelve simple modules as sums of modules V ( a , b ) V(a,b) V ( a , b ) s l 2 ⊕ s l 2 \mathfrak{sl}_2\oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2 h h h u = 1 u=1 u = 1 T = 4 T=4 T = 4 τ 1 , τ 4 ≥ 0 \tau_1,\tau_4\ge0 τ 1 , τ 4 ≥ 0 τ 6 = 2 \tau_6=2 τ 6 = 2 ( τ 1 , … , τ 6 ) = ( 0 , 10 , 30 , 0 , 4 , 2 ) (\tau_1,\ldots,\tau_6)=(0,10,30,0,4,2) ( τ 1 , … , τ 6 ) = ( 0 , 10 , 30 , 0 , 4 , 2 ) u = 35 39 u=\frac{35}{39} u = 39 35 T = 4 39 T=\frac{4}{39} T = 39 4
One-point functions of higher weight
Lemma 6.1. Let V V V c c c h ∈ V 1 h\in V_1 h ∈ V 1 L ( 1 ) h = 0 L(1)h=0 L ( 1 ) h = 0 ⟨ h , h ⟩ ≠ 0 \langle h,h\rangle\ne0 ⟨ h , h ⟩ = 0 V 1 V_1 V 1 n ≥ 0 n\ge0 n ≥ 0 v n = d e f h [ − 1 ] n 1 v_n\mathrel{\mathop{\mathop{=}\limits^{\mathrm{def}}}}h[-1]^n\mathbf{1} v n = def h [ − 1 ] n 1 v n v_n v n n n n L [ 0 ] L[0] L [ 0 ] V V V M M M
Z M ( v n , τ ) = ∑ k = 0 ⌊ n / 2 ⌋ ( n 2 k ) ( 2 k − 1 ) ! ! ( − ⟨ h , h ⟩ E 2 ( τ ) 12 ) k Tr M h ( 0 ) n − 2 k q L ( 0 ) − c 24 . (19) Z_M(v_n,\tau)=\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{2k}(2k-1)!!\left(-\frac{\langle h,h\rangle E_2(\tau)}{12}\right)^k\operatorname{Tr}_M h(0)^{n-2k}q^{L(0)-\frac{c}{24}}. \tag*{(19)} Z M ( v n , τ ) = k = 0 ∑ ⌊ n /2 ⌋ ( 2 k n ) ( 2 k − 1 )!! ( − 12 ⟨ h , h ⟩ E 2 ( τ ) ) k Tr M h ( 0 ) n − 2 k q L ( 0 ) − 24 c . (19) with the convention ( − 1 ) ! ! = 1 (-1)!!=1 ( − 1 )!! = 1 ( Z M z ( v n , τ ) ) z (Z_{M_z}(v_n,\tau))_z ( Z M z ( v n , τ ) ) z n n n
Proof: Since h h h [ L [ 0 ] , h [ m ] ] = − m h [ m ] [L[0],h[m]]=-mh[m] [ L [ 0 ] , h [ m ]] = − mh [ m ] [ h [ m ] , h [ m ′ ] ] = m ⟨ h , h ⟩ δ m + m ′ , 0 [h[m],h[m']]=m\langle h,h\rangle\delta_{m+m',0} [ h [ m ] , h [ m ′ ]] = m ⟨ h , h ⟩ δ m + m ′ , 0 [72 Section 4] . Hence v n v_n v n n n n L [ 0 ] L[0] L [ 0 ] h [ 2 j − 1 ] v n = 0 h[2j-1]v_n=0 h [ 2 j − 1 ] v n = 0 j ≥ 2 j\ge2 j ≥ 2 h [ 1 ] v n = n ⟨ h , h ⟩ v n − 1 h[1]v_n=n\langle h,h\rangle v_{n-1} h [ 1 ] v n = n ⟨ h , h ⟩ v n − 1 h ( 0 ) h(0) h ( 0 ) ⟨ h , h ⟩ ≠ 0 \langle h,h\rangle\ne0 ⟨ h , h ⟩ = 0 M = ⨁ λ F λ ⊗ Ω λ M=\bigoplus_{\lambda}F_{\lambda}\otimes\Omega_{\lambda} M = ⨁ λ F λ ⊗ Ω λ F λ F_{\lambda} F λ h ( 0 ) h(0) h ( 0 ) λ \lambda λ h h h ω h = 1 2 ⟨ h , h ⟩ h ( − 1 ) h \omega_h=\frac{1}{2\langle h,h\rangle}h(-1)h ω h = 2 ⟨ h , h ⟩ 1 h ( − 1 ) h L ( 0 ) = L h ( 0 ) + L ′ ( 0 ) L(0)=L_h(0)+L'(0) L ( 0 ) = L h ( 0 ) + L ′ ( 0 ) L ′ ( 0 ) L'(0) L ′ ( 0 ) v n v_n v n L ′ ( 0 ) = 0 L'(0)=0 L ′ ( 0 ) = 0 V V V Z M ( v n , τ ) = ∑ λ Z F λ ( v n , τ ) Tr Ω λ q L ′ ( 0 ) − c − 1 24 Z_M(v_n,\tau)=\sum_{\lambda}Z_{F_\lambda}(v_n,\tau)\operatorname{Tr}_{\Omega_\lambda}q^{L'(0)-\frac{c-1}{24}} Z M ( v n , τ ) = ∑ λ Z F λ ( v n , τ ) Tr Ω λ q L ′ ( 0 ) − 24 c − 1 F λ F_\lambda F λ [72 Proposition 4.3.5] gives
Z F λ ( v n + 1 , τ ) = λ Z F λ ( v n , τ ) + ∑ j ≥ 1 E ^ 2 j ( τ ) Z F λ ( h [ 2 j − 1 ] v n , τ ) = λ Z F λ ( v n , τ ) − n ⟨ h , h ⟩ E 2 ( τ ) 12 Z F λ ( v n − 1 , τ ) . \begin{aligned}
Z_{F_\lambda}(v_{n+1},\tau)=\lambda Z_{F_\lambda}(v_n,\tau)+\sum_{j\geq1}\widehat{E}_{2j}(\tau)Z_{F_\lambda}(h[2j-1]v_n,\tau) \\
&=\lambda Z_{F_\lambda}(v_n,\tau)-\frac{n\langle h,h\rangle E_2(\tau)}{12}Z_{F_\lambda}(v_{n-1},\tau).
\end{aligned} Z F λ ( v n + 1 , τ ) = λ Z F λ ( v n , τ ) + j ≥ 1 ∑ E 2 j ( τ ) Z F λ ( h [ 2 j − 1 ] v n , τ ) = λ Z F λ ( v n , τ ) − 12 n ⟨ h , h ⟩ E 2 ( τ ) Z F λ ( v n − 1 , τ ) . where we used that o ( h ) = h ( 0 ) o(h)=h(0) o ( h ) = h ( 0 ) λ \lambda λ j = 1 j=1 j = 1 E ^ 2 = − E 2 12 \widehat{E}_2=-\frac{E_2}{12} E 2 = − 12 E 2 n = 1 n=1 n = 1 n = 2 n=2 n = 2 P n P_n P n P 0 = 1 P_0=1 P 0 = 1 P 1 = λ P_1=\lambda P 1 = λ P n + 1 = λ P n + n e P n − 1 P_{n+1}=\lambda P_n+neP_{n-1} P n + 1 = λ P n + n e P n − 1
P n = ∑ k ( n 2 k ) ( 2 k − 1 ) ! ! e k λ n − 2 k , P_n=\sum_k \binom{n}{2k}(2k-1)!!e^k\lambda^{n-2k}, P n = k ∑ ( 2 k n ) ( 2 k − 1 )!! e k λ n − 2 k , as one checks by induction using ( n 2 k ) ( 2 k − 1 ) ! ! + n ( n − 1 2 k − 2 ) ( 2 k − 3 ) ! ! = ( n + 1 2 k ) ( 2 k − 1 ) ! ! \binom{n}{2k}(2k-1)!!+n\binom{n-1}{2k-2}(2k-3)!!=\binom{n+1}{2k}(2k-1)!! ( 2 k n ) ( 2 k − 1 )!! + n ( 2 k − 2 n − 1 ) ( 2 k − 3 )!! = ( 2 k n + 1 ) ( 2 k − 1 )!! λ \lambda λ (19) . The last assertion is Zhu’s theorem [72 ] with [49 ] .
As a numerical check, we test (19) in weights four and six for V A 2 V_{A_2} V A 2 V 6 Z V_{\sqrt{6}\mathbb{Z}} V 6 Z 5.1 × 10 − 30 5.1\times10^{-30} 5.1 × 1 0 − 30
By Lemma 6.1, for every real h h h s \mathfrak{s} s ⟨ h , h ⟩ = 1 \langle h,h\rangle=1 ⟨ h , h ⟩ = 1 n ∈ { 4 , 6 } n\in\{4,6\} n ∈ { 4 , 6 } W n ( h ) = ( Z M z ( v n , τ ) ) z W_n(h)=(Z_{M_z}(v_n,\tau))_z W n ( h ) = ( Z M z ( v n , τ ) ) z M n ( μ ) M_n(\mu) M n ( μ ) (17) are
w z = ∑ k c n , k Tr R z h ( 0 ) n − 2 k , c n , k = ( n 2 k ) ( 2 k − 1 ) ! ! ( − 1 12 ) k . (20) w_z=\sum_k c_{n,k}\operatorname{Tr}_{R_z}h(0)^{n-2k},\qquad c_{n,k}=\binom{n}{2k}(2k-1)!!\left(-\frac{1}{12}\right)^k. \tag*{(20)} w z = k ∑ c n , k Tr R z h ( 0 ) n − 2 k , c n , k = ( 2 k n ) ( 2 k − 1 )!! ( − 12 1 ) k . (20) with Tr h ( 0 ) 0 = dim \operatorname{Tr}h(0)^0=\dim Tr h ( 0 ) 0 = dim w 0 w_0 w 0 w 0 ′ w'_0 w 0 ′ Tr s ( ad h ) m \operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^m Tr s ( ad h ) m E 2 k E_2^k E 2 k h ′ h' h ′ h ′ ′ h'' h ′′ ⟨ h ′ , h ′ ′ ⟩ = 0 \langle h',h''\rangle=0 ⟨ h ′ , h ′′ ⟩ = 0 ⟨ h ′ , h ′ ⟩ = ⟨ h ′ ′ , h ′ ′ ⟩ = 1 \langle h',h'\rangle=\langle h'',h''\rangle=1 ⟨ h ′ , h ′ ⟩ = ⟨ h ′′ , h ′′ ⟩ = 1 ( x , y ) ↦ W n ( x h ′ + y h ′ ′ ) (x,y)\mapsto W_n(xh'+yh'') ( x , y ) ↦ W n ( x h ′ + y h ′′ ) n n n M n ( μ ) M_n(\mu) M n ( μ ) M n ( μ ) M_n(\mu) M n ( μ ) ⟨ x h ′ + y h ′ ′ , x h ′ + y h ′ ′ ⟩ = x 2 + y 2 \langle xh'+yh'',xh'+yh''\rangle=x^2+y^2 ⟨ x h ′ + y h ′′ , x h ′ + y h ′′ ⟩ = x 2 + y 2 x p y q x^py^q x p y q Tr R z h ′ ( 0 ) i h ′ ′ ( 0 ) j \operatorname{Tr}_{R_z}h'(0)^ih''(0)^j Tr R z h ′ ( 0 ) i h ′′ ( 0 ) j i ≤ p i\leq p i ≤ p j ≤ q j\leq q j ≤ q
Proposition 6.2 Assume γ = 0 \gamma=0 γ = 0
The coefficients (17) of the elements of M 4 ( μ ) M_4(\mu) M 4 ( μ ) M 6 ( μ ) M_6(\mu) M 6 ( μ )
For h h h T 4 = Tr s ( ad h ) 4 T_4=\operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^4 T 4 = Tr s ( ad h ) 4 Q j = Tr R d j h ( 0 ) 4 Q_j=\operatorname{Tr}_{R_{d_j}}h(0)^4 Q j = Tr R d j h ( 0 ) 4 Q 1 , … , Q 5 Q_1,\ldots,Q_5 Q 1 , … , Q 5 ( T , T 4 , T 6 , Q 6 ) (T,T_4,T_6,Q_6) ( T , T 4 , T 6 , Q 6 ) u u u
Q 5 = 12 T + 2 T 4 − 27 u 2 − 90 u + 105 26 . (21) Q_5=\frac{12T+2T_4-27u^2-90u+105}{26}. \tag*{(21)} Q 5 = 26 12 T + 2 T 4 − 27 u 2 − 90 u + 105 . (21) Proof. The relations are obtained as in Step 1 of the proof of Proposition 5.4, with the forms of weight 6 − n 6-n 6 − n E 4 Ξ i , j m E_4\Xi_{i,j}^m E 4 Ξ i , j m ψ 2 D ( Ξ i , j m ) \psi_2D(\Xi_{i,j}^m) ψ 2 D ( Ξ i , j m ) D J ⋅ D ( Ξ ‾ ∙ , j m ) DJ \cdot D(\overline{\Xi}_{\bullet,j}^{m}) D J ⋅ D ( Ξ ∙ , j m ) ( D J ) 2 Ξ ‾ ∙ , j m (DJ)^2\overline{\Xi}_{\bullet,j}^{m} ( D J ) 2 Ξ ∙ , j m 2 2 2 e 6 \mathfrak{e}_6 e 6 s l 3 \mathfrak{sl}_3 sl 3 e 6 \mathfrak{e}_6 e 6 Tr 27 h 1 3 = 0 \operatorname{Tr}_{27} h_1^3=0 Tr 27 h 1 3 = 0 Tr 27 h 1 4 = 3 u 2 \operatorname{Tr}_{27} h_1^4=3u^2 Tr 27 h 1 4 = 3 u 2 Tr e 6 ( ad h 1 ) 4 = 18 u 2 \operatorname{Tr}_{\mathfrak{e}_6}(\operatorname{ad} h_1)^4=18u^2 Tr e 6 ( ad h 1 ) 4 = 18 u 2 s l 6 ⊕ s l 2 ⊂ e 6 \mathfrak{sl}_6\oplus\mathfrak{sl}_2\subset\mathfrak{e}_6 sl 6 ⊕ sl 2 ⊂ e 6 27 = ( 15 , 1 ) ⊕ ( 6 , 2 ) \mathbf{27}=(\mathbf{15},\mathbf{1})\oplus(\mathbf{6},\mathbf{2}) 27 = ( 15 , 1 ) ⊕ ( 6 , 2 ) e 6 = ( 35 , 1 ) ⊕ ( 1 , 3 ) ⊕ ( 20 , 2 ) \mathfrak{e}_6=(\mathbf{35},\mathbf{1})\oplus(\mathbf{1},\mathbf{3})\oplus(\mathbf{20},\mathbf{2}) e 6 = ( 35 , 1 ) ⊕ ( 1 , 3 ) ⊕ ( 20 , 2 ) H H H s l 2 \mathfrak{sl}_2 sl 2 ⟨ H , H ⟩ = 2 \langle H,H\rangle=2 ⟨ H , H ⟩ = 2 27 \mathbf{27} 27 1 1 1 − 1 -1 − 1 e 6 \mathfrak{e}_6 e 6 2 2 2 − 2 -2 − 2 1 1 1 − 1 -1 − 1 12 = 3 ⋅ 2 2 12=3\cdot2^2 12 = 3 ⋅ 2 2 72 = 18 ⋅ 2 2 72=18\cdot2^2 72 = 18 ⋅ 2 2 s l 3 \mathfrak{sl}_3 sl 3 Tr 3 h 2 4 = ( 1 − u ) 2 2 \operatorname{Tr}_{3} h_2^4=\frac{(1-u)^2}{2} Tr 3 h 2 4 = 2 ( 1 − u ) 2 Tr s l 3 ( ad h 2 ) 4 = 9 ( 1 − u ) 2 \operatorname{Tr}_{\mathfrak{sl}_3}(\operatorname{ad} h_2)^4=9(1-u)^2 Tr sl 3 ( ad h 2 ) 4 = 9 ( 1 − u ) 2 (20) into the nine rational relations determines Q 1 , … , Q 5 Q_1,\ldots,Q_5 Q 1 , … , Q 5 T T T T 4 T_4 T 4 u u u τ 6 \tau_6 τ 6 Q 6 Q_6 Q 6 Q 1 , … , Q 4 Q_1,\ldots,Q_4 Q 1 , … , Q 4 (21) . We perform the elimination in rational arithmetic and independently check all nine equations after substitution and the polarization used below.
For the announced algebra, the fourth power traces computed from the announced lowest weight spaces satisfy the nine relations of weight four for both factors. The same holds for the mixed and sextic relations used below. The supplementary data include these exact trace comparisons.
The vector γ = 1 \gamma=1 γ = 1
Theorem 7.1 Let U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U)\cong\mathcal{Z}(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) U U U γ = 0 \gamma=0 γ = 0 dim U 1 = 6 \dim U_1=6 dim U 1 = 6 dim U 2 = 120 \dim U_2=120 dim U 2 = 120 dim U 3 = 956 \dim U_3=956 dim U 3 = 956
The proof uses two lemmas. Suppose γ = 1 \gamma=1 γ = 1 R d 3 R_{d_3} R d 3 s \mathfrak{s} s τ 3 = 0 \tau_3=0 τ 3 = 0 h h h [ s , s ] [\mathfrak{s},\mathfrak{s}] [ s , s ]
Lemma 7.2 Assume γ = 1 \gamma=1 γ = 1 g \mathfrak{g} g s \mathfrak{s} s i 1 = i 1 ( g ) i_1=i_1(\mathfrak{g}) i 1 = i 1 ( g ) i 2 = i 2 ( g ) i_2=i_2(\mathfrak{g}) i 2 = i 2 ( g ) k = i 1 + i 2 k=i_1+i_2 k = i 1 + i 2 ( g , i 1 , i 2 ) (\mathfrak{g},i_1,i_2) ( g , i 1 , i 2 ) ( s l 3 , 1 , 0 ) (\mathfrak{sl}_3,1,0) ( sl 3 , 1 , 0 ) ( s p 4 , 1 , 0 ) (\mathfrak{sp}_4,1,0) ( sp 4 , 1 , 0 ) ( g 2 , 1 , 0 ) (\mathfrak{g}_2,1,0) ( g 2 , 1 , 0 ) ( s l 4 , 1 , 0 ) (\mathfrak{sl}_4,1,0) ( sl 4 , 1 , 0 ) ( s l 2 , 1 , 1 ) (\mathfrak{sl}_2,1,1) ( sl 2 , 1 , 1 ) ( s l 2 , 1 , 4 ) (\mathfrak{sl}_2,1,4) ( sl 2 , 1 , 4 ) ( s l 2 , 2 , 4 ) (\mathfrak{sl}_2,2,4) ( sl 2 , 2 , 4 ) i 1 ≥ 1 i_1\geq1 i 1 ≥ 1
Proof: For h h h g \mathfrak{g} g u = i 1 k u=\frac{i_1}{k} u = k i 1 T = 2 h ∨ k T=\frac{2h^\vee}{k} T = k 2 h ∨ h ∨ h^\vee h ∨ τ 3 = 0 \tau_3=0 τ 3 = 0 τ 5 = T − 9 u + 3 \tau_5=T-9u+3 τ 5 = T − 9 u + 3 τ 4 ≥ 0 \tau_4\geq0 τ 4 ≥ 0 τ 1 ≥ 0 \tau_1\geq0 τ 1 ≥ 0 T − 9 u + 5 ≤ τ 6 ≤ − 2 T + 18 u + 10 3 T-9u+5\leq\tau_6\leq\frac{-2T+18u+10}{3} T − 9 u + 5 ≤ τ 6 ≤ 3 − 2 T + 18 u + 10 τ 5 ≥ 0 \tau_5\geq0 τ 5 ≥ 0
9 u − 3 ≤ T ≤ 9 u − 1. 9u-3\leq T\leq9u-1. 9 u − 3 ≤ T ≤ 9 u − 1. In particular u > 0 u>0 u > 0 i 1 ≥ 1 i_1\geq1 i 1 ≥ 1 2 h ∨ ≥ 6 k − 9 i 2 2h^\vee\geq6k-9i_2 2 h ∨ ≥ 6 k − 9 i 2 k ≤ 2 h ∨ + 9 i 2 6 k\leq\frac{2h^\vee+9i_2}{6} k ≤ 6 2 h ∨ + 9 i 2 dim g ≤ 19 \dim\mathfrak{g}\leq19 dim g ≤ 19 g \mathfrak{g} g A 1 A_1 A 1 A 2 A_2 A 2 B 2 B_2 B 2 G 2 G_2 G 2 A 3 A_3 A 3 i 2 = 0 i_2=0 i 2 = 0 g ≅ s l 2 \mathfrak{g}\cong\mathfrak{sl}_2 g ≅ sl 2 i 2 ∈ { 0 , 1 , 4 } i_2\in\{0,1,4\} i 2 ∈ { 0 , 1 , 4 } Proposition 4.4 . Multiplying 9 u − 3 ≤ T ≤ 9 u − 1 9u-3\leq T\leq9u-1 9 u − 3 ≤ T ≤ 9 u − 1 k k k 6 i 1 − 3 i 2 ≤ 2 h ∨ ≤ 6 i 1 − i 2 6i_1-3i_2\leq2h^\vee\leq6i_1-i_2 6 i 1 − 3 i 2 ≤ 2 h ∨ ≤ 6 i 1 − i 2 i 2 = 0 i_2=0 i 2 = 0 A 1 A_1 A 1 i 1 = 1 i_1=1 i 1 = 1 A 2 A_2 A 2 B 2 B_2 B 2 G 2 G_2 G 2 A 3 A_3 A 3 g ≅ s l 2 \mathfrak{g}\cong\mathfrak{sl}_2 g ≅ sl 2 i 2 = 1 i_2=1 i 2 = 1 i 1 = 1 i_1=1 i 1 = 1 i 2 = 4 i_2=4 i 2 = 4 i 1 ∈ { 1 , 2 } i_1\in\{1,2\} i 1 ∈ { 1 , 2 }
Lemma 7.3 Assume γ = 1 \gamma=1 γ = 1 h h h s \mathfrak{s} s ⟨ h , h ⟩ = 1 \langle h,h\rangle=1 ⟨ h , h ⟩ = 1 u ≥ 1 9 u\geq\frac{1}{9} u ≥ 9 1 p 1 p_1 p 1 s \mathfrak{s} s
Proof: Here T = 0 T=0 T = 0 τ 3 ≥ 0 \tau_3\geq0 τ 3 ≥ 0 τ 5 ≥ 3 − 9 u \tau_5\geq3-9u τ 5 ≥ 3 − 9 u τ 4 ≥ 0 \tau_4\geq0 τ 4 ≥ 0 τ 1 ≥ 0 \tau_1\geq0 τ 1 ≥ 0 11 u 18 − 7 u 2 + 19 6 ≤ τ 6 ≤ 17 3 − u − 7 τ 5 9 \frac{11u}{18}-\frac{7u}{2}+\frac{19}{6}\leq\tau_6\leq\frac{17}{3}-u-\frac{7\tau_5}{9} 18 11 u − 2 7 u + 6 19 ≤ τ 6 ≤ 3 17 − u − 9 7 τ 5 τ 6 \tau_6 τ 6 τ 5 ≤ 9 5 ( 1 + u ) \tau_5\leq\frac{9}{5}(1+u) τ 5 ≤ 5 9 ( 1 + u ) 3 − 9 u ≤ 9 5 ( 1 + u ) 3-9u\leq\frac{9}{5}(1+u) 3 − 9 u ≤ 5 9 ( 1 + u ) u ≥ 1 9 u\geq\frac{1}{9} u ≥ 9 1
We now prove Theorem 7.1 .
Proof. Suppose γ = 1 \gamma= 1 γ = 1 s = g 1 ⊕ ⋯ ⊕ g m ⊕ z \mathfrak{s} = \mathfrak{g}_1 \oplus\cdots\oplus\mathfrak{g}_m \oplus\mathfrak{z} s = g 1 ⊕ ⋯ ⊕ g m ⊕ z g i \mathfrak{g}_i g i z \mathfrak{z} z p 1 p_1 p 1 e 6 \mathfrak{e}_6 e 6 ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ p 2 p_2 p 2 p 1 p_1 p 1
rank p 1 ( s ) = ∑ i rank g i + dim z ≤ 4 \operatorname{rank} p_1(\mathfrak{s}) = \sum_i \operatorname{rank}\mathfrak{g}_i + \dim\mathfrak{z} \le4 rank p 1 ( s ) = i ∑ rank g i + dim z ≤ 4 by Proposition 4.4. By Lemma 7.2, each g i \mathfrak{g}_i g i s l 3 \mathfrak{sl}_3 sl 3 s p 4 \mathfrak{sp}_4 sp 4 g 2 \mathfrak{g}_2 g 2 s l 4 \mathfrak{sl}_4 sl 4 s l 2 \mathfrak{sl}_2 sl 2 i 2 ≠ 0 i_2 \ne0 i 2 = 0 dim s \dim\mathfrak{s} dim s s l 4 \mathfrak{sl}_4 sl 4 14 + 3 + 1 = 18 14 + 3 + 1 = 18 14 + 3 + 1 = 18 3 + 3 = 6 3 + 3 = 6 3 + 3 = 6 19 = dim U 1 19 = \dim U_1 19 = dim U 1 γ = 0 \gamma= 0 γ = 0
Remark 7.4 Theorem 7.1 does not use unitarity or the announcements [66 , 50 , 52 ] . It uses the classification of holomorphic vertex operator algebras of central charge eight, the theory of extensions [45 , 15 ] , the Witt group [23 ] , the classification of the elements of order three of E 8 E_8 E 8 [54 ] , the results of [28 , 25 , 57 , 31 ] used in § 4 , Zhu’s theorem, the Bantay–Gannon description of vector-valued modular forms, and Proposition 5.4, which rests on finitely many rational computations and on the modularity theorem used in Theorem 3.1. The vector γ = 1 \gamma= 1 γ = 1 q 60 q^{60} q 60 Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 )
Remark 7.5 For γ = 0 \gamma= 0 γ = 0 U 1 U_1 U 1 τ j ≥ 0 \tau_j \ge0 τ j ≥ 0 T ≤ 9 u − 5 T \le9u - 5 T ≤ 9 u − 5 h h h U 1 U_1 U 1 ( i 1 , i 2 ) (i_1,i_2) ( i 1 , i 2 ) 2 h ∨ ≤ 9 i 1 − 5 k 2h^\vee\le9i_1 - 5k 2 h ∨ ≤ 9 i 1 − 5 k u ≥ 5 9 u \ge\frac{5}{9} u ≥ 9 5 s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6 τ j \tau_j τ j § 8 , where U 1 U_1 U 1
The weight-one Lie algebra
Throughout this section, U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U) \cong Z(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) X ( 0 ) X^{(0)} X ( 0 ) s = U 1 ⊂ e 6 ⊕ e 6 \mathfrak{s} = U_1 \subset\mathfrak{e}_6 \oplus\mathfrak{e}_6 s = U 1 ⊂ e 6 ⊕ e 6 § 5 . For a simple ideal g ≅ s l 2 \mathfrak{g} \cong\mathfrak{sl}_2 g ≅ sl 2 s \mathfrak{s} s k k k H H H g \mathfrak{g} g ± 2 \pm2 ± 2 g \mathfrak{g} g ⟨ H , H ⟩ = 2 k \langle H,H\rangle= 2k ⟨ H , H ⟩ = 2 k h = H 2 k h = \frac{H}{\sqrt{2k}} h = 2 k H V ( a ) V(a) V ( a ) s l 2 \mathfrak{sl}_2 sl 2 a a a V ( a , b ) = V ( a ) ⊗ V ( b ) V(a,b) = V(a) \otimes V(b) V ( a , b ) = V ( a ) ⊗ V ( b ) s l 2 ⊕ s l 2 \mathfrak{sl}_2 \oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2
Three general facts
Lemma 8.1 Let g ≅ s l 2 \mathfrak{g} \cong\mathfrak{sl}_2 g ≅ sl 2 s \mathfrak{s} s k k k z z z g \mathfrak{g} g R z R_z R z V ( a ) V(a) V ( a )
Proof : By [27 Theorem 3.1 p. 4] , the vertex subalgebra of U U U g \mathfrak{g} g L s l 2 ( k , 0 ) L_{\mathfrak{sl}_2}(k,0) L sl 2 ( k , 0 ) k k k [22 Theorem 3.7 p. 7] , every weak L s l 2 ( k , 0 ) L_{\mathfrak{sl}_2}(k,0) L sl 2 ( k , 0 ) M z M_z M z ⨁ j L ( k , j ) ⊗ N j \bigoplus_j L(k,j) \otimes N_j ⨁ j L ( k , j ) ⊗ N j L ( k , j ) L(k,j) L ( k , j ) 0 ≤ j ≤ k 0 \leq j \leq k 0 ≤ j ≤ k [39 ] , tensored with multiplicity spaces. Let v ∈ R z v \in R_z v ∈ R z a a a g \mathfrak{g} g R z R_z R z M z M_z M z x ( n ) x(n) x ( n ) n n n x ( n ) v = 0 x(n)v = 0 x ( n ) v = 0 x ∈ g x \in\mathfrak{g} x ∈ g n ≥ 1 n \geq1 n ≥ 1 L ( k , j ) L(k,j) L ( k , j ) x ( n ) x(n) x ( n ) n ≥ 1 n \geq1 n ≥ 1 V ( j ) V(j) V ( j ) v ∈ ⨁ j V ( j ) ⊗ N j v \in\bigoplus_j V(j) \otimes N_j v ∈ ⨁ j V ( j ) ⊗ N j e ( 0 ) v = 0 e(0)v = 0 e ( 0 ) v = 0 H ( 0 ) v = a v H(0)v = av H ( 0 ) v = a v v v v j = a j = a j = a a ≤ k a \leq k a ≤ k
Proposition 8.2 The Lie algebra s \mathfrak{s} s s l 2 ⊕ s l 2 \mathfrak{sl}_2 \oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2
Proof : The Lie algebra s \mathfrak{s} s s l 2 \mathfrak{sl}_2 sl 2 s l 2 ⊕ s l 2 \mathfrak{sl}_2 \oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2 s \mathfrak{s} s z \mathfrak{z} z p 2 ( z ) p_2(\mathfrak{z}) p 2 ( z ) p 2 ( s ) p_2(\mathfrak{s}) p 2 ( s ) Proposition 4.4 . The kernel of p 2 ∣ z p_2|_{\mathfrak{z}} p 2 ∣ z R c 1 ≅ 3 R_{c_1} \cong3 R c 1 ≅ 3 h ∈ z h \in\mathfrak{z} h ∈ z p 2 ( h ) = 0 p_2(h) = 0 p 2 ( h ) = 0 ⟨ h , h ⟩ = 1 \langle h,h\rangle= 1 ⟨ h , h ⟩ = 1 u = 1 u = 1 u = 1 T = T 4 = 0 T = T_4 = 0 T = T 4 = 0 (21) gives Q 5 = Tr R d 5 h ( 0 ) 4 = − 6 13 < 0 Q_5 = \operatorname{Tr}_{R_{d_5}} h(0)^4 = -\frac{6}{13} < 0 Q 5 = Tr R d 5 h ( 0 ) 4 = − 13 6 < 0 h ( 0 ) h(0) h ( 0 )
Lemma 8.3 Let a ⊂ e 6 \mathfrak{a} \subset\mathfrak{e}_6 a ⊂ e 6 s l 2 \mathfrak{sl}_2 sl 2 H a H_{\mathfrak{a}} H a
After conjugation, H a H_{\mathfrak{a}} H a α i ( H a ) ∈ { 0 , 1 , 2 } \alpha_i(H_{\mathfrak{a}}) \in\{0,1,2\} α i ( H a ) ∈ { 0 , 1 , 2 } 3 6 − 1 3^6 - 1 3 6 − 1 ad H a \operatorname{ad} H_{\mathfrak{a}} ad H a e 6 \mathfrak{e}_6 e 6 H a H_{\mathfrak{a}} H a s l 2 \mathfrak{sl}_2 sl 2 ⟨ H a , H a ⟩ \langle H_{\mathfrak{a}},H_{\mathfrak{a}}\rangle ⟨ H a , H a ⟩ s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6
If a \mathfrak{a} a e 6 \mathfrak{e}_6 e 6 s l 2 \mathfrak{sl}_2 sl 2 a \mathfrak{a} a s l 2 \mathfrak{sl}_2 sl 2
Proof : 1. The first assertion is the theorem of Dynkin and Kostant on the characteristics of nilpotent orbits [17 Chapter 3] . The second one is an exact computation on the weights. We only use the resulting list as a set containing the possible indices, which is enough for the exclusion arguments below. The list of indices of the subalgebras s l 2 \mathfrak{sl}_2 sl 2 [29 Table 18] . See also [40 p. 16] .
We construct e 6 \mathfrak{e}_6 e 6 [38 ] , with a bimultiplicative cocycle, and check the Jacobi identity. For the labellings ( 2 , 1 , 1 , 0 , 1 , 2 ) (2,1,1,0,1,2) ( 2 , 1 , 1 , 0 , 1 , 2 ) ( 1 , 0 , 0 , 1 , 0 , 1 ) (1,0,0,1,0,1) ( 1 , 0 , 0 , 1 , 0 , 1 ) e e e ad H a \operatorname{ad} H_{\mathfrak{a}} ad H a [ e , f ] = H a [e,f] = H_{\mathfrak{a}} [ e , f ] = H a s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6 s l 2 \mathfrak{sl}_2 sl 2 A 5 A_5 A 5 2 A 2 + A 1 2A_2 + A_1 2 A 2 + A 1 dim g 0 + dim g 1 \dim\mathfrak{g}_0 + \dim\mathfrak{g}_1 dim g 0 + dim g 1 e e e [17 Chapter 8] . Since two subalgebras s l 2 \mathfrak{sl}_2 sl 2 [17 Chapter 3] , the result does not depend on the choice of e e e
Simple ideals
Proposition 8.4 Let g \mathfrak{g} g s \mathfrak{s} s ( i 1 , i 2 ) (i_1,i_2) ( i 1 , i 2 ) k = i 1 + i 2 k=i_1+i_2 k = i 1 + i 2 ( i 1 , i 2 ) ∈ { ( 1 , 0 ) , ( 3 , 0 ) , ( 9 , 4 ) , ( 35 , 4 ) } (i_1,i_2)\in\{(1,0),(3,0),(9,4),(35,4)\} ( i 1 , i 2 ) ∈ {( 1 , 0 ) , ( 3 , 0 ) , ( 9 , 4 ) , ( 35 , 4 )} g \mathfrak{g} g R d 1 R_{d_1} R d 1 R d 4 R_{d_4} R d 4 R d 5 R_{d_5} R d 5 R d 6 R_{d_6} R d 6 Tr R z H ( 0 ) p \operatorname{Tr}_{R_{z}} H(0)^p Tr R z H ( 0 ) p p ∈ { 2 , 4 , 6 } p\in\{2,4,6\} p ∈ { 2 , 4 , 6 } z z z
Proof. For h = H 2 k h=\frac{H}{\sqrt{2k}} h = 2 k H u = i 1 k u=\frac{i_1}{k} u = k i 1 T = 4 k T=\frac{4}{k} T = k 4 T 4 = 8 k 2 T_4=\frac{8}{k^2} T 4 = k 2 8 4 ≤ 9 i 1 − 5 k 4\leq9i_1-5k 4 ≤ 9 i 1 − 5 k i 1 ≥ 1 i_1\geq1 i 1 ≥ 1 i 1 i_1 i 1 p 1 ( g ) p_1(\mathfrak{g}) p 1 ( g ) Lemma 8.3 . By Proposition 4.4 , i 2 ∈ { 0 , 1 , 4 } i_2\in\{0,1,4\} i 2 ∈ { 0 , 1 , 4 } ( i 1 , i 2 ) (i_1,i_2) ( i 1 , i 2 ) s l 2 \mathfrak{sl}_2 sl 2 k k k Lemma 8.1 . Each structure gives τ 6 \tau_6 τ 6 Q 6 Q_6 Q 6 Proposition 6.2 then give τ j \tau_j τ j Q j Q_j Q j j j j s l 2 \mathfrak{sl}_2 sl 2 R d j R_{d_j} R d j k k k Tr H ( 0 ) 2 = 2 k τ j \operatorname{Tr} H(0)^2=2k\tau_j Tr H ( 0 ) 2 = 2 k τ j Tr H ( 0 ) 4 = 4 k 2 Q j \operatorname{Tr} H(0)^4=4k^2Q_j Tr H ( 0 ) 4 = 4 k 2 Q j e 6 \mathfrak{e}_6 e 6 i 1 i_1 i 1 Lemma 8.3 , that the highest weights occurring in R b R_b R b R a R_a R a R c 0 R_{c_0} R c 0 R c 1 R_{c_1} R c 1 R c 2 R_{c_2} R c 2 k k k R d 6 R_{d_6} R d 6
For these four pairs, the relations of weight six of Proposition 6.2 , with the sextic traces on R b , … , R c 2 R_b,\ldots,R_{c_2} R b , … , R c 2 Tr R d j H ( 0 ) 6 \operatorname{Tr}_{R_{d_j}} H(0)^6 Tr R d j H ( 0 ) 6 j ∈ { 1 , 2 , 3 } j\in\{1,2,3\} j ∈ { 1 , 2 , 3 } R d 4 R_{d_4} R d 4 R d 5 R_{d_5} R d 5 R d 6 R_{d_6} R d 6 R d 4 R_{d_4} R d 4 R d 5 R_{d_5} R d 5 R d 1 R_{d_1} R d 1
The classification
Theorem 8.5 Let U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U)\cong Z(H_3) Rep ( U ) ≅ Z ( H 3 ) U ⊂ V E 6 ⊕ A 2 U\subset V_{E_6\oplus A_2} U ⊂ V E 6 ⊕ A 2 Proposition 4.1 . Then U 1 = g ′ ⊕ g ′ ′ U_1=\mathfrak{g}'\oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ g ′ ≅ g ′ ′ ≅ s l 2 \mathfrak{g}'\cong\mathfrak{g}''\cong\mathfrak{sl}_2 g ′ ≅ g ′′ ≅ sl 2 V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2
The levels are 1 and 39. The subalgebra g ′ ′ \mathfrak{g}'' g ′′ s l 2 \mathfrak{sl}_2 sl 2 s l 6 \mathfrak{sl}_6 sl 6 e 6 \mathfrak{e}_6 e 6 s o 3 \mathfrak{so}_3 so 3 s l 3 \mathfrak{sl}_3 sl 3 g ′ ⊂ e 6 \mathfrak{g}'\subset\mathfrak{e}_6 g ′ ⊂ e 6 e 6 \mathfrak{e}_6 e 6 p 1 ( g ′ ′ ) p_1(\mathfrak{g}'') p 1 ( g ′′ )
The levels are 3 and 13. The subalgebra g ′ ′ \mathfrak{g}'' g ′′ s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6 2 A 2 + A 1 2A_2+A_1 2 A 2 + A 1 s o 3 \mathfrak{so}_3 so 3 s l 3 \mathfrak{sl}_3 sl 3 g ′ ⊂ e 6 \mathfrak{g}'\subset\mathfrak{e}_6 g ′ ⊂ e 6 e 6 \mathfrak{e}_6 e 6 p 1 ( g ′ ′ ) p_1(\mathfrak{g}'') p 1 ( g ′′ )
In both cases, the lowest weight spaces of the twelve simple U U U U 2 U_2 U 2 g ′ ⊕ g ′ ′ \mathfrak{g}'\oplus\mathfrak{g}'' g ′ ⊕ g ′′ Table 2 .
Proof. By Proposition 8.2, U 1 = g ′ ⊕ g ′ ′ U_1=\mathfrak{g}'\oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ Proposition 4.4 , at most one of the two ideals has a nonzero projection to s l 3 \mathfrak{sl}_3 sl 3 e 6 \mathfrak{e}_6 e 6 Proposition 8.4 , the pair of indices is one of { ( 1 , 0 ) , ( 1 , 0 ) } \{(1,0),(1,0)\} {( 1 , 0 ) , ( 1 , 0 )} { ( 1 , 0 ) , ( 3 , 0 ) } \{(1,0),(3,0)\} {( 1 , 0 ) , ( 3 , 0 )} { ( 3 , 0 ) , ( 3 , 0 ) } \{(3,0),(3,0)\} {( 3 , 0 ) , ( 3 , 0 )} { ( 1 , 0 ) , ( 9 , 4 ) } \{(1,0),(9,4)\} {( 1 , 0 ) , ( 9 , 4 )} { ( 1 , 0 ) , ( 35 , 4 ) } \{(1,0),(35,4)\} {( 1 , 0 ) , ( 35 , 4 )} { ( 3 , 0 ) , ( 9 , 4 ) } \{(3,0),(9,4)\} {( 3 , 0 ) , ( 9 , 4 )} { ( 3 , 0 ) , ( 35 , 4 ) } \{(3,0),(35,4)\} {( 3 , 0 ) , ( 35 , 4 )}
If an ideal has nonzero projection to s l 3 \mathfrak{sl}_3 sl 3 e 6 \mathfrak{e}_6 e 6 s l 2 \mathfrak{sl}_2 sl 2 Lemma 8.3 . The other ideal is three-dimensional and contained in this centralizer, hence equals it. This excludes the pairs { ( 1 , 0 ) , ( 9 , 4 ) } \{(1,0),(9,4)\} {( 1 , 0 ) , ( 9 , 4 )} { ( 3 , 0 ) , ( 35 , 4 ) } \{(3,0),(35,4)\} {( 3 , 0 ) , ( 35 , 4 )} s l 3 \mathfrak{sl}_3 sl 3 (21) . Let H ′ H' H ′ H ′ ′ H'' H ′′ k ′ , k ′ ′ ∈ { 1 , 3 } k', k'' \in\{1,3\} k ′ , k ′′ ∈ { 1 , 3 } h = x H ′ + y H ′ ′ h=xH'+yH'' h = x H ′ + y H ′′
⟨ h , h ⟩ = 2 k ′ x 2 + 2 k ′ ′ y 2 , Tr s ( ad h ) 2 = 8 ( x 2 + y 2 ) , Tr s ( ad h ) 4 = 32 ( x 4 + y 4 ) . \langle h,h\rangle= 2k'x^2+2k''y^2,\qquad\operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^2=8(x^2+y^2),\qquad\operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^4=32(x^4+y^4). ⟨ h , h ⟩ = 2 k ′ x 2 + 2 k ′′ y 2 , Tr s ( ad h ) 2 = 8 ( x 2 + y 2 ) , Tr s ( ad h ) 4 = 32 ( x 4 + y 4 ) . Since p 2 ( h ) = 0 p_2(h)=0 p 2 ( h ) = 0 (21) from normalized elements to
26 Tr R d 5 h ( 0 ) 4 = 12 ⟨ h , h ⟩ Tr s ( ad h ) 2 + 2 Tr s ( ad h ) 4 − 12 ⟨ h , h ⟩ 2 . 26\operatorname{Tr}_{R_{d_5}}h(0)^4=12\langle h,h\rangle\operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^2+2\operatorname{Tr}_{\mathfrak{s}}(\operatorname{ad}h)^4-12\langle h,h\rangle^2. 26 Tr R d 5 h ( 0 ) 4 = 12 ⟨ h , h ⟩ Tr s ( ad h ) 2 + 2 Tr s ( ad h ) 4 − 12 ⟨ h , h ⟩ 2 . Comparing the coefficients of x 2 y 2 x^2y^2 x 2 y 2
Tr R d 5 H ′ ( 0 ) 2 H ′ ′ ( 0 ) 2 = 8 ( 2 ( k ′ + k ′ ′ ) − k ′ k ′ ′ ) 13 . (22) \operatorname{Tr}_{R_{d_5}}H'(0)^2H''(0)^2=\frac{8(2(k'+k'')-k'k'')}{13}. \tag*{(22)} Tr R d 5 H ′ ( 0 ) 2 H ′′ ( 0 ) 2 = 13 8 ( 2 ( k ′ + k ′′ ) − k ′ k ′′ ) . (22) The left-hand side is an integer, because H ′ ( 0 ) H'(0) H ′ ( 0 ) H ′ ′ ( 0 ) H''(0) H ′′ ( 0 ) s l 2 ⊕ s l 2 \mathfrak{sl}_2\oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2 R d 5 R_{d_5} R d 5 ( k ′ , k ′ ′ ) = ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 3 ) (k',k'')=(1,1),(1,3),(3,3) ( k ′ , k ′′ ) = ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 3 ) 24 / 13 24/13 24/13 40 / 13 40/13 40/13 24 / 13 24/13 24/13 R d 6 R_{d_6} R d 6 ( 1 , 1 ) (1,1) ( 1 , 1 ) Tr R d 4 H ′ ( 0 ) 2 H ′ ′ ( 0 ) 2 = − 40 / 39 \operatorname{Tr}_{R_{d_4}}H'(0)^2H''(0)^2=-40/39 Tr R d 4 H ′ ( 0 ) 2 H ′′ ( 0 ) 2 = − 40/39 − 44 / 195 -44/195 − 44/195 ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 3 , 3 ) (3,3) ( 3 , 3 ) − 36 / 65 -36/65 − 36/65 124 / 65 124/65 124/65
For the two remaining pairs, the projection p 1 ( g ′ ′ ) p_1(\mathfrak{g}'') p 1 ( g ′′ ) p 2 ( g ′ ′ ) p_2(\mathfrak{g}'') p 2 ( g ′′ ) s o 3 \mathfrak{so}_3 so 3 g ′ \mathfrak{g}' g ′ e 6 \mathfrak{e}_6 e 6 p 1 ( g ′ ′ ) p_1(\mathfrak{g}'') p 1 ( g ′′ ) p 1 ( g ′ ′ ) p_1(\mathfrak{g}'') p 1 ( g ′′ ) e 6 \mathfrak{e}_6 e 6 Lemma 8.3 . Since the subalgebras s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6 [17 Chapter 3] , and the same holds for the principal s o 3 \mathfrak{so}_3 so 3 s l 3 \mathfrak{sl}_3 sl 3 U 1 U_1 U 1 V E 6 ⊕ A 2 V_{\mathfrak{E}_6\oplus A_2} V E 6 ⊕ A 2 e 6 ⊕ s l 3 \mathfrak{e}_6\oplus\mathfrak{sl}_3 e 6 ⊕ sl 3 p 1 ( g ′ ′ ) \mathfrak{p}_1(\mathfrak{g}^{\prime\prime}) p 1 ( g ′′ ) p 2 ( g ′ ′ ) \mathfrak{p}_2(\mathfrak{g}^{\prime\prime}) p 2 ( g ′′ ) s l 2 \mathfrak{sl}_2 sl 2 s l 3 \mathfrak{sl}_3 sl 3 s o 3 \mathfrak{so}_3 so 3
Now we determine the lowest weight spaces. For R b , … , R c 2 R_b,\ldots,R_{c_2} R b , … , R c 2 e 8 ⊃ e 6 ⊕ s l 3 \mathfrak{e}_8\supset\mathfrak{e}_6\oplus\mathfrak{sl}_3 e 8 ⊃ e 6 ⊕ sl 3 g ′ \mathfrak{g}^{\prime} g ′ g ′ ′ \mathfrak{g}^{\prime\prime} g ′′ e 6 \mathfrak{e}_6 e 6 27 27 27 s l 3 \mathfrak{sl}_3 sl 3 d j d_j d j x 4 y 2 x^4y^2 x 4 y 2 x 2 y 4 x^2y^4 x 2 y 4 W 6 ( x h ′ + y h ′ ′ ) W_6(xh^{\prime}+yh^{\prime\prime}) W 6 ( x h ′ + y h ′′ ) R d 4 R_{d_4} R d 4 R d 5 R_{d_5} R d 5 R d 6 R_{d_6} R d 6 Tr R d j H ′ ( 0 ) p H ′ ′ ( 0 ) q \operatorname{Tr}_{R_{d_j}}H^{\prime}(0)^pH^{\prime\prime}(0)^q Tr R d j H ′ ( 0 ) p H ′′ ( 0 ) q p + q ≤ 6 p+q\leq6 p + q ≤ 6 R d 4 R_{d_4} R d 4 R d 5 R_{d_5} R d 5 R d 6 R_{d_6} R d 6 R d 1 R_{d_1} R d 1 R d 2 R_{d_2} R d 2 R d 3 R_{d_3} R d 3 R R R ⨁ a = 0 k ′ ⨁ b = 0 k ′ ′ m a b V ( a , b ) \bigoplus_{a=0}^{k^{\prime}}\bigoplus_{b=0}^{k^{\prime\prime}}m_{ab}V(a,b) ⨁ a = 0 k ′ ⨁ b = 0 k ′′ m ab V ( a , b ) 0 ≤ m a b ≤ ⌊ dim R / ( ( a + 1 ) ( b + 1 ) ) ⌋ 0\leq m_{ab}\leq\left\lfloor\dim R/((a+1)(b+1))\right\rfloor 0 ≤ m ab ≤ ⌊ dim R / (( a + 1 ) ( b + 1 )) ⌋ ∑ a , b ( a + 1 ) ( b + 1 ) m a b = dim R \sum_{a,b}(a+1)(b+1)m_{ab}=\dim R ∑ a , b ( a + 1 ) ( b + 1 ) m ab = dim R
Finally, augment the coefficient map (17) by the fourteenth functional w 0 ′ ′ = [ q 5 / 3 ] W 0 w_0^{\prime\prime}=[q^5/3]W_0 w 0 ′′ = [ q 5 /3 ] W 0 M n ( μ ) \mathcal{M}_n(\mu) M n ( μ ) C 14 \mathbb{C}^{14} C 14 n ∈ { 2 , 4 , 6 } n\in\{2,4,6\} n ∈ { 2 , 4 , 6 } W = W n ( x h ′ + y h ′ ′ ) W=W_n(xh^{\prime}+yh^{\prime\prime}) W = W n ( x h ′ + y h ′′ ) w ′ ′ w^{\prime\prime} w ′′ Tr U 2 H ′ ( 0 ) p H ′ ′ ( 0 ) q \operatorname{Tr}_{U_2}H^{\prime}(0)^pH^{\prime\prime}(0)^q Tr U 2 H ′ ( 0 ) p H ′′ ( 0 ) q U 0 U_0 U 0 U 1 U_1 U 1 x p y q x^py^q x p y q p , q p,q p , q 2 ≤ p + q ≤ 6 2\leq p+q\leq6 2 ≤ p + q ≤ 6 J m J^m J m 0 ≤ m ≤ 2 0\leq m\leq2 0 ≤ m ≤ 2 53 × 144 53\times144 53 × 144 41 × 72 41\times72 41 × 72 29 × 36 29\times36 29 × 36 w 0 ′ ′ w_0^{\prime\prime} w 0 ′′ U 2 U_2 U 2 U 1 U_1 U 1 U U U ( V E 6 ⊕ A 2 ) 2 ≅ ( 27 ⊗ 27 ‾ , 1 ) ⊕ ( 78 , 8 ) ⊕ ( 1 , 1 ⊕ 8 ⊕ 8 ) (V_{\mathfrak{E}_6\oplus A_2})_2\cong(27\otimes\overline{27},1)\oplus(78,8)\oplus(1,1\oplus8\oplus8) ( V E 6 ⊕ A 2 ) 2 ≅ ( 27 ⊗ 27 , 1 ) ⊕ ( 78 , 8 ) ⊕ ( 1 , 1 ⊕ 8 ⊕ 8 ) L e 6 ( 1 , 0 ) L_{\mathfrak{e}_6}(1,0) L e 6 ( 1 , 0 ) L s l 3 ( 1 , 0 ) L_{\mathfrak{sl}_3}(1,0) L sl 3 ( 1 , 0 ) ⨁ a , b M a b V ( a , b ) \bigoplus_{a,b}M_{ab}V(a,b) ⨁ a , b M ab V ( a , b ) ⨁ a , b l a b V ( a , b ) \bigoplus_{a,b}l_{ab}V(a,b) ⨁ a , b l ab V ( a , b ) U 2 U_2 U 2 l a b ≤ m a b ≤ M a b l_{ab}\leq m_{ab}\leq M_{ab} l ab ≤ m ab ≤ M ab ∑ a , b ( a + 1 ) ( b + 1 ) m a b = 120 \sum_{a,b}(a+1)(b+1)m_{ab}=120 ∑ a , b ( a + 1 ) ( b + 1 ) m ab = 120 V 2 ′ V'_2 V 2 ′
■ \blacksquare ■
Remark 8.6 Case (I) is the structure announced in [66 , 50 ] : the level one factor comes from a root s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6 A 5 A_5 A 5 A 2 A_2 A 2 [67 Section 1] . The lowest weight spaces of Table 2 in case (I) coincide with those listed in the working manuscript [67 ] , and the decomposition of U 2 U_2 U 2 [67 Appendix H.1] . In the present work these data are consequences of the hypotheses of Theorem 8.5 , and the announced data serve only as a test.
Remark 8.7 The relations of this section do not exclude case (II), whose explicit realization is constructed in § 10 and shown to be a Haagerup algebra in [5 Theorem 6.1] .
The remaining arguments determine its weight-two generating space and identify the resulting algebra. Finite-degree consistency checks alone do not establish existence. The lowest weight spaces of Table 2 are also compatible with the bound Δ a + Δ b ≤ h z \Delta_a+\Delta_b\le h_z Δ a + Δ b ≤ h z Δ a \Delta_a Δ a z ∈ { 0 , b , a , c 0 , c 1 , c 2 } z\in\{0,b,a,c_0,c_1,c_2\} z ∈ { 0 , b , a , c 0 , c 1 , c 2 } U U U Lemma 4.3 . Choose the compact form for the affine embedding. Its Sugawara operators and those of the coset are then unitary on these modules, and the coset conformal weights are nonnegative. The central charge of the commutant of the affine subalgebra is 8 − 9 5 − 13 5 = 18 5 8-\frac{9}{5}-\frac{13}{5}=\frac{18}{5} 8 − 5 9 − 5 13 = 5 18 U 2 U_2 U 2 Corollary 9.2 ). Note that the products of the levels are equal in the two cases, 1 ⋅ 39 = 3 ⋅ 13 1\cdot39=3\cdot13 1 ⋅ 39 = 3 ⋅ 13 U 2 U_2 U 2 Theorem 10.5 and Proposition 10.7 ).
Constraints on the weight-two space
We keep the hypotheses of Theorem 8.5 , we write A ⊂ U A\subset U A ⊂ U U 1 U_1 U 1 V ′ = V E 6 ⊕ A 2 V'=V_{E_6\oplus A_2} V ′ = V E 6 ⊕ A 2
Proposition 9.1 In both cases of Theorem 8.5 , U 2 U_2 U 2 U 1 U_1 U 1 V 2 ′ V'_2 V 2 ′ Table 2 , containing ( A ) 2 (A)_2 ( A ) 2 V ′ V' V ′ V 2 ′ V'_2 V 2 ′
Proof. The first assertion is Theorem 8.5 . A U 1 U_1 U 1 V 2 ′ V'_2 V 2 ′ u a b u_{ab} u ab ( a , b ) (a,b) ( a , b ) u a b u_{ab} u ab V ( a , b ) V(a,b) V ( a , b ) V 2 ′ V'_2 V 2 ′ v a b v_{ab} v ab l a b l_{ab} l ab ( A ) 2 (A)_2 ( A ) 2 C ω \mathbb{C}\omega C ω ( u a b − l a b ) ( v a b − u a b ) (u_{ab}-l_{ab})(v_{ab}-u_{ab}) ( u ab − l ab ) ( v ab − u ab ) v a b v_{ab} v ab
Proposition 9.1 turns the determination of U 2 U_2 U 2 x ( 1 ) y ∈ U 2 x_{(1)}y\in U_2 x ( 1 ) y ∈ U 2 x ( 2 ) y ∈ U 1 x_{(2)}y\in U_1 x ( 2 ) y ∈ U 1 x , y ∈ U 2 x,y\in U_2 x , y ∈ U 2 V 2 ′ ⊖ U 2 V'_2\ominus U_2 V 2 ′ ⊖ U 2 U U U M b M_b M b Theorems 10.29 and 10.38 .
Corollary 9.2 In both cases, A ≅ L s l 2 ( k ′ , 0 ) ⊗ L s l 2 ( k ′ ′ , 0 ) A\cong L_{\mathfrak{sl}_2}(k',0)\otimes L_{\mathfrak{sl}_2}(k'',0) A ≅ L sl 2 ( k ′ , 0 ) ⊗ L sl 2 ( k ′′ , 0 ) ( k ′ , k ′ ′ ) = ( 1 , 39 ) (k',k'')=(1,39) ( k ′ , k ′′ ) = ( 1 , 39 ) ( 3 , 13 ) (3,13) ( 3 , 13 ) Com U ( A ) \operatorname{Com}_U(A) Com U ( A ) 170 41 \frac{170}{41} 41 170 185 ? \frac{185}{?} ? 185 ω − ω k ′ − ω k ′ ′ \omega-\omega_{k'}-\omega_{k''} ω − ω k ′ − ω k ′′
Proof. The first assertion follows from Theorem 8.5 and [Dom06, Theorem 3.1, p. 4]. The weight-two space of Com U ( A ) \operatorname{Com}_U(A) Com U ( A ) U 2 U_2 U 2 x ( n ) x(n) x ( n ) n ≥ 0 n\ge0 n ≥ 0 x ∈ U 1 x\in U_1 x ∈ U 1 U 1 U_1 U 1 Table 2 , the trivial isotypic component of U 2 U_2 U 2 ω k ′ \omega_{k'} ω k ′ ω k ′ ′ \omega_{k''} ω k ′′ x ( 1 ) x(1) x ( 1 ) x x x ( A ) 2 (A)_2 ( A ) 2 U 1 U_1 U 1 U 2 U_2 U 2 C ω k ′ ⊕ C ω k ′ ′ \mathbb{C}\omega_{k'}\oplus\mathbb{C}\omega_{k''} C ω k ′ ⊕ C ω k ′′ x ( 1 ) x(1) x ( 1 ) x ∈ U 1 x\in U_1 x ∈ U 1 w w w U 2 U_2 U 2 x ( n ) w = 0 x(n)w=0 x ( n ) w = 0 n ≥ 3 n\ge3 n ≥ 3 x ↦ x ( 2 ) w x\mapsto x(2)w x ↦ x ( 2 ) w U 1 U_1 U 1 U 1 U_1 U 1 U 0 = C 1 U_0=\mathbb{C}1 U 0 = C 1 U 1 U_1 U 1 x ( 0 ) w = 0 x(0)w=0 x ( 0 ) w = 0 x ( 1 ) w ∈ U 1 x(1)w\in U_1 x ( 1 ) w ∈ U 1 U 1 U_1 U 1 x x x x ↦ x ( 1 ) w x\mapsto x(1)w x ↦ x ( 1 ) w U 1 U_1 U 1 g ′ ⊕ g ′ ′ \mathfrak{g}'\oplus\mathfrak{g}'' g ′ ⊕ g ′′ g ′ \mathfrak{g}' g ′ g ′ ′ \mathfrak{g}'' g ′′ 8 − 1 − 117 41 8-1-\frac{117}{41} 8 − 1 − 41 117 8 − 9 5 − 13 5 8-\frac{9}{5}-\frac{13}{5} 8 − 5 9 − 5 13
The commutant of the currents and the space U₂
We keep the hypotheses of Theorem 8.5. We write V ′ = V E 6 ⊕ A 2 V' = V_{E_6\oplus A_2} V ′ = V E 6 ⊕ A 2 V ′ = U ⊕ M b V' = U \oplus M_b V ′ = U ⊕ M b A ⊂ U A \subset U A ⊂ U U 1 = g ′ ⊕ g ′ ′ U_1 = \mathfrak{g}' \oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ C ^ = Com V ′ ( A ) \widehat{C} = \operatorname{Com}_{V'}(A) C = Com V ′ ( A ) C = Com U ( A ) ⊂ C ^ C = \operatorname{Com}_U(A) \subset\widehat{C} C = Com U ( A ) ⊂ C C ^ \widehat{C} C C C C ω C : = ω − ω k ′ − ω k ′ ′ \omega_C \mathrel{\mathop:}= \omega-\omega_{k'}-\omega_{k''} ω C : = ω − ω k ′ − ω k ′′ c C = 170 41 c_C = \frac{170}{41} c C = 41 170 18 5 \frac{18}{5} 5 18 Corollary 9.2 ). In this section we determine, by exact computations in V ′ V' V ′ C C C U 2 U_2 U 2 [52 ] in Theorem 10.24. We denote the normalized invariant bilinear form on V ′ V' V ′ β \beta β β ( 1 , 1 ) = 1 \beta(1,1)=1 β ( 1 , 1 ) = 1 H χ H_\chi H χ ⟨ ⋅ , ⋅ ⟩ L \langle\cdot,\cdot\rangle_L ⟨ ⋅ , ⋅ ⟩ L
An exact model of the lattice algebra
We use the construction of [37 Chapter 8] . The lattice L = E 6 ⊕ A 2 L=E_6\oplus A_2 L = E 6 ⊕ A 2 α 1 , … , α 8 \alpha_1,\ldots,\alpha_8 α 1 , … , α 8 E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2 E 6 E_6 E 6 V ′ V' V ′ α i 1 ( − n 1 ) ⋯ α i k ( − n k ) e β \alpha_{i_1}(-n_1)\cdots\alpha_{i_k}(-n_k)e^\beta α i 1 ( − n 1 ) ⋯ α i k ( − n k ) e β β ∈ L \beta\in L β ∈ L n 1 ≥ ⋯ ≥ n k ≥ 1 n_1\ge\cdots\ge n_k\ge1 n 1 ≥ ⋯ ≥ n k ≥ 1 i j ≤ i j + 1 i_j\le i_{j+1} i j ≤ i j + 1 n j = n j + 1 n_j=n_{j+1} n j = n j + 1 ε ( α i , α j ) = ( − 1 ) ⟨ α i , α j ⟩ L \varepsilon(\alpha_i,\alpha_j)=(-1)^{\langle\alpha_i,\alpha_j\rangle_L} ε ( α i , α j ) = ( − 1 ) ⟨ α i , α j ⟩ L i < j i<j i < j ε ( α i , α i ) = − 1 \varepsilon(\alpha_i,\alpha_i)=-1 ε ( α i , α i ) = − 1 ε ( α i , α j ) = 1 \varepsilon(\alpha_i,\alpha_j)=1 ε ( α i , α j ) = 1 i > j i>j i > j
Y ( e α , z ) = exp ( ∑ n > 0 α ( − n ) n z n ) exp ( − ∑ n > 0 α ( n ) n z − n ) e α z α ( 0 ) . (23) Y(e^\alpha,z)=\exp\left(\sum_{n>0}\frac{\alpha(-n)}{n}z^n\right)\exp\left(-\sum_{n>0}\frac{\alpha(n)}{n}z^{-n}\right)e_\alpha z^{\alpha(0)}. \tag*{(23)} Y ( e α , z ) = exp ( n > 0 ∑ n α ( − n ) z n ) exp ( − n > 0 ∑ n α ( n ) z − n ) e α z α ( 0 ) . (23) The modes of a general basis vector are obtained from those of e α e^\alpha e α
( a ( − m ) w ) ( k ) = ∑ j ≥ 0 ( ( m + j − 1 j ) ( a ( − m − j ) w ) ( k + j ) − ( − 1 ) m w ( k − m − j ) a ( j ) ) (a(-m)w)_{(k)}=\sum_{j\ge0}\left(\binom{m+j-1}{j}(a(-m-j)w)_{(k+j)}-(-1)^m w_{(k-m-j)}a_{(j)}\right) ( a ( − m ) w ) ( k ) = j ≥ 0 ∑ ( ( j m + j − 1 ) ( a ( − m − j ) w ) ( k + j ) − ( − 1 ) m w ( k − m − j ) a ( j ) ) for a = α i ( − 1 ) 1 a=\alpha_i(-1)1 a = α i ( − 1 ) 1 1 , 86 , 1370 , 11586 1,86,1370,11586 1 , 86 , 1370 , 11586 L ( 0 ) = L(0)= L ( 0 ) = c = 8 c=8 c = 8 [ u ( m ) , v ( n ) ] = ∑ j ( m j ) ( u ( j ) v ) ( m + n − j ) [u_{(m)},v_{(n)}]=\sum_j\binom{m}{j}(u_{(j)}v)_{(m+n-j)} [ u ( m ) , v ( n ) ] = ∑ j ( j m ) ( u ( j ) v ) ( m + n − j ) ω \omega ω e 6 ⊕ s l 3 \mathfrak{e}_6\oplus\mathfrak{sl}_3 e 6 ⊕ sl 3 β ( ω , ω ) = 4 \beta(\omega,\omega)=4 β ( ω , ω ) = 4
The enumeration of lattice momenta in a ball may be justified independently of the implementation. In a simply laced root lattice, write a nonzero vector v = r 1 + ⋯ + r m v=r_1+\cdots+r_m v = r 1 + ⋯ + r m m m m − 1 -1 − 1 − 2 -2 − 2 ⟨ v , r 1 ⟩ L ≥ 2 \langle v,r_1\rangle_L\ge2 ⟨ v , r 1 ⟩ L ≥ 2 ∥ v − r 1 ∥ 2 = ∥ v ∥ 2 − 2 ⟨ v , r 1 ⟩ L + 2 ≤ ∥ v ∥ 2 − 2 \|v-r_1\|^2=\|v\|^2-2\langle v,r_1\rangle_L+2\le\|v\|^2-2 ∥ v − r 1 ∥ 2 = ∥ v ∥ 2 − 2 ⟨ v , r 1 ⟩ L + 2 ≤ ∥ v ∥ 2 − 2 v v v R R R E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2 R = 8 R=8 R = 8 L D L T L D L^{\mathsf T} L D L T 1 , 78 , 702 , 2346 1,78,702,2346 1 , 78 , 702 , 2346 5694 5694 5694 0 , 2 , 4 , 6 0,2,4,6 0 , 2 , 4 , 6 8 8 8
In both cases of Theorem 8.5 we realize U 1 U_1 U 1 V ′ V' V ′ H ′ H' H ′ H ′ ′ H'' H ′′ h = L ⊗ C \mathfrak{h}=L\otimes\mathbb{C} h = L ⊗ C s l 2 ⊕ s l 2 \mathfrak{sl}_2\oplus\mathfrak{sl}_2 sl 2 ⊕ sl 2 H ( 1 ) H = 2 k 1 H(1)H=2k1 H ( 1 ) H = 2 k 1 e ( 1 ) f = k 1 e(1)f=k1 e ( 1 ) f = k 1 Table 2 , the weight-one A A A V ′ V' V ′ R b R_b R b g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ R b R_b R b
Weight window of the lattice calculation. The mode implementation stores lattice momenta of squared norm at most 8 8 8 d = wt u + wt v − n − 1 d = \operatorname{wt} u + \operatorname{wt} v - n - 1 d = wt u + wt v − n − 1 γ \gamma γ ∣ γ ∣ 2 / 2 \lvert\gamma\rvert^2/2 ∣ γ ∣ 2 /2 u = h ( − m ) u ′ u = h_{(-m)}u' u = h ( − m ) u ′ m ≥ 1 m \ge1 m ≥ 1 d − m − j d-m-j d − m − j d d d j ≥ 0 j \ge0 j ≥ 0
The Griess algebra of the commutant
Lemma 10.1 Let B = C ^ 2 B = \widehat{C}_2 B = C 2 C ^ 1 = 0 \widehat{C}_1 = 0 C 1 = 0 dim B = 4 \dim B = 4 dim B = 4 ( I ) (I) ( I ) dim B = 5 \dim B = 5 dim B = 5 ( I I ) (II) ( I I ) a ⋅ b : = a ( 1 ) b a \mathbin{\cdot} b \mathrel{\mathop:}= a_{(1)}b a ⋅ b : = a ( 1 ) b B B B e : = 1 2 ω C e \mathrel{\mathop:}= \frac{1}{2}\omega_C e : = 2 1 ω C β ( ⋅ , ⋅ ) \beta(\mathord{\cdot},\mathord{\cdot}) β ( ⋅ , ⋅ ) β ( a ⋅ b , c ) = β ( a , b ⋅ c ) \beta(a\mathbin{\cdot}b,c) = \beta(a,b\mathbin{\cdot}c) β ( a ⋅ b , c ) = β ( a , b ⋅ c ) B = C 2 ⊕ Y B = C_2 \oplus Y B = C 2 ⊕ Y Y = B ∩ M b Y = B \cap M_b Y = B ∩ M b C 2 C_2 C 2 B B B e e e
Proof: The space C ^ 1 \widehat{C}_1 C 1 U 1 U_1 U 1 e 6 ⊕ s l 3 \mathfrak{e}_6 \oplus\mathfrak{sl}_3 e 6 ⊕ sl 3 Theorem 8.5 . The dimensions are computed by solving exactly x ( n ) v = 0 x_{(n)}v = 0 x ( n ) v = 0 n ∈ { 0 , 1 , 2 } n \in\{0,1,2\} n ∈ { 0 , 1 , 2 } x ∈ U 1 x \in U_1 x ∈ U 1 V 2 ′ V'_2 V 2 ′ g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ V 2 ′ V'_2 V 2 ′ A A A ∑ a x a ( − 1 ) w a \sum_a x^a(-1)w_a ∑ a x a ( − 1 ) w a V ( 0 , 2 ) V(0,2) V ( 0 , 2 ) R b R_b R b C ^ 1 = 0 \widehat{C}_1 = 0 C 1 = 0 e e e L ( 0 ) − L A ( 0 ) = ( ω C ) ( 1 ) L(0)-L_A(0) = (\omega_C)_{(1)} L ( 0 ) − L A ( 0 ) = ( ω C ) ( 1 ) B B B B B B L ( 1 ) B ⊂ C ^ 1 = 0 L(1)B \subset\widehat{C}_1 = 0 L ( 1 ) B ⊂ C 1 = 0 M b M_b M b U U U U U U U × M b U \times M_b U × M b V ′ = U ⊕ M b V' = U \oplus M_b V ′ = U ⊕ M b A A A B = ( B ∩ U ) ⊕ ( B ∩ M b ) B = (B \cap U) \oplus(B \cap M_b) B = ( B ∩ U ) ⊕ ( B ∩ M b ) B ∩ U = C 2 B \cap U = C_2 B ∩ U = C 2 Corollary 9.2 .
Lemma 10.2 Let D ⊂ B D \subset B D ⊂ B e e e D = C e ⊕ C x C D = \mathbb{C}e \oplus\mathbb{C}x_C D = C e ⊕ C x C β ( x C , e ) = 0 \beta(x_C,e) = 0 β ( x C , e ) = 0 x C ⋅ x C = λ x C + μ e x_C \mathbin{\cdot} x_C = \lambda x_C + \mu e x C ⋅ x C = λ x C + μ e μ = β ( x C , x C ) β ( e , e ) \mu= \frac{\beta(x_C,x_C)}{\beta(e,e)} μ = β ( e , e ) β ( x C , x C ) Δ : = λ 2 + 4 μ ≠ 0 \Delta\mathrel{\mathop:}= \lambda^2 + 4\mu\ne0 Δ : = λ 2 + 4 μ = 0 D D D f f f e − f e-f e − f ω 1 = 2 f \omega_1 = 2f ω 1 = 2 f ω 2 = 2 ( e − f ) \omega_2 = 2(e-f) ω 2 = 2 ( e − f ) ω 1 + ω 2 = ω C \omega_1 + \omega_2 = \omega_C ω 1 + ω 2 = ω C
c 1 , 2 = c C 2 ( 1 ∓ λ Δ ) . (24) c_{1,2} = \frac{c_C}{2}\left(1 \mp\frac{\lambda}{\sqrt{\Delta}}\right). \tag*{(24)} c 1 , 2 = 2 c C ( 1 ∓ Δ λ ) . (24) Conversely, x C ∈ e ⊥ x_C \in e^\perp x C ∈ e ⊥ e e e x C x_C x C q ( x C ) = ( x C ⋅ x C ) ⊥ q(x_C) = (x_C \mathbin{\cdot} x_C)^{\perp} q ( x C ) = ( x C ⋅ x C ) ⊥ e ⊥ e^\perp e ⊥
Proof: Since e e e D D D x C ⋅ x C = λ x C + μ e x_C \mathbin{\cdot} x_C = \lambda x_C + \mu e x C ⋅ x C = λ x C + μ e β ( x C ⋅ x C , e ) = β ( x C , x C ⋅ e ) = β ( x C , x C ) \beta(x_C \mathbin{\cdot} x_C,e) = \beta(x_C,x_C \mathbin{\cdot} e) = \beta(x_C,x_C) β ( x C ⋅ x C , e ) = β ( x C , x C ⋅ e ) = β ( x C , x C ) μ \mu μ f = a e + b x C f = ae + bx_C f = a e + b x C 2 a + b λ = 1 2a + b\lambda= 1 2 a + bλ = 1 a 2 + b 2 μ = a a^2 + b^2\mu= a a 2 + b 2 μ = a b 2 Δ = 1 b^2\Delta= 1 b 2 Δ = 1 a = 1 − b λ 2 a = \frac{1-b\lambda}{2} a = 2 1 − bλ ( e − f ) ⋅ ( e − f ) = e − f (e-f)\mathbin{\cdot}(e-f) = e-f ( e − f ) ⋅ ( e − f ) = e − f f ⋅ ( e − f ) = 0 f\mathbin{\cdot}(e-f) = 0 f ⋅ ( e − f ) = 0 C ^ 1 = 0 \widehat{C}_1 = 0 C 1 = 0 ω 1 \omega_1 ω 1 ( ω 1 ) ( 1 ) ω 1 = 2 ω 1 (\omega_1)_{(1)}\omega_1 = 2\omega_1 ( ω 1 ) ( 1 ) ω 1 = 2 ω 1 2 β ( ω 1 , ω 1 ) 2\beta(\omega_1,\omega_1) 2 β ( ω 1 , ω 1 ) [60 ] ]. Put v = 2 f v = 2f v = 2 f w = ω C − v w = \omega_C-v w = ω C − v C ^ 1 = 0 \widehat{C}_1=0 C 1 = 0 v ( 0 ) v = L ( − 1 ) v v_{(0)}v=L(-1)v v ( 0 ) v = L ( − 1 ) v v ( 0 ) ω C = L ( − 1 ) v v_{(0)}\omega_C=L(-1)v v ( 0 ) ω C = L ( − 1 ) v v ( 0 ) w = 0 v_{(0)}w=0 v ( 0 ) w = 0 v ( 1 ) w v_{(1)}w v ( 1 ) w v ( 2 ) w v_{(2)}w v ( 2 ) w v ( 3 ) w v_{(3)}w v ( 3 ) w c 1 = 2 β ( 2 f , 2 f ) = 8 ( a 2 β ( e , e ) + b 2 μ β ( e , e ) ) = c C ( a 2 + b 2 μ ) = c C a c_1=2\beta(2f,2f)=8(a^2\beta(e,e)+b^2\mu\beta(e,e))=c_C(a^2+b^2\mu)=c_Ca c 1 = 2 β ( 2 f , 2 f ) = 8 ( a 2 β ( e , e ) + b 2 μ β ( e , e )) = c C ( a 2 + b 2 μ ) = c C a β ( e , e ) = c C 8 \beta(e,e)=\frac{c_C}{8} β ( e , e ) = 8 c C
Proposition 10.3 The two-dimensional subalgebras of B B B e e e ( I ) (I) ( I ) ( I I ) (II) ( I I ) Δ ≠ 0 \Delta\ne0 Δ = 0 ( c 1 , c 2 ) (c_1,c_2) ( c 1 , c 2 ) (24) are listed in Table 3 . In case ( I ) (I) ( I ) 2400 1271 \frac{2400}{1271} 1271 2400 70 31 \frac{70}{31} 31 70 ( I I ) (II) ( I I ) 182 95 \frac{182}{95} 95 182 32 19 \frac{32}{19} 19 32 13 ↦ − 13 \sqrt{13}\mapsto-\sqrt{13} 13 ↦ − 13 Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
Table 3. The two-dimensional subalgebras of B B B e e e Theorem 10.5 and Propositions 10.34 and 10.35 . The supplementary data use the same numbering.
Proof : We compute the structure constants of B B B e , x 1 , … , x n − 1 e,x_1,\ldots,x_{n-1} e , x 1 , … , x n − 1 x i ⊥ e x_i\perp e x i ⊥ e q ( x ) = λ x q(x)=\lambda x q ( x ) = λ x P ( e ⊥ ) \mathbb{P}(e^\perp) P ( e ⊥ ) Q \mathbb{Q} Q [12 Theorem 1.2 p. 2] a quadratic map of C n \mathbb{C}^n C n 2 n − 1 2^n-1 2 n − 1 7 = 2 3 − 1 7=2^3-1 7 = 2 3 − 1 15 = 2 4 − 1 15=2^4-1 15 = 2 4 − 1 ( 7 , 0 , 0 ) (7,0,0) ( 7 , 0 , 0 ) ( I ) (I) ( I ) ( 11 , 2 , 1 , 1 ) (11,2,1,1) ( 11 , 2 , 1 , 1 ) ( I I ) (II) ( I I )
Linear and quadratic conditions on the primary lines
For a pair ( a , b ) (a,b) ( a , b ) Π a b ⊂ V 2 ′ \Pi_{ab}\subset V'_2 Π ab ⊂ V 2 ′ A A A V ( a , b ) V(a,b) V ( a , b ) v v v ( a , b ) (a,b) ( a , b ) e ′ ( 0 ) v = e ′ ′ ( 0 ) v = 0 e'(0)v=e''(0)v=0 e ′ ( 0 ) v = e ′′ ( 0 ) v = 0 x ( n ) v = 0 x(n)v=0 x ( n ) v = 0 x ∈ U x\in U x ∈ U n ≥ 1 n\ge1 n ≥ 1 U 2 U_2 U 2 Table 2 not accounted for by ( A ) 2 ⊕ C 2 (A)_2\oplus C_2 ( A ) 2 ⊕ C 2 ( I ) (I) ( I ) ( 0 , 4 ) (0,4) ( 0 , 4 ) ( 0 , 8 ) (0,8) ( 0 , 8 ) ( 0 , 12 ) (0,12) ( 0 , 12 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 5 ) (1,5) ( 1 , 5 ) ( 1 , 7 ) (1,7) ( 1 , 7 ) ( 1 , 13 ) (1,13) ( 1 , 13 ) ( I I ) (II) ( I I ) ( 0 , 4 ) (0,4) ( 0 , 4 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 1 , 5 ) (1,5) ( 1 , 5 ) ( 1 , 7 ) (1,7) ( 1 , 7 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) ( 2 , 6 ) (2,6) ( 2 , 6 ) ( 3 , 3 ) (3,3) ( 3 , 3 ) U = ⨁ λ L A ( λ ) ⊗ C λ U=\bigoplus_{\lambda}L_A(\lambda)\otimes\mathbb{C}^{\lambda} U = ⨁ λ L A ( λ ) ⊗ C λ λ ≠ 0 \lambda\ne0 λ = 0 L A ( λ ) L_A(\lambda) L A ( λ ) U 1 = A 1 U_1=A_1 U 1 = A 1 ( A ) 2 ⊂ C 2 (A)_2\subset C_2 ( A ) 2 ⊂ C 2 P a b = U 2 ∩ Π a b P_{ab}=U_2\cap\Pi_{ab} P ab = U 2 ∩ Π ab Π a b \Pi_{ab} Π ab P a b ( u , v ) = β ( u , θ a b v ) P_{ab}(u,v)=\beta(u,\theta_{ab}v) P ab ( u , v ) = β ( u , θ ab v ) θ a b v = f ′ ( 0 ) a f ′ ′ ( 0 ) b v \theta_{ab}v=f'(0)^af''(0)^bv θ ab v = f ′ ( 0 ) a f ′′ ( 0 ) b v
Lemma 10.4 Let C 2 = C e ⊕ C x C C_2=\mathbb{C}e\oplus\mathbb{C}x_C C 2 = C e ⊕ C x C Y = B ∩ C 2 ⊥ Y=B\cap C_2^\perp Y = B ∩ C 2 ⊥ ( a , b ) (a,b) ( a , b ) u , u ′ ∈ P a b u,u'\in P_{ab} u , u ′ ∈ P ab
e ( 2 ) u = ( x C ) ( 2 ) u = 0 e_{(2)}u=(x_C)_{(2)}u=0 e ( 2 ) u = ( x C ) ( 2 ) u = 0 u u u ω 1 \omega_1 ω 1 ω 2 \omega_2 ω 2
( x C ) ( 1 ) p a b ⊂ p a b (x_C)_{(1)}\mathfrak{p}_{ab}\subset\mathfrak{p}_{ab} ( x C ) ( 1 ) p ab ⊂ p ab p a b \mathfrak{p}_{ab} p ab ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) ( ω 1 ) ( 1 ) , ( ω 2 ) ( 1 ) (\omega_1)_{(1)},(\omega_2)_{(1)} ( ω 1 ) ( 1 ) , ( ω 2 ) ( 1 ) h 1 , h 2 h_1,h_2 h 1 , h 2 u u u h 1 + h 2 = 2 − h a b h_1+h_2=2-h_{ab} h 1 + h 2 = 2 − h ab h a b h_{ab} h ab
P a b ( u , y ( 1 ) u ′ ) = 0 P_{ab}(u,y_{(1)}u^{\prime})=0 P ab ( u , y ( 1 ) u ′ ) = 0 y ∈ Y y\in Y y ∈ Y p a b \mathfrak{p}_{ab} p ab
for every primary type ( c , d ) (c,d) ( c , d ) w w w g ′ ⊕ g ′ ′ \mathfrak{g}^{\prime}\oplus\mathfrak{g}^{\prime\prime} g ′ ⊕ g ′′ p c d \mathfrak{p}_{cd} p c d u ( 1 ) w ∈ U 2 u_{(1)}w\in U_2 u ( 1 ) w ∈ U 2 u ( 2 ) w ∈ U 1 u_{(2)}w\in U_1 u ( 2 ) w ∈ U 1
Proof. 1. The vectors e ( 2 ) u e_{(2)}u e ( 2 ) u ( x C ) ( 2 ) u (x_C)_{(2)}u ( x C ) ( 2 ) u U 1 U_1 U 1 z ( n ) z_{(n)} z ( n ) z ∈ U 1 z\in U_1 z ∈ U 1 n ≥ 1 n\geq1 n ≥ 1 e e e x C x_C x C A A A ( a , b ) ≠ ( 0 , 0 ) (a,b)\ne(0,0) ( a , b ) = ( 0 , 0 ) U 1 = A 1 U_1=A_1 U 1 = A 1 A A A ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) U 2 U_2 U 2 A A A y ∈ M b y\in M_b y ∈ M b M b M_b M b U U U y ( 1 ) u ′ ∈ M b y_{(1)}u^{\prime}\in M_b y ( 1 ) u ′ ∈ M b U × M b U\times M_b U × M b θ a b \theta_{ab} θ ab y ( 1 ) y_{(1)} y ( 1 ) β \beta β A A A A A A ( A ) 2 (A)_2 ( A ) 2 s l 2 \mathfrak{sl}_2 sl 2 U 2 U_2 U 2 θ a b \theta_{ab} θ ab U U U
Case (II)
Theorem 10.5 Assume the hypotheses of Theorem 8.5 in case (II). Then C 2 C_2 C 2 Table 3 , so that ω C \omega_C ω C 182 95 \frac{182}{95} 95 182 32 19 \frac{32}{19} 19 32 U 2 U_2 U 2 V 2 V_2 V 2 Q ( 13 , − 3 ) \mathbb{Q}(\sqrt{13},\sqrt{-3}) Q ( 13 , − 3 )
Proof. The proof consists of four finite steps. Exact linear algebra and Gröbner bases construct the admissible spaces. Certified interval evaluations are used only for exclusions.
Step 1: exclusion of thirteen subalgebras. It suffices to use the primary type ( 3 , 3 ) (3,3) ( 3 , 3 ) U 2 U_2 U 2 Π 33 \Pi_{33} Π 33 x C x_C x C B B B X X X ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) Π 33 \Pi_{33} Π 33 P P P P 33 P_{33} P 33 Y 1 , Y 2 , Y 3 Y_1,Y_2,Y_3 Y 1 , Y 2 , Y 3 y ( 1 ) y_{(1)} y ( 1 ) B ∩ C 2 ⊥ B\cap C_2^{\perp} B ∩ C 2 ⊥ u ∈ C 3 u\in\mathbb{C}^3 u ∈ C 3
rank [ u , X u ] = 1 , u T P Y j u = 0 ( j = 1 , 2 , 3 ) , u T P u ≠ 0. \operatorname{rank}[u,Xu]=1,\qquad u^{T}PY_j u=0\ (j=1,2,3),\qquad u^{T}Pu\ne0. rank [ u , X u ] = 1 , u T P Y j u = 0 ( j = 1 , 2 , 3 ) , u T P u = 0. These are only some of the necessary conditions in Lemma 10.4. Their inconsistency is enough. On the projective chart whose first nonzero coordinate is u k u_k u k u 0 = ⋯ = u k − 1 = 0 u_0=\cdots=u_{k-1}=0 u 0 = ⋯ = u k − 1 = 0 u k = 1 u_k=1 u k = 1 x x x x u T P u − 1 = 0 xu^{T}Pu-1=0 x u T P u − 1 = 0 [ u , X u ] [u,Xu] [ u , X u ] Q \mathbb{Q} Q { 1 } \{1\} { 1 }
chart polynomials and all 39 unit bases. Hence none of these thirteen candidates admits the required primary line. No eigenvalue approximation is used in this step.
Step 2: the admissible spaces. We call admissible space of type ( a , b ) (a,b) ( a , b ) E ξ E_{\xi} E ξ ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) Π a b ∩ Ker e ( 2 ) ∩ Ker ( x C ) ( 2 ) \Pi_{ab}\cap\operatorname{Ker} e_{(2)}\cap\operatorname{Ker}(x_C)_{(2)} Π ab ∩ Ker e ( 2 ) ∩ Ker ( x C ) ( 2 ) u u u P a b ( u , u ) ≠ 0 P_{ab}(u,u)\ne0 P ab ( u , u ) = 0 P a b ( u , y ( 1 ) u ) = 0 P_{ab}(u,y_{(1)}u)=0 P ab ( u , y ( 1 ) u ) = 0 y ∈ Y y\in Y y ∈ Y p a b p_{ab} p ab E ξ E_{\xi} E ξ ξ \xi ξ
Step 3: the lines. In each two-dimensional E ξ E_{\xi} E ξ P a b ( u , y ( 1 ) u ) P_{ab}(u,y_{(1)}u) P ab ( u , y ( 1 ) u ) y ∈ Y y\in Y y ∈ Y Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 ) P a b ( u , u ) ≠ 0 P_{ab}(u,u)\ne0 P ab ( u , u ) = 0 2 8 = 256 2^8=256 2 8 = 256 U 2 U_2 U 2 Q ( 13 , − 3 ) \mathbb{Q}(\sqrt{13},\sqrt{-3}) Q ( 13 , − 3 ) 6 − 7 + 13 = − 3 ( 13 − 1 ) \sqrt{6}\sqrt{-7+\sqrt{13}}=\sqrt{-3}(\sqrt{13}-1) 6 − 7 + 13 = − 3 ( 13 − 1 )
Step 4: pairwise closure. For two lines of types ( a , b ) (a,b) ( a , b ) ( c , d ) (c,d) ( c , d ) u ( 2 ) w ∈ U 1 u_{(2)}w\in U_1 u ( 2 ) w ∈ U 1 u ( 1 ) w u_{(1)}w u ( 1 ) w ( A ) 2 (A)_2 ( A ) 2 C 2 C_2 C 2 U 1 U_1 U 1 u ( 1 ) w u_{(1)}w u ( 1 ) w u ( 2 ) w u_{(2)}w u ( 2 ) w [53 ] . For each exclusion, the certificate records the rational annihilator, the product tested and a nonzero interval, giving 62 certified rejections. The algebraic inputs are evaluated directly, without an intermediate rounded decimal approximation. The finite coverage check uses only the 62 pairs marked as certified. The ten inconclusive pairs are not used. In the order of primary types displayed in the theorem, the two remaining choices are ( 0 , 1 , 1 , 0 , 0 , 1 , 0 , 1 ) (0,1,1,0,0,1,0,1) ( 0 , 1 , 1 , 0 , 0 , 1 , 0 , 1 ) ( 1 , 0 , 0 , 1 , 1 , 0 , 1 , 0 ) (1,0,0,1,1,0,1,0) ( 1 , 0 , 0 , 1 , 1 , 0 , 1 , 0 ) 13 ↦ − 13 \sqrt{13}\mapsto-\sqrt{13} 13 ↦ − 13
Reduction conventions. For a prime p p p p \mathfrak{p} p O K \mathcal{O}_K O K 13 \sqrt{13} 13 − 3 \sqrt{-3} − 3 F p \mathbb{F}_p F p red p : ( O K ) p → F p \operatorname{red}_{\mathfrak{p}}:(\mathcal{O}_K)_{\mathfrak{p}}\to\mathbb{F}_p red p : ( O K ) p → F p p \mathfrak{p} p p p p M M M rank F p ( M ‾ ) ≤ rank K ( M ) \operatorname{rank}_{\mathbb{F}_p}(\overline{M})\leq\operatorname{rank}_K(M) rank F p ( M ) ≤ rank K ( M ) K K K
Proposition 10.7. Let U 2 U_2 U 2 Theorem 10.5 , and let S ⊂ V ′ S \subset V' S ⊂ V ′ U 1 ⊕ U 2 U_1 \oplus U_2 U 1 ⊕ U 2 p 1 = 134217613 p_1 = 134217613 p 1 = 134217613 p 2 = 133217647 p_2 = 133217647 p 2 = 133217647 p i p_i p i p i p_i p i
dim U 2 = 120 \dim U_2 = 120 dim U 2 = 120 U 2 U_2 U 2 u ( 1 ) w ∈ U 2 u_{(1)}w \in U_2 u ( 1 ) w ∈ U 2 u ( 2 ) w ∈ U 1 u_{(2)}w \in U_1 u ( 2 ) w ∈ U 1 u , w ∈ U 2 u,w \in U_2 u , w ∈ U 2
The vectors x ( − 3 ) 1 x(-3)1 x ( − 3 ) 1 x ( − 2 ) y x(-2)y x ( − 2 ) y x ( − 1 ) u x(-1)u x ( − 1 ) u L ( − 1 ) u L(-1)u L ( − 1 ) u u ( 0 ) w u_{(0)}w u ( 0 ) w x , y ∈ U 1 x,y \in U_1 x , y ∈ U 1 u , w ∈ U 2 u,w \in U_2 u , w ∈ U 2 u u u g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ S 3 g e n S_3^{\mathrm{gen}} S 3 gen 956 = dim U 3 956 = \dim U_3 956 = dim U 3 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′
4 V ( 0 , 0 ) + 11 V ( 0 , 2 ) + 4 V ( 0 , 4 ) + 6 V ( 0 , 6 ) + V ( 0 , 10 ) + 6 V ( 1 , 1 ) + 8 V ( 1 , 3 ) + 7 V ( 1 , 5 ) + 5 V ( 1 , 7 ) + V ( 1 , 9 ) + 8 V ( 2 , 0 ) + 6 V ( 2 , 2 ) + 7 V ( 2 , 4 ) + 4 V ( 2 , 6 ) + 2 V ( 2 , 8 ) + 2 V ( 3 , 1 ) + 5 V ( 3 , 3 ) + 2 V ( 3 , 5 ) + V ( 3 , 7 ) + V ( 4 , 0 ) + 2 V ( 4 , 2 ) + V ( 4 , 6 ) + V ( 6 , 0 ) . 4V(0,0) + 11V(0,2) + 4V(0,4) + 6V(0,6) + V(0,10) + 6V(1,1) + 8V(1,3) + 7V(1,5)
+ 5V(1,7) + V(1,9) + 8V(2,0) + 6V(2,2) + 7V(2,4) + 4V(2,6) + 2V(2,8) + 2V(3,1)
+ 5V(3,3) + 2V(3,5) + V(3,7) + V(4,0) + 2V(4,2) + V(4,6) + V(6,0). 4 V ( 0 , 0 ) + 11 V ( 0 , 2 ) + 4 V ( 0 , 4 ) + 6 V ( 0 , 6 ) + V ( 0 , 10 ) + 6 V ( 1 , 1 ) + 8 V ( 1 , 3 ) + 7 V ( 1 , 5 ) + 5 V ( 1 , 7 ) + V ( 1 , 9 ) + 8 V ( 2 , 0 ) + 6 V ( 2 , 2 ) + 7 V ( 2 , 4 ) + 4 V ( 2 , 6 ) + 2 V ( 2 , 8 ) + 2 V ( 3 , 1 ) + 5 V ( 3 , 3 ) + 2 V ( 3 , 5 ) + V ( 3 , 7 ) + V ( 4 , 0 ) + 2 V ( 4 , 2 ) + V ( 4 , 6 ) + V ( 6 , 0 ) . Its nine joint traces Tr H ′ ( 0 ) p H ′ ′ ( 0 ) q \operatorname{Tr} H'(0)^p H''(0)^q Tr H ′ ( 0 ) p H ′′ ( 0 ) q p , q p,q p , q 2 ≤ p + q ≤ 6 2 \le p+q \le6 2 ≤ p + q ≤ 6 U 3 U_3 U 3 Proposition 6.2 .
For every u ∈ U 2 u \in U_2 u ∈ U 2 w w w u ( 2 ) w ∈ U 2 u_{(2)}w \in U_2 u ( 2 ) w ∈ U 2 u ( 3 ) w ∈ U 1 u_{(3)}w \in U_1 u ( 3 ) w ∈ U 1 y ∈ U 1 y \in U_1 y ∈ U 1 y ( 1 ) w ∈ U 2 y_{(1)}w \in U_2 y ( 1 ) w ∈ U 2 y ( 2 ) w ∈ U 1 y_{(2)}w \in U_1 y ( 2 ) w ∈ U 1
Proof: Exact computations in F p i \mathbb{F}_{p_i} F p i F p i \mathbb{F}_{p_i} F p i e ′ e' e ′ f ′ f' f ′ e ′ ′ e'' e ′′ f ′ ′ f'' f ′′ u = e u=e u = e u = x C u=x_C u = x C y y y U 1 U_1 U 1 p 1 p_1 p 1 U 2 U_2 U 2 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ ( A ) 2 (A)_2 ( A ) 2 x ( − 1 ) y x(-1)y x ( − 1 ) y x ( − 2 ) 1 x(-2)1 x ( − 2 ) 1
Case (II): exact weight-two closure, unitarity and symmetries
From now on, and until the end of § 10.7 , U 1 = g ′ ⊕ g ′ ′ U_1 = \mathfrak{g}' \oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ § 10 (its raising currents are displayed in (27) ), and U 2 U_2 U 2 V 2 ′ V'_2 V 2 ′ ( A ) 2 (A)_2 ( A ) 2 Table 3 and by the g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ Theorem 10.5 . This construction is a finite computation in V ′ V' V ′ Corollaries 10.22 and 10.25 and Theorem 10.29 , the results of § 10.5 –§ 10.7 neither assert the existence of a Haagerup algebra nor ( H J ) (H_J) ( H J ) Theorems 10.9 , 10.10 , 10.21 and 10.24 , which are used in [ArXiv], do not depend on [ArXiv].
Let K = Q ( 13 , − 3 ) \mathbb{K} = \mathbb{Q}(\sqrt{13},\sqrt{-3}) K = Q ( 13 , − 3 ) V K ′ V'_{\mathbb{K}} V K ′ K \mathbb{K} K U j , K = U j ∩ V K ′ U_{j,\mathbb{K}} = U_j \cap V'_{\mathbb{K}} U j , K = U j ∩ V K ′ K \mathbb{K} K C \mathbb{C} C Theorem 10.5 are spanned by vectors of V 2 ′ V_2' V 2 ′ K \mathbb{K} K c ↦ c ˉ c \mapsto\bar{c} c ↦ c ˉ K \mathbb{K} K − 3 ↦ − − 3 \sqrt{-3} \mapsto-\sqrt{-3} − 3 ↦ − − 3 v ↦ v ˉ v \mapsto\bar{v} v ↦ v ˉ V ′ V' V ′ σ \sigma σ 13 ↦ − 13 \sqrt{13} \mapsto-\sqrt{13} 13 ↦ − 13 K \mathbb{K} K − 3 \sqrt{-3} − 3
Theorem 10.9 Each of the four subspaces U 2 U_2 U 2 Theorem 10.5 has dimension 120 and is closed in weight two over K \mathbb{K} K u , w ∈ U 2 u,w \in U_2 u , w ∈ U 2 u ( 1 ) w ∈ U 2 u_{(1)}w \in U_2 u ( 1 ) w ∈ U 2 u ( 2 ) w ∈ U 1 u_{(2)}w \in U_1 u ( 2 ) w ∈ U 1
Proof. We work in exact arithmetic in K \mathbb{K} K 1 , 13 , − 3 , − 39 1,\sqrt{13},\sqrt{-3},\sqrt{-39} 1 , 13 , − 3 , − 39 K ≅ Q 4 \mathbb{K} \cong\mathbb{Q}^4 K ≅ Q 4 K \mathbb{K} K Q \mathbb{Q} Q g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ U 2 U_2 U 2 = 4 ⋅ 120 =4\cdot120 = 4 ⋅ 120 u ( n ) w u_{(n)}w u ( n ) w n ∈ { 1 , 2 } n \in\{1,2\} n ∈ { 1 , 2 } u u u e e e x C x_C x C w w w e e e x C x_C x C g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ [ y ( 0 ) , u ( n ) ] = ( y ( 0 ) u ) ( n ) [y_{(0)},u_{(n)}]=(y_{(0)}u)_{(n)} [ y ( 0 ) , u ( n ) ] = ( y ( 0 ) u ) ( n ) U 1 U_1 U 1 U 2 U_2 U 2 z ∈ U 1 z \in U_1 z ∈ U 1 z ( n ) U 2 ⊂ U 2 − n z_{(n)}U_2 \subset U_{2-n} z ( n ) U 2 ⊂ U 2 − n n ≥ 0 n \geq0 n ≥ 0 U 0 = C 1 U_0=\mathbb{C}1 U 0 = C 1 U j = 0 U_j=0 U j = 0 j < 0 j<0 j < 0 ( ( A 2 ) ( 2 ) ) ( 1 ) U 2 ⊂ U 2 ((A_2)_{(2)})(1)U_2 \subset U_2 (( A 2 ) ( 2 ) ) ( 1 ) U 2 ⊂ U 2 ( ( A 2 ) ( 2 ) ) ( 2 ) U 2 ⊂ U 1 ((A_2)_{(2)})(2)U_2 \subset U_1 (( A 2 ) ( 2 ) ) ( 2 ) U 2 ⊂ U 1 U 1 ( − 1 ) U 1 + U 1 ( − 2 ) U 0 ⊂ ( A 2 ) U_1(-1)U_1+U_1(-2)U_0\subset(A_2) U 1 ( − 1 ) U 1 + U 1 ( − 2 ) U 0 ⊂ ( A 2 ) ( x ( − 2 ) 1 ) ( n ) = − n x ( n − 1 ) (x_{(-2)}1)_{(n)}=-nx_{(n-1)} ( x ( − 2 ) 1 ) ( n ) = − n x ( n − 1 ) σ \sigma σ V ′ V' V ′
This proves part 1 of Proposition 10.7 over K \mathbb{K} K
We now show that the algebra generated is unitary, in the sense of Dong and Lin [24 ] and Carpi, Kawahigashi, Longo and Weiner [14 Section 5] . Recall that V ′ V' V ′ h 1 ( − n 1 ) ⋯ h k ( − n k ) e λ h_1(-n_1)\cdots h_k(-n_k)e^\lambda h 1 ( − n 1 ) ⋯ h k ( − n k ) e λ h i h_i h i ± 1 \pm1 ± 1 χ : L → R > 0 \chi:L\to\mathbb{R}_{>0} χ : L → R > 0 L = E 6 ⊕ A 2 L=E_6\oplus A_2 L = E 6 ⊕ A 2 ϕ χ \phi_\chi ϕ χ
ϕ χ ( c h 1 ( − n 1 ) ⋯ h k ( − n k ) e λ ) = ( − 1 ) k χ ( λ ) c ˉ h 1 ( − n 1 ) ⋯ h k ( − n k ) e − λ , c ∈ C . (25) \phi_\chi\left(c h_1(-n_1)\cdots h_k(-n_k)e^\lambda\right)=(-1)^k\chi(\lambda)\bar{c}h_1(-n_1)\cdots h_k(-n_k)e^{-\lambda},\qquad c\in\mathbb{C}. \tag*{(25)} ϕ χ ( c h 1 ( − n 1 ) ⋯ h k ( − n k ) e λ ) = ( − 1 ) k χ ( λ ) c ˉ h 1 ( − n 1 ) ⋯ h k ( − n k ) e − λ , c ∈ C . (25) Theorem 10.10 Let χ \chi χ 2 15 , 1 , 1 , 3 5 , 9 25 , 5 24 \frac{2}{15},1,1,\frac{3}{5},\frac{9}{25},\frac{5}{24} 15 2 , 1 , 1 , 5 3 , 25 9 , 24 5 α 1 , … , α 6 \alpha_1,\ldots,\alpha_6 α 1 , … , α 6 E 6 E_6 E 6 2 , 2 2,2 2 , 2 A 2 A_2 A 2
ϕ χ \phi_\chi ϕ χ V ′ V' V ′ H χ ( u , v ) = β ( u , ϕ χ ( v ) ) H_\chi(u,v)=\beta(u,\phi_\chi(v)) H χ ( u , v ) = β ( u , ϕ χ ( v )) ( V ′ , ϕ χ ) (V',\phi_\chi) ( V ′ , ϕ χ )
ϕ χ ( e ′ ) = − f ′ \phi_\chi(e')=-f' ϕ χ ( e ′ ) = − f ′ ϕ χ ( f ′ ) = − e ′ \phi_\chi(f')=-e' ϕ χ ( f ′ ) = − e ′ ϕ χ ( H ′ ) = − H ′ \phi_\chi(H')=-H' ϕ χ ( H ′ ) = − H ′ g ′ ′ \mathfrak{g}'' g ′′ ϕ χ ( U 1 ) = U 1 \phi_\chi(U_1)=U_1 ϕ χ ( U 1 ) = U 1
ϕ χ ( U 2 ) = U 2 \phi_\chi(U_2)=U_2 ϕ χ ( U 2 ) = U 2 Theorem 10.5 .
Consequently, the vertex subalgebra S ⊂ V ′ S\subset V' S ⊂ V ′ U 1 ⊕ U 2 U_1\oplus U_2 U 1 ⊕ U 2 S ≅ S ′ S\cong S' S ≅ S ′
Proof. 1. Let ϕ 1 \phi_1 ϕ 1 ϵ ( − α , − β ) = ϵ ( α , β ) \epsilon(-\alpha,-\beta)=\epsilon(\alpha,\beta) ϵ ( − α , − β ) = ϵ ( α , β ) ϕ 1 \phi_1 ϕ 1 [24 Theorem 4.12] , [14 Example 5.9] . We have checked our sign conventions directly: ϕ 1 ( u ( n ) v ) = ϕ 1 ( u ) ( n ) ϕ 1 ( v ) \phi_1(u(n)v)=\phi_1(u)(n)\phi_1(v) ϕ 1 ( u ( n ) v ) = ϕ 1 ( u ) ( n ) ϕ 1 ( v ) H 1 ( u , v ) = β ( u , ϕ 1 ( v ) ) H_1(u,v)=\beta(u,\phi_1(v)) H 1 ( u , v ) = β ( u , ϕ 1 ( v )) V 1 ′ V'_1 V 1 ′ V 2 ′ V'_2 V 2 ′ V 3 ′ V'_3 V 3 ′ χ ( λ ) = e ⟨ h χ , λ ⟩ L \chi(\lambda)=e^{\langle h_\chi,\lambda\rangle_L} χ ( λ ) = e ⟨ h χ , λ ⟩ L h χ h_\chi h χ g = exp ( h χ ( 0 ) ) g=\exp(h_\chi(0)) g = exp ( h χ ( 0 )) V ′ V' V ′ χ ( λ ) \chi(\lambda) χ ( λ ) M ( 1 ) ⊗ e λ M(1)\otimes e^\lambda M ( 1 ) ⊗ e λ ϕ χ = ϕ 1 ∘ g = g − 1 / 2 ∘ ϕ 1 ∘ g 1 / 2 \phi_\chi=\phi_1\circ g=g^{-1/2}\circ\phi_1\circ g^{1/2} ϕ χ = ϕ 1 ∘ g = g − 1/2 ∘ ϕ 1 ∘ g 1/2 ( V ′ , ϕ χ ) (V',\phi_\chi) ( V ′ , ϕ χ ) ( V ′ , ϕ 1 ) (V',\phi_1) ( V ′ , ϕ 1 ) g − 1 / 2 g^{-1/2} g − 1/2 H χ ( u , v ) = H 1 ( g 1 / 2 u , g 1 / 2 v ) H_\chi(u,v)=H_1(g^{1/2}u,g^{1/2}v) H χ ( u , v ) = H 1 ( g 1/2 u , g 1/2 v )
Direct computation. The character χ \chi χ χ ( α ) = − κ d α / c α \chi(\alpha)=-\kappa d_\alpha/c_\alpha χ ( α ) = − κ d α / c α κ > 0 \kappa>0 κ > 0 κ ′ \kappa' κ ′ g ′ g' g ′ κ ′ ′ \kappa'' κ ′′ g ′ ′ g'' g ′′ c α c_\alpha c α d α d_\alpha d α e α e^\alpha e α e ′ e' e ′ e ′ ′ e'' e ′′ e − α e^{-\alpha} e − α f ′ f' f ′ f ′ ′ f'' f ′′ d α / c α d_\alpha/c_\alpha d α / c α
Exact computation over K: for each of the 120 spanning vectors v v v U 2 U_2 U 2 Theorem 10.9 , the restriction of scalars of ϕ χ ( v ) \phi_\chi(v) ϕ χ ( v ) U 2 U_2 U 2 χ \chi χ ϕ χ \phi_\chi ϕ χ σ \sigma σ
Finally, S S S ϕ χ \phi_\chi ϕ χ ω = ω A + 2 e \omega=\omega_A+2e ω = ω A + 2 e V ′ V' V ′ L ( 1 ) S ⊂ S L(1)S\subset S L ( 1 ) S ⊂ S [14 Proposition 5.23] , S S S ( V ′ , ϕ χ ) (V',\phi_\chi) ( V ′ , ϕ χ ) S 0 = C 1 S_0=\mathbb{C}1 S 0 = C 1 [14 Proposition 5.3] . The bilinear form β \beta β S S S β ( s , ϕ χ ( s ) ) > 0 \beta(s,\phi_\chi(s))>0 β ( s , ϕ χ ( s )) > 0 s ≠ 0 s\ne0 s = 0 ϕ χ ( s ) ∈ S \phi_\chi(s)\in S ϕ χ ( s ) ∈ S S ≅ S ′ S\cong S' S ≅ S ′ [36 ] .
Remark 10.11 Theorem Theorem 10.10 does not use any information on S S S S S S [67 Section 3] for case (I). The trivial character does not work: ϕ 1 ( e ′ ) \phi_1(e') ϕ 1 ( e ′ ) f ′ f' f ′ 13 \sqrt{13} 13 C 2 C_2 C 2 d = 3 ± 13 2 d=\frac{3\pm\sqrt{13}}{2} d = 2 3 ± 13 S S S σ ( S ) \sigma(S) σ ( S ) Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [34 ] cannot be the representation category of a strongly rational algebra of CFT type whose nonvacuum simple modules have positive lowest weights, since its categorical dimensions are then positive [19 Lemma 4.2] , [49 ] .
Proposition 10.12 1. The two subspaces U 2 U_2 U 2 ϕ χ : v ↦ ϕ χ ( v ) \phi_\chi:v\mapsto\phi_\chi(v) ϕ χ : v ↦ ϕ χ ( v ) V ′ V' V ′ U 1 U_1 U 1
The group of automorphisms of the algebra B B B e e e s s s Table 3 , as well as 0 and 14, 1 and 8, 4 and 5, and fixes the seven others.
Consequently the four subspaces give at most two isomorphism classes of vertex operator algebras, and exactly one if s s s V ′ V' V ′ U 1 U_1 U 1 Proposition 10.27 .
Proof. 1. Exact computation: the conjugate of each spanning vector of the first subspace lies in the second. Since χ \chi χ ϕ χ \phi_\chi ϕ χ W W W Theorem 10.10 gives ϕ χ ( W ) = ϕ χ ( W ‾ ) = ϕ χ ( W ) ‾ = W ‾ \phi_\chi(W)=\phi_\chi(\overline{W})=\overline{\phi_\chi(W)}=\overline{W} ϕ χ ( W ) = ϕ χ ( W ) = ϕ χ ( W ) = W
An automorphism g g g B B B e e e g g g C x C \mathbb{C}x_C C x C g ( x C ) = x C g(x_C)=x_C g ( x C ) = x C g ( x C ⋅ x C ) = g ( x C ) ⋅ g ( x C ) g(x_C\cdot x_C)=g(x_C)\cdot g(x_C) g ( x C ⋅ x C ) = g ( x C ) ⋅ g ( x C ) λ ≠ 0 \lambda\ne0 λ = 0 c 1 ≠ c 2 c_1\ne c_2 c 1 = c 2 (24) .
These seven vectors span the hyperplane H = span { x 1 , x 2 , x 3 } H = \operatorname{span}\{x_1,x_2,x_3\} H = span { x 1 , x 2 , x 3 } e ⊥ e^\perp e ⊥ β \beta β H H H 147968 2025 \frac{147968}{2025} 2025 147968 B B B B B B ϕ χ \phi_\chi ϕ χ H χ H_\chi H χ N = H ⊥ ∩ e ⊥ N = H^\perp\cap e^\perp N = H ⊥ ∩ e ⊥ e ⊥ = H ⊕ N e^\perp= H \oplus N e ⊥ = H ⊕ N H H H 1 1 1 − 1 -1 − 1 N N N g g g s s s H H H s s s V ′ V' V ′ U 1 U_1 U 1 B B B e e e
Proposition 10.27 constructs an automorphism of V ′ V' V ′ U 1 U_1 U 1 B B B s s s
Proposition 10.13. Let U 2 U_2 U 2 9 9 9 p 3 = 8388571 p_3 = 8388571 p 3 = 8388571 p 3 p_3 p 3
The space S 3 g e n S_3^{\mathrm{gen}} S 3 gen Proposition 10.7 has dimension 956 and u ( 1 ) S 3 g e n ⊂ S 3 g e n u(1)S_3^{\mathrm{gen}} \subset S_3^{\mathrm{gen}} u ( 1 ) S 3 gen ⊂ S 3 gen u ∈ U 2 u \in U_2 u ∈ U 2 Proposition 10.7 , this gives the finite mode closure y ( n ) S i ⊂ S i − n y(n)S_i \subset S_{i-n} y ( n ) S i ⊂ S i − n n ≥ 0 n \ge0 n ≥ 0 u ( n ) S i ⊂ S i + 1 − n u(n)S_i \subset S_{i+1-n} u ( n ) S i ⊂ S i + 1 − n n ≥ 1 n \ge1 n ≥ 1 y ∈ U 1 y \in U_1 y ∈ U 1 u ∈ U 2 u \in U_2 u ∈ U 2 i ∈ { 2 , 3 } i \in\{2,3\} i ∈ { 2 , 3 } S 2 = U 2 S_2 = U_2 S 2 = U 2 S 3 = S 3 g e n S_3 = S_3^{\mathrm{gen}} S 3 = S 3 gen [67 Proposition 4.1] .
Let R b ⊂ V 1 ′ R_b \subset V'_1 R b ⊂ V 1 ′ U 1 U_1 U 1 M M M R b R_b R b U 1 U_1 U 1 L ( − 1 ) L(-1) L ( − 1 ) u ( 0 ) u(0) u ( 0 ) u ( − 1 ) u(-1) u ( − 1 ) u ∈ U 2 u \in U_2 u ∈ U 2 Proposition 10.7 . Then dim M 2 = 1250 \dim M_2 = 1250 dim M 2 = 1250 dim M 3 = 10630 \dim M_3 = 10630 dim M 3 = 10630 U 2 + M 2 = V 2 ′ U_2 + M_2 = V'_2 U 2 + M 2 = V 2 ′ S 3 g e n + M 3 = V 3 ′ S_3^{\mathrm{gen}} + M_3 = V'_3 S 3 gen + M 3 = V 3 ′ M b M_b M b
The space S 4 g e n S_4^{\mathrm{gen}} S 4 gen y ( − 1 ) S 3 y(-1)S_3 y ( − 1 ) S 3 L ( − 1 ) S 3 L(-1)S_3 L ( − 1 ) S 3 y ( − 2 ) U 2 y(-2)U_2 y ( − 2 ) U 2 y ( − 3 ) U 1 y(-3)U_1 y ( − 3 ) U 1 u ( − 1 ) U 2 u(-1)U_2 u ( − 1 ) U 2 u ( − 2 ) U 1 u(-2)U_1 u ( − 2 ) U 1 u ( 0 ) S 3 u(0)S_3 u ( 0 ) S 3 y ∈ U 1 y \in U_1 y ∈ U 1 u = e u = e u = e u = x C u = x_C u = x C g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ q 11 / 3 q^{11/3} q 11/3
24 V ( 0 , 0 ) + 27 V ( 0 , 2 ) + 35 V ( 0 , 4 ) + 18 V ( 0 , 6 ) + 11 V ( 0 , 8 ) + 3 V ( 0 , 10 ) + V ( 0 , 12 ) + 28 V ( 1 , 1 ) + 43 V ( 1 , 3 ) + 37 V ( 1 , 5 ) + 24 V ( 1 , 7 ) + 9 V ( 1 , 9 ) + 2 V ( 1 , 11 ) + 17 V ( 2 , 0 ) + 39 V ( 2 , 2 ) + 31 V ( 2 , 4 ) + 28 V ( 2 , 6 ) + 10 V ( 2 , 8 ) + 4 V ( 2 , 10 ) + 15 V ( 3 , 1 ) + 23 V ( 3 , 3 ) + 16 V ( 3 , 5 ) + 10 V ( 3 , 7 ) + 2 V ( 3 , 9 ) + V ( 3 , 13 ) + 9 V ( 4 , 0 ) + 7 V ( 4 , 2 ) + 7 V ( 4 , 4 ) + 5 V ( 4 , 6 ) + 2 V ( 4 , 8 ) + V ( 5 , 1 ) + 2 V ( 5 , 3 ) + V ( 5 , 5 ) + V ( 5 , 7 ) + V ( 6 , 0 ) + V ( 6 , 2 ) . 24V(0,0) + 27V(0,2) + 35V(0,4) + 18V(0,6) + 11V(0,8) + 3V(0,10) + V(0,12)
+ 28V(1,1) + 43V(1,3) + 37V(1,5) + 24V(1,7) + 9V(1,9) + 2V(1,11) + 17V(2,0)
+ 39V(2,2) + 31V(2,4) + 28V(2,6) + 10V(2,8) + 4V(2,10) + 15V(3,1) + 23V(3,3)
+ 16V(3,5) + 10V(3,7) + 2V(3,9) + V(3,13) + 9V(4,0) + 7V(4,2) + 7V(4,4)
+ 5V(4,6) + 2V(4,8) + V(5,1) + 2V(5,3) + V(5,5) + V(5,7) + V(6,0)
+ V(6,2). 24 V ( 0 , 0 ) + 27 V ( 0 , 2 ) + 35 V ( 0 , 4 ) + 18 V ( 0 , 6 ) + 11 V ( 0 , 8 ) + 3 V ( 0 , 10 ) + V ( 0 , 12 ) + 28 V ( 1 , 1 ) + 43 V ( 1 , 3 ) + 37 V ( 1 , 5 ) + 24 V ( 1 , 7 ) + 9 V ( 1 , 9 ) + 2 V ( 1 , 11 ) + 17 V ( 2 , 0 ) + 39 V ( 2 , 2 ) + 31 V ( 2 , 4 ) + 28 V ( 2 , 6 ) + 10 V ( 2 , 8 ) + 4 V ( 2 , 10 ) + 15 V ( 3 , 1 ) + 23 V ( 3 , 3 ) + 16 V ( 3 , 5 ) + 10 V ( 3 , 7 ) + 2 V ( 3 , 9 ) + V ( 3 , 13 ) + 9 V ( 4 , 0 ) + 7 V ( 4 , 2 ) + 7 V ( 4 , 4 ) + 5 V ( 4 , 6 ) + 2 V ( 4 , 8 ) + V ( 5 , 1 ) + 2 V ( 5 , 3 ) + V ( 5 , 5 ) + V ( 5 , 7 ) + V ( 6 , 0 ) + V ( 6 , 2 ) . In particular it contains the g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ V ( 3 , 13 ) V(3,13) V ( 3 , 13 )
For y ∈ U 1 y \in U_1 y ∈ U 1 y ( 1 ) S 4 g e n ⊂ S 3 g e n y(1)S_4^{\mathrm{gen}} \subset S_3^{\mathrm{gen}} y ( 1 ) S 4 gen ⊂ S 3 gen y ( 2 ) S 4 g e n ⊂ U 2 y(2)S_4^{\mathrm{gen}} \subset U_2 y ( 2 ) S 4 gen ⊂ U 2 u u u ( 0 , 4 ) (0,4) ( 0 , 4 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 1 , 5 ) (1,5) ( 1 , 5 ) ( 1 , 7 ) (1,7) ( 1 , 7 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) ( 2 , 6 ) (2,6) ( 2 , 6 ) ( 3 , 3 ) (3,3) ( 3 , 3 ) u = e u = e u = e u = x C u = x_C u = x C u ( 1 ) S 4 g e n ⊂ S 4 g e n u(1)S_4^{\mathrm{gen}} \subset S_4^{\mathrm{gen}} u ( 1 ) S 4 gen ⊂ S 4 gen u ( 2 ) S 4 g e n ⊂ S 3 g e n u(2)S_4^{\mathrm{gen}} \subset S_3^{\mathrm{gen}} u ( 2 ) S 4 gen ⊂ S 3 gen u ( 3 ) S 4 g e n ⊂ U 2 u(3)S_4^{\mathrm{gen}} \subset U_2 u ( 3 ) S 4 gen ⊂ U 2
The subspace N 4 ′ N'_4 N 4 ′ L ( − 1 ) S 3 L(-1)S_3 L ( − 1 ) S 3 y ( − 1 ) S 3 y(-1)S_3 y ( − 1 ) S 3 y ( − 2 ) U 2 y(-2)U_2 y ( − 2 ) U 2 y ( − 3 ) U 1 y(-3)U_1 y ( − 3 ) U 1 u ( − 1 ) U 2 u(-1)U_2 u ( − 1 ) U 2 u u u g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ e ( − 1 ) e e(-1)e e ( − 1 ) e x C ( − 1 ) x C x_C(-1)x_C x C ( − 1 ) x C e ( − 1 ) x C e(-1)x_C e ( − 1 ) x C C 2 C_2 C 2 Theorem 10.21 .
Remark 10.14 The ranks modulo p 3 p_3 p 3 K \mathbb{K} K p 3 p_3 p 3 K \mathbb{K} K V ′ = S ⊕ M b V' = S \oplus M_b V ′ = S ⊕ M b Proposition 4.1 : by unitarity, $V' = S ⊕ S ⊥ S \oplus S^{\perp} S ⊕ S ⊥ M M M S S S S 1 = U 1 S_1 = U_1 S 1 = U 1 H χ ( a ( n ) r , s ) H_{\chi}(a_{(n)}r,s) H χ ( a ( n ) r , s ) r ∈ R b r \in R_b r ∈ R b S 1 S_1 S 1 K K K L ( 3 , 3 ) ⊗ L ( 13 , 13 ) L(3,3) \otimes L(13,13) L ( 3 , 3 ) ⊗ L ( 13 , 13 ) 3 4 + 13 4 = 4 \frac{3}{4} + \frac{13}{4} = 4 4 3 + 4 13 = 4 Proposition 11.4 : an explicit weight-four vector is shown to be affine-primary. Thus no inference from the modular module decomposition to characteristic-zero occurrence or affine primarity is needed.
Case (II): strong generation and C2-cofiniteness
In this subsection U 2 U_2 U 2 9 9 9 σ \sigma σ ( A ) 2 ⊂ U 2 (A)_2 \subset U_2 ( A ) 2 ⊂ U 2 x ( − 1 ) y x(-1)y x ( − 1 ) y x ( − 2 ) 1 x(-2)1 x ( − 2 ) 1 ( x , y ∈ U 1 ) (x,y \in U_1) ( x , y ∈ U 1 )
N 3 = L ( − 1 ) U 2 + U 1 ( − 1 ) U 2 ⊂ V 3 ′ , S 3 = N 3 + ( U 2 ) ( 0 ) U 2 , N_3 = L(-1)U_2 + U_1(-1)U_2 \subset V'_3,\qquad S_3 = N_3 + (U_2)_{(0)}U_2, N 3 = L ( − 1 ) U 2 + U 1 ( − 1 ) U 2 ⊂ V 3 ′ , S 3 = N 3 + ( U 2 ) ( 0 ) U 2 , N 4 ′ = L ( − 1 ) S 3 + U 1 ( − 1 ) S 3 + U 1 ( − 2 ) U 2 + U 1 ( − 3 ) U 1 + ∑ τ M τ ( − 1 ) U 2 ⊂ V 4 ′ . N'_4 = L(-1)S_3 + U_1(-1)S_3 + U_1(-2)U_2 + U_1(-3)U_1 + \sum_{\tau} M_{\tau}(-1)U_2 \subset V'_4. N 4 ′ = L ( − 1 ) S 3 + U 1 ( − 1 ) S 3 + U 1 ( − 2 ) U 2 + U 1 ( − 3 ) U 1 + τ ∑ M τ ( − 1 ) U 2 ⊂ V 4 ′ . where M τ ⊂ U 2 M_{\tau} \subset U_2 M τ ⊂ U 2 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ τ \tau τ Z ( n ) W Z(n)W Z ( n ) W z ( n ) w z(n)w z ( n ) w K K K g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ S S S x ( − 2 ) y x(-2)y x ( − 2 ) y x ( − 3 ) 1 x(-3)1 x ( − 3 ) 1 N 3 N_3 N 3
Lemma 10.15 For each of the four choices of U 2 U_2 U 2 dim N 3 = 725 \dim N_3 = 725 dim N 3 = 725
Proof. We use the positive Hermitian form of Theorem 10.10 . By part 2 of that theorem, its restriction to the vertex subalgebra generated by U 1 U_1 U 1 A ≅ L s l 2 ( 3 , 0 ) ⊗ L s l 2 ( 13 , 0 ) A \cong L_{\mathfrak{sl}_2}(3,0) \otimes L_{\mathfrak{sl}_2}(13,0) A ≅ L sl 2 ( 3 , 0 ) ⊗ L sl 2 ( 13 , 0 ) S S S A A A [22 Theorem 3.7 p. 7] . The regularity of A A A L A ( − 1 ) L_A(-1) L A ( − 1 ) L C ( − 1 ) L_C(-1) L C ( − 1 ) L ( − 1 ) = L A ( − 1 ) + L C ( − 1 ) L(-1) = L_A(-1) + L_C(-1) L ( − 1 ) = L A ( − 1 ) + L C ( − 1 ) 6 + 6 2 + ( 8 3 ) = 98 6+6^2+\binom{8}{3}=98 6 + 6 2 + ( 3 8 ) = 98
The two vectors of C 2 C_2 C 2 6 ⋅ 2 6 \cdot2 6 ⋅ 2 L C ( − 1 ) C 2 L_C(-1)C_2 L C ( − 1 ) C 2 5 + 4 + 8 + 12 + 16 + 9 + 21 + 16 = 91 5+4+8+12+16+9+21+16=91 5 + 4 + 8 + 12 + 16 + 9 + 21 + 16 = 91 ( a , b ) (a,b) ( a , b ) ( g ′ ⊕ g ′ ′ ) ⊗ V ( a , b ) (\mathfrak{g}' \oplus\mathfrak{g}'') \otimes V(a,b) ( g ′ ⊕ g ′′ ) ⊗ V ( a , b ) ( 3 , 3 ) (3,3) ( 3 , 3 ) e ′ ( − 1 ) v e'(-1)v e ′ ( − 1 ) v V ( 5 , 3 ) V(5,3) V ( 5 , 3 ) 4 − a 4-a 4 − a 14 − b 14-b 14 − b 6 ⋅ 91 − 24 6 \cdot91 - 24 6 ⋅ 91 − 24
The Sugawara formula gives L A ( − 1 ) M τ ⊂ U 1 ( − 1 ) M τ L_A(-1)M_{\tau} \subset U_1(-1)M_{\tau} L A ( − 1 ) M τ ⊂ U 1 ( − 1 ) M τ L C ( − 1 ) v L_C(-1)v L C ( − 1 ) v v ∈ M τ v \in M_{\tau} v ∈ M τ v v v h C = 2 − h a b > 0 h_C = 2-h_{ab} > 0 h C = 2 − h ab > 0 ( L C ( n ) ) ∗ = L C ( − n ) (L_C(n))^* = L_C(-n) ( L C ( n ) ) ∗ = L C ( − n )
∥ L C ( − 1 ) v ∥ H x 2 = ∥ L C ( 1 ) v ∥ H x 2 + 2 h C ∥ v ∥ H x 2 . \left\|L_C(-1)v\right\|_{H_x}^2 = \left\|L_C(1)v\right\|_{H_x}^2 + 2h_C\left\|v\right\|_{H_x}^2. ∥ L C ( − 1 ) v ∥ H x 2 = ∥ L C ( 1 ) v ∥ H x 2 + 2 h C ∥ v ∥ H x 2 . Consequently L C ( − 1 ) L_C(-1) L C ( − 1 ) C 2 C_2 C 2 h C = 2 h_C=2 h C = 2 L A ( 0 ) L_A(0) L A ( 0 ) A 3 A_3 A 3 U 1 ( − 1 ) C 2 U_1(-1)C_2 U 1 ( − 1 ) C 2 L C ( − 1 ) C 2 L_C(-1)C_2 L C ( − 1 ) C 2 0 0 0 N 3 N_3 N 3 dim N 3 = 98 + 14 + ( 6 ⋅ 91 − 24 ) + 91 = 725 \dim N_3 = 98 + 14 + (6 \cdot91 - 24) + 91 = 725 dim N 3 = 98 + 14 + ( 6 ⋅ 91 − 24 ) + 91 = 725
Definition 10.16 (weight-four certificate condition) Fix the specified bases of U 1 U_1 U 1 U 2 U_2 U 2 S 3 S_3 S 3 e e e x C x_C x C B 4 B_4 B 4 N 4 ′ N'_4 N 4 ′
the vectors in B 4 B_4 B 4 p 3 p_3 p 3
every u ( 0 ) s u_{(0)}s u ( 0 ) s u u u s s s S 3 S_3 S 3 e ( − 1 ) w e_{(-1)}w e ( − 1 ) w ( x C ) ( − 1 ) w (x_C)_{(-1)}w ( x C ) ( − 1 ) w w w w U 2 U_2 U 2 K \mathbb{K} K B 4 B_4 B 4
every word of the five defining families of N 4 ′ N'_4 N 4 ′
The membership assertions mean the exact coordinate identities B c = v Bc=v B c = v
Proposition 10.17 Condition (F4) holds for the first subspace attached to the subalgebra ϑ \vartheta ϑ Table 3 . The characteristic-zero identities and dimensions are transported to the other three subspaces by coefficient conjugation and the Galois automorphism σ \sigma σ
Proof : We use the exact lattice model over K = Q ( 13 , − 3 ) \mathbb{K} = \mathbb{Q}(\sqrt{13},\sqrt{-3}) K = Q ( 13 , − 3 ) U 1 U_1 U 1 U 2 U_2 U 2 Theorem 10.19 , does not use any weight-four assertion. The verification of weight four is independent of any identification with an externally constructed algebra.
All mode evaluations below have homogeneous inputs and outputs of weights at most four, so the weight-window argument in § 10 applies. There are 956 basis vectors in S 3 S_3 S 3 U 2 U_2 U 2 N 4 ′ N'_4 N 4 ′
956 + 6 ⋅ 956 + 6 ⋅ 120 + 6 2 + 91 ⋅ 120 = 18368 956 + 6 \cdot956 + 6 \cdot120 + 6^2 + 91 \cdot120 = 18368 956 + 6 ⋅ 956 + 6 ⋅ 120 + 6 2 + 91 ⋅ 120 = 18368 labelled input words. All of them are evaluated from the lattice mode formulas, including the zero products. Among the nonzero values, the calculation selects 6010 words with independent reductions at the chosen prime above p 3 = 8388571 p_3 = 8388571 p 3 = 8388571 13 ↦ 1649360 \sqrt{13} \mapsto1649360 13 ↦ 1649360 − 3 ↦ 2036966 \sqrt{-3} \mapsto2036966 − 3 ↦ 2036966 p 3 p_3 p 3 B 4 B_4 B 4
The remaining membership assertions have 10 ⋅ 956 + 2 ⋅ 120 = 9800 10 \cdot956 + 2 \cdot120 = 9800 10 ⋅ 956 + 2 ⋅ 120 = 9800 r r r 4 r 4r 4 r 4 r × 4 r 4r \times4r 4 r × 4 r p 3 p_3 p 3 Q \mathbb{Q} Q K \mathbb{K} K
The complete verification covers all 131 Cartan blocks carrying nonzero input vectors. The families contain 18194 nonzero words and 174 zero words, and the additional products contain 9088 nonzero values and 712 zero values. All 886 zero inputs are independently checked from the mode formulas or from the Cartan energy bound in the lattice. The complete verification reports no failure, and the exact coordinate data and verification results are included in the supplementary material. Consequently all three requirements in Definition 10.16 hold.
Finally, coefficient conjugation and σ \sigma σ U 2 U_2 U 2 K K K Theorem 10.24 .
Theorem 10.19 The following statements hold over K K K
dim S 3 = 956 \dim S_3=956 dim S 3 = 956 S 3 S_3 S 3 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′
L ( 1 ) S 3 ⊂ U 2 L(1)S_3 \subset U_2 L ( 1 ) S 3 ⊂ U 2 L ( 2 ) S 3 ⊂ U 1 L(2)S_3 \subset U_1 L ( 2 ) S 3 ⊂ U 1 y ( 1 ) S 3 ⊂ U 2 y(1)S_3 \subset U_2 y ( 1 ) S 3 ⊂ U 2 y ( 2 ) S 3 ⊂ U 1 y(2)S_3 \subset U_1 y ( 2 ) S 3 ⊂ U 1 y ∈ U 1 y \in U_1 y ∈ U 1
u ( 0 ) S 3 ⊂ N 4 ′ u(0)S_3 \subset N'_4 u ( 0 ) S 3 ⊂ N 4 ′ u ∈ U 2 u \in U_2 u ∈ U 2 e ( − 1 ) U 2 + x C ( − 1 ) U 2 ⊂ N 4 ′ e(-1)U_2+x_C(-1)U_2 \subset N'_4 e ( − 1 ) U 2 + x C ( − 1 ) U 2 ⊂ N 4 ′ ( U 2 ) ( 0 ) S 3 + ( U 2 ) ( − 1 ) U 2 ⊂ N 4 ′ (U_2)_{(0)}S_3+(U_2)_{(-1)}U_2 \subset N'_4 ( U 2 ) ( 0 ) S 3 + ( U 2 ) ( − 1 ) U 2 ⊂ N 4 ′
dim N 4 ′ = 6010 \dim N'_4=6010 dim N 4 ′ = 6010 N 4 ′ = L ( − 1 ) S 3 + U 1 ( − 1 ) S 3 + U 1 ( − 2 ) U 2 + U 1 ( − 3 ) U 1 + ( U 2 ) ( − 1 ) U 2 + ( U 2 ) ( 0 ) S 3 N'_4=L(-1)S_3+U_1(-1)S_3+U_1(-2)U_2+U_1(-3)U_1+(U_2)_{(-1)}U_2+(U_2)_{(0)}S_3 N 4 ′ = L ( − 1 ) S 3 + U 1 ( − 1 ) S 3 + U 1 ( − 2 ) U 2 + U 1 ( − 3 ) U 1 + ( U 2 ) ( − 1 ) U 2 + ( U 2 ) ( 0 ) S 3
Proof: All vectors are represented as K K K § 10 . Membership of a vector v v v b 1 , … , b r b_1,\ldots,b_r b 1 , … , b r v v v g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ K ≅ Q 4 K \cong\mathbb{Q}^4 K ≅ Q 4 b i , 13 b i , − 3 b i , − 39 b i b_i,\sqrt{13}b_i,\sqrt{-3}b_i,\sqrt{-39}b_i b i , 13 b i , − 3 b i , − 39 b i 4 r 4r 4 r 4 r 4r 4 r p 3 = 8388571 p_3=8388571 p 3 = 8388571 Q \mathbb{Q} Q Q \mathbb{Q} Q B c = v Bc=v B c = v
We choose 956 vectors among the generating words x ( − 1 ) w x(-1)w x ( − 1 ) w L ( − 1 ) w L(-1)w L ( − 1 ) w x ( − 2 ) y x(-2)y x ( − 2 ) y x ( − 3 ) 1 x(-3)1 x ( − 3 ) 1 x , y x,y x , y U 1 U_1 U 1 w w w U 2 U_2 U 2 u ( 0 ) w u_{(0)}w u ( 0 ) w u = e u=e u = e x C x_C x C e ( 0 ) ′ , f ( 0 ) ′ , e ( 0 ) ′ ′ , f ( 0 ) ′ ′ e'_{(0)},f'_{(0)},e''_{(0)},f''_{(0)} e ( 0 ) ′ , f ( 0 ) ′ , e ( 0 ) ′′ , f ( 0 ) ′′ p 3 p_3 p 3 K \mathbb{K} K Σ \Sigma Σ g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ N 3 N_3 N 3 u ( 0 ) U 2 u_{(0)}U_2 u ( 0 ) U 2 u u u [ y ( 0 ) , u ( 0 ) ] = ( y ( 0 ) u ) ( 0 ) [y_{(0)},u_{(0)}]=(y_{(0)}u)_{(0)} [ y ( 0 ) , u ( 0 ) ] = ( y ( 0 ) u ) ( 0 ) M τ ( 0 ) U 2 ⊂ Σ M_{\tau(0)}U_2 \subset\Sigma M τ ( 0 ) U 2 ⊂ Σ u ∈ ( A ) 2 u \in(A)_2 u ∈ ( A ) 2 u ( 0 ) u_{(0)} u ( 0 ) u ( 0 ) U 2 ⊂ U 1 ( − 1 ) U 2 + U 1 ( − 2 ) U 1 + L ( − 1 ) U 2 ⊂ N 3 u_{(0)}U_2 \subset U_1(-1)U_2+U_1(-2)U_1+L(-1)U_2 \subset N_3 u ( 0 ) U 2 ⊂ U 1 ( − 1 ) U 2 + U 1 ( − 2 ) U 1 + L ( − 1 ) U 2 ⊂ N 3 Σ = S 3 \Sigma=S_3 Σ = S 3
These inclusions also follow without a matrix computation. Put T = L ( − 1 ) T = L(-1) T = L ( − 1 ) Theorem 10.9 , applied to the conformal vector ω ∈ U 2 \omega\in U_2 ω ∈ U 2 L ( 1 ) U 2 ⊂ U 1 L(1)U_2 \subset U_1 L ( 1 ) U 2 ⊂ U 1 L ( 2 ) U 2 ⊂ C 1 L(2)U_2 \subset C1 L ( 2 ) U 2 ⊂ C 1 [ L ( j ) , T ] = ( j + 1 ) L ( j − 1 ) [L(j),T] = (j + 1)L(j - 1) [ L ( j ) , T ] = ( j + 1 ) L ( j − 1 ) N 3 = T U 2 + U 1 ( − 1 ) U 2 N_3 = TU_2 + U_1(-1)U_2 N 3 = T U 2 + U 1 ( − 1 ) U 2 y , z ∈ U 1 y,z \in U_1 y , z ∈ U 1 u ∈ U 2 u \in U_2 u ∈ U 2 [ L ( j ) , y ( − 1 ) ] = y ( j − 1 ) [L(j),y(-1)] = y(j-1) [ L ( j ) , y ( − 1 )] = y ( j − 1 ) [ z ( j ) , T ] = j z ( j − 1 ) [z(j),T] = jz(j-1) [ z ( j ) , T ] = j z ( j − 1 ) [ z ( j ) , y ( − 1 ) ] = [ z , y ] ( j − 1 ) + δ j , 1 κ ( z , y ) Id [z(j),y(-1)] = [z,y](j-1) + \delta_{j,1}\kappa(z,y)\operatorname{Id} [ z ( j ) , y ( − 1 )] = [ z , y ] ( j − 1 ) + δ j , 1 κ ( z , y ) Id j = 1 , 2 j = 1,2 j = 1 , 2 z ( 1 ) y = κ ( z , y ) 1 z(1)y = \kappa(z,y)1 z ( 1 ) y = κ ( z , y ) 1
For u , v ∈ U 2 u,v \in U_2 u , v ∈ U 2
L ( 1 ) ( u ( 0 ) v ) = 2 u ( 1 ) v + ( L ( 1 ) u ) ( 0 ) v + u ( 0 ) L ( 1 ) v , L ( 2 ) ( u ( 0 ) v ) = 3 u ( 2 ) v + 3 ( L ( 1 ) u ) ( 1 ) v . \begin{aligned}
L(1)(u(0)v) &= 2u(1)v + (L(1)u)(0)v + u(0)L(1)v,\\
L(2)(u(0)v) &= 3u(2)v + 3(L(1)u)(1)v.
\end{aligned} L ( 1 ) ( u ( 0 ) v ) L ( 2 ) ( u ( 0 ) v ) = 2 u ( 1 ) v + ( L ( 1 ) u ) ( 0 ) v + u ( 0 ) L ( 1 ) v , = 3 u ( 2 ) v + 3 ( L ( 1 ) u ) ( 1 ) v . The terms involving L ( 2 ) u L(2)u L ( 2 ) u L ( 2 ) v L(2)v L ( 2 ) v u ( 0 ) 1 = 0 u(0)1 = 0 u ( 0 ) 1 = 0 y ∈ U 1 y \in U_1 y ∈ U 1
y ( 1 ) ( u ( 0 ) v ) = ( y ( 0 ) u ) ( 1 ) v + ( y ( 1 ) u ) ( 0 ) v + u ( 0 ) y ( 1 ) v , y ( 2 ) ( u ( 0 ) v ) = ( y ( 0 ) u ) ( 2 ) v + 2 ( y ( 1 ) u ) ( 1 ) v . \begin{aligned}
y(1)(u(0)v) &= (y(0)u)(1)v + (y(1)u)(0)v + u(0)y(1)v,\\
y(2)(u(0)v) &= (y(0)u)(2)v + 2(y(1)u)(1)v.
\end{aligned} y ( 1 ) ( u ( 0 ) v ) y ( 2 ) ( u ( 0 ) v ) = ( y ( 0 ) u ) ( 1 ) v + ( y ( 1 ) u ) ( 0 ) v + u ( 0 ) y ( 1 ) v , = ( y ( 0 ) u ) ( 2 ) v + 2 ( y ( 1 ) u ) ( 1 ) v . All the terms belong to the spaces asserted in part 2, by weight-two closure and skew symmetry. This proves the inclusions on S 3 = N 3 + ( U 2 ) ( 0 ) U 2 S_3 = N_3 + (U_2)(0)U_2 S 3 = N 3 + ( U 2 ) ( 0 ) U 2
For the ten vectors u u u s s s S 3 S_3 S 3 u ∈ { e , x C } u \in\{e,x_C\} u ∈ { e , x C } w w w U 2 U_2 U 2 u ( 0 ) s u(0)s u ( 0 ) s u ( − 1 ) w u(-1)w u ( − 1 ) w N 4 ′ N'_4 N 4 ′ p 3 p_3 p 3 Proposition 10.17 . Its complete verification covers 9088 nonzero tests and 712 zero tests, with no failure. The extension to all u ∈ U 2 u \in U_2 u ∈ U 2 N 4 ′ N'_4 N 4 ′ g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ S 3 S_3 S 3 u ∈ ( A ) 2 u \in(A)_2 u ∈ ( A ) 2 u ( 0 ) u(0) u ( 0 ) u ( − 1 ) u(-1) u ( − 1 ) S 3 S_3 S 3 U 2 U_2 U 2 N 4 ′ N'_4 N 4 ′ M τ ( − 1 ) U 2 ⊂ N 4 ′ M_\tau(-1)U_2 \subset N'_4 M τ ( − 1 ) U 2 ⊂ N 4 ′
Proposition 10.17 gives membership for all 18368 words of the five defining families and independence of the 6010 pivots, hence dim N 4 ′ = 6010 \dim N'_4 = 6010 dim N 4 ′ = 6010
The next lemma gives the passage from these finite identities to all conformal weights. It is a variant of [67 Lemma 4.3] , where N ( G ) N(G) N ( G ) A = G A = G A = G
Lemma 10.20 (triangular strong generation). Let W = ⨁ n ≥ 0 W n W = \bigoplus_{n\geq0} W_n W = ⨁ n ≥ 0 W n wt ( a ( n ) b ) = wt a + wt b − n − 1 \operatorname{wt}(a_{(n)}b) = \operatorname{wt}a + \operatorname{wt}b - n - 1 wt ( a ( n ) b ) = wt a + wt b − n − 1 W 0 = C 1 W_0 = \mathbb{C}1 W 0 = C 1 A ⊂ G A \subset G A ⊂ G N ( G ) N(G) N ( G ) 1 1 1 g ( − n 1 ) 1 ⋯ g ( − n k ) k 1 g^1_{(-n_1)}\cdots g^k_{(-n_k)}1 g ( − n 1 ) 1 ⋯ g ( − n k ) k 1 g i ∈ G g^i \in G g i ∈ G n i ≥ 1 n_i \geq1 n i ≥ 1
a ( n ) g ∈ N ( G ) a_{(n)}g \in N(G) a ( n ) g ∈ N ( G ) a ∈ A a \in A a ∈ A g ∈ G g \in G g ∈ G n ≥ 0 n \geq0 n ≥ 0
every g ∈ G ∖ A g \in G \setminus A g ∈ G ∖ A d d d N ( G < d ) N(G_{<d}) N ( G < d ) a ( 0 ) b a_{(0)}b a ( 0 ) b a ∈ A a \in A a ∈ A wt a ≥ 2 \operatorname{wt}a \geq2 wt a ≥ 2 b ∈ N ( G < d ) b \in N(G_{<d}) b ∈ N ( G < d ) d − wt a + 1 d - \operatorname{wt}a + 1 d − wt a + 1 G < d G_{<d} G < d G G G d d d
Then N ( G ) N(G) N ( G ) W W W G G G
Proof. All vectors are homogeneous, and a monomial is a word g ( − n 1 ) 1 ⋯ g ( − n k ) k 1 g^1_{(-n_1)}\cdots g^k_{(-n_k)}1 g ( − n 1 ) 1 ⋯ g ( − n k ) k 1 k k k T ≥ 0 T \geq0 T ≥ 0 P ( T ) P(T) P ( T ) g ( n ) w ∈ N ( G ) g_{(n)}w \in N(G) g ( n ) w ∈ N ( G ) g ∈ G g \in G g ∈ G n ≥ 0 n \ge0 n ≥ 0 w ∈ N ( G ) w \in N(G) w ∈ N ( G ) wt g + wt w ≤ T \operatorname{wt} g+\operatorname{wt} w \le T wt g + wt w ≤ T
Q ( T ) Q(T) Q ( T ) z ( s ) w ∈ N ( G ) z_{(s)}w \in N(G) z ( s ) w ∈ N ( G ) z , w ∈ N ( G ) z,w \in N(G) z , w ∈ N ( G ) s ∈ Z s \in\mathbb{Z} s ∈ Z wt z + wt w ≤ T \operatorname{wt} z+\operatorname{wt} w \le T wt z + wt w ≤ T
and P < d ( T ) P_{<d}(T) P < d ( T ) Q < d ( T ) Q_{<d}(T) Q < d ( T ) g ∈ G < d g \in G_{<d} g ∈ G < d z ∈ N ( G < d ) z \in N(G_{<d}) z ∈ N ( G < d )
Step 1: if P ( T ) P(T) P ( T ) G ′ ∈ { G , G < d } G' \in\{G,G_{<d}\} G ′ ∈ { G , G < d } Q ( T ) Q(T) Q ( T ) z ∈ N ( G ′ ) z \in N(G') z ∈ N ( G ′ ) z z z z = 1 z=1 z = 1 z = k ( − p ) z ′ z=k_{(-p)}z' z = k ( − p ) z ′
z ( s ) v = ∑ i ≥ 0 ( p + i − 1 i ) ( k ( − p − i ) ( z ( s + i ) ′ v ) − ( − 1 ) p z ( s − p − i ) ′ ( k ( i ) v ) ) . z_{(s)}v=\sum_{i\geq0}\binom{p+i-1}{i}\left(k_{(-p-i)}\left(z'_{(s+i)}v\right)-(-1)^p z'_{(s-p-i)}\left(k_{(i)}v\right)\right). z ( s ) v = i ≥ 0 ∑ ( i p + i − 1 ) ( k ( − p − i ) ( z ( s + i ) ′ v ) − ( − 1 ) p z ( s − p − i ) ′ ( k ( i ) v ) ) . a finite sum. Here z ( s + i ) ′ v ∈ N ( G ) z'_{(s+i)}v \in N(G) z ( s + i ) ′ v ∈ N ( G ) z ′ z' z ′ k ( − p − i ) k_{(-p-i)} k ( − p − i ) N ( G ) N(G) N ( G ) k ( i ) v ∈ N ( G ) k_{(i)}v \in N(G) k ( i ) v ∈ N ( G ) P ( T ) P(T) P ( T ) z ( s − p − i ) ′ ( k ( i ) v ) ∈ N ( G ) z'_{(s-p-i)}\left(k_{(i)}v\right) \in N(G) z ( s − p − i ) ′ ( k ( i ) v ) ∈ N ( G ) z ′ z' z ′ wt z ′ + wt ( k ( i ) v ) = wt z + wt v − p − i ≤ T \operatorname{wt} z'+\operatorname{wt}\left(k_{(i)}v\right)=\operatorname{wt} z+\operatorname{wt} v-p-i \le T wt z ′ + wt ( k ( i ) v ) = wt z + wt v − p − i ≤ T
Step 2: Q ( T − 1 ) Q(T-1) Q ( T − 1 ) P ( T ) P(T) P ( T ) g ∈ A g \in A g ∈ A w = h ( − m ) w ′ w=h_{(-m)}w' w = h ( − m ) w ′
g ( n ) h ( − m ) w ′ = h ( − m ) g ( n ) w ′ + ∑ j ≥ 0 ( n j ) ( g ( j ) h ) ( n − m − j ) w ′ . g_{(n)}h_{(-m)}w'=h_{(-m)}g_{(n)}w'+\sum_{j\geq0}\binom{n}{j}(g_{(j)}h)_{(n-m-j)}w'. g ( n ) h ( − m ) w ′ = h ( − m ) g ( n ) w ′ + j ≥ 0 ∑ ( j n ) ( g ( j ) h ) ( n − m − j ) w ′ . The first term lies in N ( G ) N(G) N ( G ) P ( T − 1 ) P(T-1) P ( T − 1 ) Q ( T − 1 ) Q(T-1) Q ( T − 1 ) z = g ( j ) h ∈ N ( G ) z=g_{(j)}h \in N(G) z = g ( j ) h ∈ N ( G ) wt z + wt w ′ ≤ T − m − j ≤ T − 1 \operatorname{wt} z+\operatorname{wt} w' \le T-m-j \le T-1 wt z + wt w ′ ≤ T − m − j ≤ T − 1 z ( n − m − j ) w ′ ∈ N ( G ) z_{(n-m-j)}w' \in N(G) z ( n − m − j ) w ′ ∈ N ( G ) Q ( T − 1 ) Q(T-1) Q ( T − 1 ) n − m − j n-m-j n − m − j
Step 3: Q ( T − 1 ) Q(T-1) Q ( T − 1 ) P ( T ) P(T) P ( T ) A A A G < d G_{<d} G < d P ( T ) P(T) P ( T ) G ∖ A G\setminus A G ∖ A d d d Q < d ( T ) Q_{<d}(T) Q < d ( T ) g = ν + ∑ a ( 0 ) b g=\nu+\sum a_{(0)}b g = ν + ∑ a ( 0 ) b ν , b ∈ N ( G < d ) \nu,b\in N(G_{<d}) ν , b ∈ N ( G < d ) b b b a ( 0 ) b = 0 a_{(0)}b=0 a ( 0 ) b = 0 wt b ≥ 1 \operatorname{wt} b\ge1 wt b ≥ 1 wt a ≤ d \operatorname{wt} a\le d wt a ≤ d ν ( n ) w ∈ N ( G ) \nu_{(n)}w\in N(G) ν ( n ) w ∈ N ( G ) Q < d ( T ) Q_{<d}(T) Q < d ( T )
( a ( 0 ) b ) ( n ) w = a ( 0 ) ( b ( n ) w ) − b ( n ) ( a ( 0 ) w ) , (a_{(0)}b)_{(n)}w=a_{(0)}(b_{(n)}w)-b_{(n)}(a_{(0)}w), ( a ( 0 ) b ) ( n ) w = a ( 0 ) ( b ( n ) w ) − b ( n ) ( a ( 0 ) w ) , where b ( n ) w ∈ N ( G ) b_{(n)}w\in N(G) b ( n ) w ∈ N ( G ) Q < d ( T ) Q_{<d}(T) Q < d ( T ) a ( 0 ) ( b ( n ) w ) ∈ N ( G ) a_{(0)}(b_{(n)}w)\in N(G) a ( 0 ) ( b ( n ) w ) ∈ N ( G ) a ( 0 ) w ∈ N ( G ) a_{(0)}w\in N(G) a ( 0 ) w ∈ N ( G ) P ( T ) P(T) P ( T ) A A A b ( n ) ( a ( 0 ) w ) ∈ N ( G ) b_{(n)}(a_{(0)}w)\in N(G) b ( n ) ( a ( 0 ) w ) ∈ N ( G ) Q < d ( T ) Q_{<d}(T) Q < d ( T ) wt b + wt ( a ( 0 ) w ) = wt g + wt w ≤ T \operatorname{wt} b+\operatorname{wt}(a_{(0)}w)=\operatorname{wt} g+\operatorname{wt} w\le T wt b + wt ( a ( 0 ) w ) = wt g + wt w ≤ T wt a + wt ( b ( n ) w ) = wt g + wt w − n ≤ T \operatorname{wt} a+\operatorname{wt}(b_{(n)}w)=\operatorname{wt} g+\operatorname{wt} w-n\le T wt a + wt ( b ( n ) w ) = wt g + wt w − n ≤ T
Step 4. Q ( 0 ) Q(0) Q ( 0 ) N ( G ) 0 = C 1 N(G)_0=\mathbb{C}1 N ( G ) 0 = C 1 Q ( T − 1 ) Q(T-1) Q ( T − 1 ) d d d P ( T ) P(T) P ( T ) Q ( T ) Q(T) Q ( T ) N ( G ) ( s ) N ( G ) ⊂ N ( G ) N(G)_{(s)}N(G)\subset N(G) N ( G ) ( s ) N ( G ) ⊂ N ( G ) s s s 1 ∈ N ( G ) 1\in N(G) 1 ∈ N ( G ) N ( G ) N(G) N ( G ) G G G
Theorem 10.21 Let S ⊂ V ′ S\subset V' S ⊂ V ′ U 1 ⊕ U 2 U_1\oplus U_2 U 1 ⊕ U 2 U 2 U_2 U 2 Theorem 10.5 .
S S S U 1 ⊕ U 2 ⊕ D 3 U_1\oplus U_2\oplus D_3 U 1 ⊕ U 2 ⊕ D 3 D 3 D_3 D 3 N 3 N_3 N 3 S 3 S_3 S 3 6 + 93 6+93 6 + 93 231 231 231
S 1 = U 1 S_1=U_1 S 1 = U 1 S 2 = U 2 S_2=U_2 S 2 = U 2 dim S 3 = 956 \dim S_3=956 dim S 3 = 956 dim S 4 = 6010 \dim S_4=6010 dim S 4 = 6010 S S S X 0 ( 0 ) X_0^{(0)} X 0 ( 0 ) § 3 up to q 11 / 3 q^{11/3} q 11/3
S S S C 2 C_2 C 2
Proof: 1–2. Apply Lemma 10.20 in W = S W=S W = S A A A U 1 ⊕ U 2 U_1\oplus U_2 U 1 ⊕ U 2 G G G D 3 D_3 D 3 u ( 0 ) w u_{(0)}w u ( 0 ) w N 3 N_3 N 3 w , u ∈ U 2 w,u\in U_2 w , u ∈ U 2 a ∈ U 1 a\in U_1 a ∈ U 1 g ∈ U 1 ⊕ U 2 g\in U_1\oplus U_2 g ∈ U 1 ⊕ U 2 U 2 U_2 U 2 ( A 2 ) (A_2) ( A 2 ) y ( n ) e = y ( n ) x C = 0 y_{(n)}e=y_{(n)}x_C=0 y ( n ) e = y ( n ) x C = 0 y ∈ U 1 y\in U_1 y ∈ U 1 n ≥ 0 n\ge0 n ≥ 0 M r M_r M r A A A a , g ∈ U 2 a,g\in U_2 a , g ∈ U 2 a ( 0 ) g ∈ S 3 = N 3 + D 3 ⊂ N ( G ) a_{(0)}g\in S_3=N_3+D_3\subset N(G) a ( 0 ) g ∈ S 3 = N 3 + D 3 ⊂ N ( G ) a ( 1 ) g ∈ U 2 a_{(1)}g\in U_2 a ( 1 ) g ∈ U 2 a ( 2 ) g ∈ U 1 a_{(2)}g\in U_1 a ( 2 ) g ∈ U 1 Theorem 10.9 ). For a ∈ U 2 a\in U_2 a ∈ U 2 g ∈ U 1 g\in U_1 g ∈ U 1 a ∈ U 1 a \in U_1 a ∈ U 1 Theorem 10.19 and the stability of S 3 S_3 S 3 g ∈ D 3 g \in D_3 g ∈ D 3 a ∈ U 2 a \in U_2 a ∈ U 2 a ( 0 ) g ∈ N 4 ′ ⊂ N ( G ) a_{(0)}g \in N'_4 \subset N(G) a ( 0 ) g ∈ N 4 ′ ⊂ N ( G ) Theorem 10.19 , every term of N 4 ′ N'_4 N 4 ′ a ( n ) g a_{(n)}g a ( n ) g n = 1 , 2 , 3 n=1,2,3 n = 1 , 2 , 3 [ L ( 1 ) , a ( n ) ] = ( 2 − n ) a ( n + 1 ) [L(1),a_{(n)}]=(2-n)a_{(n+1)} [ L ( 1 ) , a ( n ) ] = ( 2 − n ) a ( n + 1 ) [ L ( 2 ) , a ( 1 ) ] = 2 a ( 3 ) [L(2),a_{(1)}]=2a_{(3)} [ L ( 2 ) , a ( 1 ) ] = 2 a ( 3 )
2 a ( 1 ) g = L ( 1 ) a ( 0 ) g − a ( 0 ) L ( 1 ) g , a ( 2 ) g = L ( 1 ) a ( 1 ) g − a ( 1 ) L ( 1 ) g , 2 a ( 3 ) g = L ( 2 ) a ( 1 ) g − a ( 1 ) L ( 2 ) g . 2a_{(1)}g=L(1)a_{(0)}g-a_{(0)}L(1)g,\qquad a_{(2)}g=L(1)a_{(1)}g-a_{(1)}L(1)g,\qquad2a_{(3)}g=L(2)a_{(1)}g-a_{(1)}L(2)g. 2 a ( 1 ) g = L ( 1 ) a ( 0 ) g − a ( 0 ) L ( 1 ) g , a ( 2 ) g = L ( 1 ) a ( 1 ) g − a ( 1 ) L ( 1 ) g , 2 a ( 3 ) g = L ( 2 ) a ( 1 ) g − a ( 1 ) L ( 2 ) g . Since L ( 1 ) N 4 ′ ⊂ S 3 L(1)N'_4 \subset S_3 L ( 1 ) N 4 ′ ⊂ S 3 [ L ( 1 ) , y ( n ) ] = − n y ( n + 1 ) [L(1),y_{(n)}]=-ny_{(n+1)} [ L ( 1 ) , y ( n ) ] = − n y ( n + 1 ) [ L ( 1 ) , m ( − 1 ) ] = 3 m ( 0 ) [L(1),m_{(-1)}]=3m_{(0)} [ L ( 1 ) , m ( − 1 ) ] = 3 m ( 0 ) m m m L ( 1 ) L ( − 1 ) = L ( − 1 ) L ( 1 ) + 2 L ( 0 ) L(1)L(-1)=L(-1)L(1)+2L(0) L ( 1 ) L ( − 1 ) = L ( − 1 ) L ( 1 ) + 2 L ( 0 ) Theorem 10.19 ), L ( 1 ) S 3 ⊂ U 2 L(1)S_3 \subset U_2 L ( 1 ) S 3 ⊂ U 2 L ( 2 ) S 3 ⊂ U 1 L(2)S_3 \subset U_1 L ( 2 ) S 3 ⊂ U 1 ( U 2 ) ( n ) U k ⊂ U k + 1 − n (U_2)_{(n)}U_k \subset U_{k+1-n} ( U 2 ) ( n ) U k ⊂ U k + 1 − n k = 1 , 2 k=1,2 k = 1 , 2 n ≥ 1 n \ge1 n ≥ 1 ( U 2 ) ( 0 ) U 2 ⊂ S 3 (U_2)_{(0)}U_2 \subset S_3 ( U 2 ) ( 0 ) U 2 ⊂ S 3 ( U 2 ) ( 0 ) U 1 ⊂ U 2 (U_2)_{(0)}U_1 \subset U_2 ( U 2 ) ( 0 ) U 1 ⊂ U 2 a ( 1 ) g ∈ S 3 a_{(1)}g \in S_3 a ( 1 ) g ∈ S 3 a ( 2 ) g ∈ U 2 a_{(2)}g \in U_2 a ( 2 ) g ∈ U 2 a ( 3 ) g ∈ U 1 a_{(3)}g \in U_1 a ( 3 ) g ∈ U 1 U 2 = ( U 2 ∩ Ker L ( 1 ) ) + L ( − 1 ) U 1 U_2=(U_2\cap\operatorname{Ker}L(1))+L(-1)U_1 U 2 = ( U 2 ∩ Ker L ( 1 )) + L ( − 1 ) U 1 L ( 1 ) U 2 ⊂ U 1 L(1)U_2\subset U_1 L ( 1 ) U 2 ⊂ U 1 L ( 1 ) L ( − 1 ) = 2 L(1)L(-1)=2 L ( 1 ) L ( − 1 ) = 2 U 1 U_1 U 1 a = L ( − 1 ) y a=L(-1)y a = L ( − 1 ) y y ∈ U 1 y\in U_1 y ∈ U 1 a ( n ) = − n y ( n − 1 ) a_{(n)}=-ny_{(n-1)} a ( n ) = − n y ( n − 1 ) a ∈ U 1 a\in U_1 a ∈ U 1 n ≥ 4 n\ge4 n ≥ 4 a ( n ) g a_{(n)}g a ( n ) g S 4 − n ⊂ C 1 S_{4-n}\subset C_1 S 4 − n ⊂ C 1 M τ ( − 1 ) U 1 M_\tau(-1)U_1 M τ ( − 1 ) U 1 D 3 ( − 1 ) U 1 D_3(-1)U_1 D 3 ( − 1 ) U 1 U 2 ( − 2 ) U 1 U_2(-2)U_1 U 2 ( − 2 ) U 1 N ( G ) N(G) N ( G ) U 1 ⊕ U 2 U_1\oplus U_2 U 1 ⊕ U 2 N ( G ) = S N(G)=S N ( G ) = S S 1 = U 1 S_1=U_1 S 1 = U 1 S 2 = U 2 S_2=U_2 S 2 = U 2 S 3 = N 3 + D 3 S_3=N_3+D_3 S 3 = N 3 + D 3 S 4 S_4 S 4 L ( − 1 ) S 3 L(-1)S_3 L ( − 1 ) S 3 U 1 ( − 1 ) S 3 U_1(-1)S_3 U 1 ( − 1 ) S 3 U 1 ( − 2 ) U 2 U_1(-2)U_2 U 1 ( − 2 ) U 2 U 1 ( − 3 ) U 1 U_1(-3)U_1 U 1 ( − 3 ) U 1 U 2 ( − 1 ) U 2 U_2(-1)U_2 U 2 ( − 1 ) U 2 N 4 ′ N'_4 N 4 ′ Theorem 10.19 . Since ( A ) 2 ⊂ N ( U 1 ) (A)_2\subset N(U_1) ( A ) 2 ⊂ N ( U 1 ) 120 − 27 = 93 120-27=93 120 − 27 = 93 Lemma 10.15 gives dim N 3 = 725 \dim N_3=725 dim N 3 = 725 D 3 D_3 D 3 956 − 725 = 231 956-725=231 956 − 725 = 231 p 3 p_3 p 3 dim N 3 ≥ 725 \dim N_3\ge725 dim N 3 ≥ 725
The initial choice of representatives does not restrict the complement in the statement. Indeed, for any other complement D 3 ′ D'_3 D 3 ′ N 3 N_3 N 3 S 3 S_3 S 3 D 3 ′ D'_3 D 3 ′ u ( 0 ) w u_{(0)}w u ( 0 ) w N 3 N_3 N 3 N 3 ⊂ N ( U 1 ⊕ U 2 ) N_3\subset N(U_1\oplus U_2) N 3 ⊂ N ( U 1 ⊕ U 2 ) Lemma 10.20 still holds. The verification of hypothesis 1 above uses inclusions on the whole space S 3 S_3 S 3
We follow [67 Section 5] . Let R = S / C 2 ( S ) R=S/C_2(S) R = S / C 2 ( S ) C 2 C_2 C 2 [ a ] [ b ] = [ a ( − 1 ) b ] [a][b]=[a_{(-1)}b] [ a ] [ b ] = [ a ( − 1 ) b ] { [ a ] , [ b ] } = [ a ( 0 ) b ] \{[a],[b]\}=[a_{(0)}b] {[ a ] , [ b ]} = [ a ( 0 ) b ] R ‾ \overline{R} R n \mathfrak n n [3 Corollary 2.4.2] . By part 1, R R R G G G [3 Corollary 2.6.2] (only the elementary direction is used: C 2 ( S ) C_2(S) C 2 ( S ) a ( − n ) b a_{(-n)}b a ( − n ) b n ≥ 2 n\ge2 n ≥ 2 [ a ] [ b ] = [ a ( − 1 ) b ] [a][b]=[a_{(-1)}b] [ a ] [ b ] = [ a ( − 1 ) b ] R ‾ \overline{R} R v = e ′ ( − 1 ) 4 1 v=e'(-1)^4\mathbf{1} v = e ′ ( − 1 ) 4 1 f ′ ( 1 ) v = 0 f'(1)v=0 f ′ ( 1 ) v = 0 ϕ x ( e ′ ) = − f ′ \phi_x(e')=-f' ϕ x ( e ′ ) = − f ′ H x ( e ( m ) ′ u , u ′ ) = H x ( u , f ( − m ) ′ u ′ ) H_x(e'_{(m)}u,u')=H_x(u,f'_{(-m)}u') H x ( e ( m ) ′ u , u ′ ) = H x ( u , f ( − m ) ′ u ′ ) Theorem 10.10 . Hence H x ( v , v ) = H x ( e ′ ( − 1 ) 3 1 , f ′ ( 1 ) v ) = 0 H_x(v,v)=H_x(e'(-1)^3\mathbf{1},f'(1)v)=0 H x ( v , v ) = H x ( e ′ ( − 1 ) 3 1 , f ′ ( 1 ) v ) = 0 v = 0 v=0 v = 0 e ′ ′ ( − 1 ) 14 1 = 0 e''(-1)^{14}\mathbf{1}=0 e ′′ ( − 1 ) 14 1 = 0 [ e ′ ] 14 = [ e ′ ′ ] 14 = 0 [e']^{14}=[e'']^{14}=0 [ e ′ ] 14 = [ e ′′ ] 14 = 0 f ′ , f ′ ′ f',f'' f ′ , f ′′ g ⊕ g ′ \mathfrak g\oplus\mathfrak g' g ⊕ g ′ M M M M = ( g ′ ⊕ g ′ ′ ) ⋅ M M=(\mathfrak g'\oplus\mathfrak g'')\cdot M M = ( g ′ ⊕ g ′′ ) ⋅ M R ‾ \overline{R} R U 2 U_2 U 2 R ‾ \overline{R} R ( A ) 2 (A)_2 ( A ) 2 U 2 U_2 U 2 ( A ) 2 (A)_2 ( A ) 2 e e e x C x_C x C Theorem 10.19 , e ( − 1 ) e e(-1)e e ( − 1 ) e x C ( − 1 ) x C x_C(-1)x_C x C ( − 1 ) x C N 4 ′ N'_4 N 4 ′ n \mathfrak n n L ( − 1 ) S 3 L(-1)S_3 L ( − 1 ) S 3 U 1 ( − 2 ) U 2 U_1(-2)U_2 U 1 ( − 2 ) U 2 U 1 ( − 3 ) U 1 U_1(-3)U_1 U 1 ( − 3 ) U 1 C 2 ( S ) C_2(S) C 2 ( S ) U 1 ( − 1 ) S 3 U_1(-1)S_3 U 1 ( − 1 ) S 3 M τ ( − 1 ) U 2 M_\tau(-1)U_2 M τ ( − 1 ) U 2 [ e ] 2 = [ x C ] 2 = 0 [e]^2=[x_C]^2=0 [ e ] 2 = [ x C ] 2 = 0 R ‾ \overline{R} R [ e ] = [ x C ] = 0 [e]=[x_C]=0 [ e ] = [ x C ] = 0 D 3 ⊂ N 3 + ( U 2 ) ( 0 ) U 2 D_3\subset N_3+(U_2)_{(0)}U_2 D 3 ⊂ N 3 + ( U 2 ) ( 0 ) U 2 R R R R R R
Corollary 10.22 Independently of the membership verification in Proposition 10.17 , condition (F4) follows from a dimension argument in either of the following situations:
the subspace U 2 U_2 U 2 U ⊂ V ′ U \subset V' U ⊂ V ′
the existence and character assertion of [52 ] is granted, and S S S Theorem 10.24 .
Consequently the conclusions of Theorem 10.21 hold in either situation without a separate weight-four membership certificate.
Proof. The rank computation of Proposition 10.13 , part 5, supplies 6010 words of N 4 ′ N'_4 N 4 ′ p 3 p_3 p 3 dim K N 4 ′ ≥ 6010 \dim_K N'_4 \ge6010 dim K N 4 ′ ≥ 6010 S ⊂ U S \subset U S ⊂ U dim U 4 = 6010 \dim U_4 = 6010 dim U 4 = 6010 N 4 ′ ⊂ S 4 ⊂ U 4 N'_4 \subset S_4 \subset U_4 N 4 ′ ⊂ S 4 ⊂ U 4 dim S 4 = 6010 \dim S_4 = 6010 dim S 4 = 6010 S 4 S_4 S 4 S 4 S_4 S 4 N 4 ′ N'_4 N 4 ′ C \mathbb{C} C Theorem 10.24 uses weight-two closure and generation, not Theorem 10.21 , so the second argument is not circular.
Remark 10.23 For S S S [67 Proposition 4.2] is needed. In case (I), the weight-four space of U R U^{\mathrm{R}} U R N 4 ′ = S 4 N'_4 = S_4 N 4 ′ = S 4 Theorem 10.21 uses the finite input of Theorems 10.9 and 10.19 , whose weight-four identities are established over K in Proposition 10.17 . Corollary 10.22 gives an alternative dimension argument when an ambient Haagerup algebra or the stated external existence result is available. It cannot be applied with U = S U = S U = S [5 Theorem 6.1] , since the proof of that theorem uses Theorem 10.21 , which rests on Proposition 10.17 . Modular computations select pivots. The dimension dim N 3 = 725 \dim N_3 = 725 dim N 3 = 725 Lemma 10.15 , and the modular minor supplies an independent lower-bound check.
Case (II): identification with Jiao’s algebra and uniqueness
Jiao [52 ] defines a vertex subalgebra V = ⟨ ω , P E , P F , P H , B E , B F , B H , u 26 , u 17 ⟩ V = \langle\omega, P_E, P_F, P_H, B_E, B_F, B_H, u_{26}, u_{17} \rangle V = ⟨ ω , P E , P F , P H , B E , B F , B H , u 26 , u 17 ⟩ E 6 ⊕ A 2 E_6 \oplus A_2 E 6 ⊕ A 2 s l 2 \mathfrak{sl}_2 sl 2 [52 Section 3] . He gives a computer-assisted proof, which we have not checked in full, that V V V Rep ( V ) ≃ Z ( H 3 ) \operatorname{Rep}(V) \simeq\mathcal{Z}(\mathcal{H}_3) Rep ( V ) ≃ Z ( H 3 ) [52 Theorem 1.1] , and that every simple V V V V V V [52 Corollary 6.8] . His construction is dated October 4, 2026. It precedes the present work and the announcements [66 , 50 ] , and it makes no uniqueness assertion [52 Section 1.2] . Below, “assuming (HJ _J J Theorem 10.24 , they hold as soon as S S S [5 Theorem 6.1] without using [52 ] . Hence Corollary 10.25 and Theorem 10.29 hold unconditionally.
Theorem 10.24 Let S S S U 1 ⊕ U 2 U_1 \oplus U_2 U 1 ⊕ U 2 U 2 U_2 U 2 G \mathcal{G} G Theorem 10.9 . There is an isomorphism of vertex operator algebras ψ \psi ψ [52 ] onto V ′ = V E 6 ⊕ A 2 V' = V_{E_6 \oplus A_2} V ′ = V E 6 ⊕ A 2 ψ ( V ) = S \psi(V) = S ψ ( V ) = S ( P E , P F , P H ) (P_E, P_F, P_H) ( P E , P F , P H ) ( B E , B F , B H ) (B_E, B_F, B_H) ( B E , B F , B H ) ( e ′ , f ′ , h ′ ) (e', f', h') ( e ′ , f ′ , h ′ ) ( e ′ ′ , f ′ ′ , h ′ ′ ) (e'', f'', h'') ( e ′′ , f ′′ , h ′′ )
Proof: We compute the transport, membership and generation in exact arithmetic with the normalized coefficients described below. Both models use the convention
Y ( e p , x ) e q = ε ( p , q ) x ( p , q ) ( e p + q + O ( x ) ) , Y(e^{p},x)e^{q}=\varepsilon(p,q)x^{(p,q)}(e^{p+q}+O(x)), Y ( e p , x ) e q = ε ( p , q ) x ( p , q ) ( e p + q + O ( x )) , with bimultiplicative cocycles with values ± 1 \pm1 ± 1 ( − 1 ) ( p , q ) (-1)^{(p,q)} ( − 1 ) ( p , q ) ( α i , α j ) (\alpha_i,\alpha_j) ( α i , α j ) i > j i>j i > j [52 Section 3.1] .
The isometry. Let φ \varphi φ 1 , … , 8 1,\ldots,8 1 , … , 8 [52 ] (edges 12 , 23 , 34 , 45 , 36 , 78 12,23,34,45,36,78 12 , 23 , 34 , 45 , 36 , 78 1 , 3 , 4 , 5 , 6 , 2 , 7 , 8 1,3,4,5,6,2,7,8 1 , 3 , 4 , 5 , 6 , 2 , 7 , 8 w = s 6 s 1 s 3 s 2 s 4 s 5 s 4 s 3 s 2 s 4 w=s_6s_1s_3s_2s_4s_5s_4s_3s_2s_4 w = s 6 s 1 s 3 s 2 s 4 s 5 s 4 s 3 s 2 s 4 E 6 E_6 E 6 ψ 0 = w ∘ φ \psi_0=w\circ\varphi ψ 0 = w ∘ φ P H P_H P H B H B_H B H H ′ H' H ′ H ′ ′ H'' H ′′
The cocycle. Put r ( p , q ) = ε ( ψ 0 p , ψ 0 q ) ε J i a o ( p , q ) r(p,q)=\varepsilon(\psi_0p,\psi_0q)\varepsilon_{\mathrm{Jiao}}(p,q) r ( p , q ) = ε ( ψ 0 p , ψ 0 q ) ε Jiao ( p , q ) ψ 0 \psi_0 ψ 0 r ( p , q ) r ( q , p ) = ( − 1 ) ( ψ 0 p , ψ 0 q ) + ( p , q ) = 1 r(p,q)r(q,p)=(-1)^{(\psi_0p,\psi_0q)+(p,q)}=1 r ( p , q ) r ( q , p ) = ( − 1 ) ( ψ 0 p , ψ 0 q ) + ( p , q ) = 1 r ( α i , α i ) = 1 r(\alpha_i,\alpha_i)=1 r ( α i , α i ) = 1 ε ( α , α ) = ( − 1 ) ( α , α ) / 2 \varepsilon(\alpha,\alpha)=(-1)^{(\alpha,\alpha)/2} ε ( α , α ) = ( − 1 ) ( α , α ) /2 r i j = r ( α i , α j ) r_{ij}=r(\alpha_i,\alpha_j) r ij = r ( α i , α j ) η ( p ) = ∏ i < j r i j p i p j \eta(p)=\prod_{i<j}r_{ij}^{p_ip_j} η ( p ) = ∏ i < j r ij p i p j η ( p + q ) = η ( p ) η ( q ) r ( p , q ) \eta(p+q)=\eta(p)\eta(q)r(p,q) η ( p + q ) = η ( p ) η ( q ) r ( p , q ) e p ↦ η ( p ) e ψ 0 p e^p\mapsto\eta(p)e^{\psi_0p} e p ↦ η ( p ) e ψ 0 p h ( − n ) ↦ ( ψ 0 h ) ( − n ) h(-n)\mapsto(\psi_0h)(-n) h ( − n ) ↦ ( ψ 0 h ) ( − n )
The torus. The images of P E P_E P E B E B_E B E e ′ e' e ′ e ′ ′ e'' e ′′ t t t P E P_E P E B E B_E B E t ( α i ) = 2 a i 3 b i 5 c i t(\alpha_i)=2^{a_i}3^{b_i}5^{c_i} t ( α i ) = 2 a i 3 b i 5 c i exp ( h t ( 0 ) ) \exp(h_t(0)) exp ( h t ( 0 )) M ( 1 ) ⊗ e λ M(1)\otimes e^\lambda M ( 1 ) ⊗ e λ t ( λ ) t(\lambda) t ( λ ) ψ \psi ψ ψ ( P E ) = e ′ \psi(P_E)=e' ψ ( P E ) = e ′ ψ ( B E ) = e ′ ′ \psi(B_E)=e'' ψ ( B E ) = e ′′ ψ ( P H ) = H ′ \psi(P_H)=H' ψ ( P H ) = H ′ ψ ( B H ) = H ′ ′ \psi(B_H)=H'' ψ ( B H ) = H ′′ ψ ( P F ) = f ′ \psi(P_F)=f' ψ ( P F ) = f ′ ψ ( B F ) = f ′ ′ \psi(B_F)=f'' ψ ( B F ) = f ′′ f f f s l 2 \mathfrak{sl}_2 sl 2 e e e H H H ψ \psi ψ K \mathbb{K} K
Inclusion. The vectors ψ ( u 26 ) \psi(u_{26}) ψ ( u 26 ) ( − 5 ) − 1 / 2 ψ ( u 17 ) (-5)^{-1/2}\psi(u_{17}) ( − 5 ) − 1/2 ψ ( u 17 ) K \mathbb{K} K U 2 U_2 U 2 9 \mathcal{9} 9 K \mathbb{K} K ψ ( V ) ⊂ S \psi(V)\subset S ψ ( V ) ⊂ S
Equality. Put u ^ 26 = ψ ( u 26 ) \widehat{u}_{26}=\psi(u_{26}) u 26 = ψ ( u 26 ) u ^ 17 = ( − 5 ) − 1 / 2 ψ ( u 17 ) \widehat{u}_{17}=(-5)^{-1/2}\psi(u_{17}) u 17 = ( − 5 ) − 1/2 ψ ( u 17 ) U 2 , K = U 2 ∩ V K ′ U_{2,\mathbb{K}}=U_2\cap V'_{\mathbb{K}} U 2 , K = U 2 ∩ V K ′ ( A ) 2 , K = ( A ) 2 ∩ V K ′ (A)_{2,\mathbb{K}}=(A)_2\cap V'_{\mathbb{K}} ( A ) 2 , K = ( A ) 2 ∩ V K ′ K \mathbb{K} K V ′ V' V ′ W K W_{\mathbb{K}} W K K \mathbb{K} K U 2 , K U_{2,\mathbb{K}} U 2 , K ( A ) 2 , K (A)_{2,\mathbb{K}} ( A ) 2 , K ω \omega ω u ^ 26 \widehat{u}_{26} u 26 u ^ 17 \widehat{u}_{17} u 17 W K ⊂ U 2 , K W_{\mathbb{K}}\subset U_{2,\mathbb{K}} W K ⊂ U 2 , K ψ ( V ) \psi(V) ψ ( V )
At p = 8388571 p=8388571 p = 8388571 13 ↦ 1649360 \sqrt{13}\mapsto1649360 13 ↦ 1649360 − 3 ↦ 2036966 \sqrt{-3}\mapsto2036966 − 3 ↦ 2036966
A nonzero 120 × 120 120\times120 120 × 120 K \mathbb{K} K dim K W K ≥ 120 = dim K U 2 , K \dim_{\mathbb{K}}W_{\mathbb{K}}\ge120=\dim_{\mathbb{K}}U_{2,\mathbb{K}} dim K W K ≥ 120 = dim K U 2 , K W K = U 2 , K W_{\mathbb{K}}=U_{2,\mathbb{K}} W K = U 2 , K U 2 ⊂ ψ ( V ) U_2\subset\psi(V) U 2 ⊂ ψ ( V ) U 1 ⊂ ψ ( V ) U_1\subset\psi(V) U 1 ⊂ ψ ( V ) S ⊂ ψ ( V ) S\subset\psi(V) S ⊂ ψ ( V )
For this ψ \psi ψ ψ ( u 26 ) \psi(u_{26}) ψ ( u 26 ) ψ ( u 17 ) \psi(u_{17}) ψ ( u 17 ) ψ \psi ψ V ′ V' V ′ U 1 U_1 U 1
Corollary 10.25 Assuming (HJ _J J 9 \mathcal{9} 9 Rep ( S ) ≃ Z ( H 3 ) \operatorname{Rep}(S)\simeq Z(\mathcal{H}_3) Rep ( S ) ≃ Z ( H 3 )
Proof: A vertex operator algebra isomorphism induces a braided equivalence of the module categories and preserves the lowest weights. Apply Theorem 10.24 and Proposition 10.12. ■
The six currents e ′ , f ′ , H ′ , e ′ ′ , f ′ ′ , H ′ ′ e',f',H',e'',f'',H'' e ′ , f ′ , H ′ , e ′′ , f ′′ , H ′′ Q \mathbb{Q} Q V Q ′ V'_{\mathbb{Q}} V Q ′ U 2 U_2 U 2 K \mathbb{K} K σ ( 13 ) = − 13 \sigma(\sqrt{13})=-\sqrt{13} σ ( 13 ) = − 13 σ ( − 3 ) = − 3 \sigma(\sqrt{-3})=\sqrt{-3} σ ( − 3 ) = − 3 V K ′ = V Q ′ ⊗ Q K V'_{\mathbb{K}}=V'_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{K} V K ′ = V Q ′ ⊗ Q K σ \sigma σ σ ~ = id ⊗ σ \widetilde{\sigma}=\operatorname{id}\otimes\sigma σ = id ⊗ σ ω \omega ω K \mathbb{K} K S ( 9 , c ) S^{(9,c)} S ( 9 , c ) S ( 10 , c ) S^{(10,c)} S ( 10 , c ) c ∈ { 0 , 1 } c\in\{0,1\} c ∈ { 0 , 1 } S ( 9 , c ) S^{(9,c)} S ( 9 , c ) σ \sigma σ C \mathbb{C} C V ′ V' V ′ χ χ \chi_{\chi} χ χ S ( 9 , 0 ) S^{(9,0)} S ( 9 , 0 ) S ( 9 , 1 ) S^{(9,1)} S ( 9 , 1 )
Lemma 10.26. In either case of Theorem 8.5 , let Z Z Z U 1 U_1 U 1 Aut ( V ′ ) \operatorname{Aut}(V') Aut ( V ′ ) h \mathfrak{h} h E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2 Z Z Z h \mathfrak{h} h h \mathfrak{h} h ∣ Z ∣ ≤ 8 |Z|\leq8 ∣ Z ∣ ≤ 8 ρ \rho ρ χ 0 \chi_0 χ 0
( χ 0 ( α 1 ) , … , χ 0 ( α 8 ) ) = { ( 1 , − 1 , − 1 , 1 , − 1 , 1 , 1 , 1 ) in case (I) , ( − 1 , 1 , 1 , − 1 , 1 , − 1 , 1 , 1 ) in case (II) . (\chi_0(\alpha_1),\ldots,\chi_0(\alpha_8))=
\begin{cases}
(1,-1,-1,1,-1,1,1,1) & \text{in case (I)},\\
(-1,1,1,-1,1,-1,1,1) & \text{in case (II)}.
\end{cases} ( χ 0 ( α 1 ) , … , χ 0 ( α 8 )) = { ( 1 , − 1 , − 1 , 1 , − 1 , 1 , 1 , 1 ) ( − 1 , 1 , 1 , − 1 , 1 , − 1 , 1 , 1 ) in case (I) , in case (II) . Thus ρ ( e λ ) = χ 0 ( λ ) e λ \rho(e^\lambda)=\chi_0(\lambda)e^\lambda ρ ( e λ ) = χ 0 ( λ ) e λ ρ \rho ρ
Proof. We give an argument using the roots and the supports of the currents. An automorphism of V ′ V' V ′ V 1 ′ = e 6 ⊕ s l 3 V'_1=e_6\oplus\mathfrak{sl}_3 V 1 ′ = e 6 ⊕ sl 3 V 1 ′ V'_1 V 1 ′ V ′ V' V ′ E 6 E_6 E 6 H ′ ′ H'' H ′′ ( 2 , 1 , 1 , 0 , 1 , 2 ) (2,1,1,0,1,2) ( 2 , 1 , 1 , 0 , 1 , 2 ) ( 1 , 0 , 0 , 1 , 0 , 1 ) (1,0,0,1,0,1) ( 1 , 0 , 0 , 1 , 0 , 1 ) α 4 \alpha_4 α 4 α 2 , α 3 , α 5 \alpha_2,\alpha_3,\alpha_5 α 2 , α 3 , α 5 H ′ = α 4 H'=\alpha_4 H ′ = α 4 H ′ = − α 2 + α 3 + α 5 H'=-\alpha_2+\alpha_3+\alpha_5 H ′ = − α 2 + α 3 + α 5 − 2 , 2 , 2 -2,2,2 − 2 , 2 , 2 A 2 A_2 A 2 H ′ ′ H'' H ′′ ( 2 , 2 ) (2,2) ( 2 , 2 ) H ′ H' H ′ H ′ ′ H'' H ′′ V 1 ′ V'_1 V 1 ′ h \mathfrak{h} h Z Z Z h \mathfrak{h} h
An element of Z Z Z H ′ H' H ′ H ′ ′ H'' H ′′ H ′ H' H ′ H ′ ′ H'' H ′′
For completeness, the raising currents of the chosen rational triples are, in case (I),
e ′ = e α 4 , e'=e^{\alpha_4}, e ′ = e α 4 , e ′ ′ = 5 e α 1 + 4 e α 6 + 3 e α 2 + α 3 + α 4 + e α 2 + α 4 + α 5 + 4 e α 3 + α 4 + α 5 + e α 7 + e α 8 . (26) e''=5e^{\alpha_1}+4e^{\alpha_6}+3e^{\alpha_2+\alpha_3+\alpha_4}+e^{\alpha_2+\alpha_4+\alpha_5}
+4e^{\alpha_3+\alpha_4+\alpha_5}+e^{\alpha_7}+e^{\alpha_8}. \tag*{(26)} e ′′ = 5 e α 1 + 4 e α 6 + 3 e α 2 + α 3 + α 4 + e α 2 + α 4 + α 5 + 4 e α 3 + α 4 + α 5 + e α 7 + e α 8 . (26) and in case (II),
e ′ = e − α 2 + e α 3 − 5 3 e α 5 , e'=e^{-\alpha_2}+e^{\alpha_3}-\frac{5}{3}e^{\alpha_5}, e ′ = e − α 2 + e α 3 − 3 5 e α 5 , e ′ ′ = 4 e α 4 + α 5 + α 6 + 4 e α 2 + α 3 + α 4 + α 5 + α 6 + e α 2 + α 3 + 2 α 4 + α 5 + 3 e α 1 + α 3 + α 4 + 5 e α 1 + α 2 + α 3 + α 4 + α 5 + e α 7 + e α 8 . (27) e''=4e^{\alpha_4+\alpha_5+\alpha_6}+4e^{\alpha_2+\alpha_3+\alpha_4+\alpha_5+\alpha_6}+e^{\alpha_2+\alpha_3+2\alpha_4+\alpha_5}
+3e^{\alpha_1+\alpha_3+\alpha_4}+5e^{\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5}+e^{\alpha_7}+e^{\alpha_8}. \tag*{(27)} e ′′ = 4 e α 4 + α 5 + α 6 + 4 e α 2 + α 3 + α 4 + α 5 + α 6 + e α 2 + α 3 + 2 α 4 + α 5 + 3 e α 1 + α 3 + α 4 + 5 e α 1 + α 2 + α 3 + α 4 + α 5 + e α 7 + e α 8 . (27) An automorphism acting trivially on h \mathfrak{h} h t t t t i = t ( α i ) t_i=t(\alpha_i) t i = t ( α i ) t ( β ) = 1 t(\beta)=1 t ( β ) = 1 (26) gives t 1 = t 4 = t 6 = t 7 = t 8 = 1 t_1=t_4=t_6=t_7=t_8=1 t 1 = t 4 = t 6 = t 7 = t 8 = 1 t 2 t 3 = t 2 t 5 = t 3 t 5 = 1 t_2t_3=t_2t_5=t_3t_5=1 t 2 t 3 = t 2 t 5 = t 3 t 5 = 1 t 2 = t 3 = t 5 ∈ { 1 , − 1 } t_2=t_3=t_5\in\{1,-1\} t 2 = t 3 = t 5 ∈ { 1 , − 1 } (27) gives t 2 = t 3 = t 5 = t 7 = t 8 = 1 t_2=t_3=t_5=t_7=t_8=1 t 2 = t 3 = t 5 = t 7 = t 8 = 1 t 4 t 6 = t 4 2 = t 1 t 4 = 1 t_4t_6=t_4^2=t_1t_4=1 t 4 t 6 = t 4 2 = t 1 t 4 = 1 t 1 = t 4 = t 6 ∈ { 1 , − 1 } t_1=t_4=t_6\in\{1,-1\} t 1 = t 4 = t 6 ∈ { 1 , − 1 } s l 2 \mathfrak{sl}_2 sl 2 − 2 -2 − 2 s l 2 \mathfrak{sl}_2 sl 2 U 1 U_1 U 1 ∣ Z ∣ ≤ 2 ⋅ 4 = 8 |Z| \le2 \cdot4 = 8 ∣ Z ∣ ≤ 2 ⋅ 4 = 8
Proposition 10.27. In case (II), let Z Z Z V ′ V' V ′ U 1 U_1 U 1
Z ≅ ( Z 2 ) 3 Z \cong(\mathbb{Z}_2)^3 Z ≅ ( Z 2 ) 3 Z Z Z E 6 ⊕ A 2 E_6 \oplus A_2 E 6 ⊕ A 2
Four elements of Z Z Z B B B s s s
Let g g g E 6 E_6 E 6 α 1 ↔ α 6 \alpha_1 \leftrightarrow\alpha_6 α 1 ↔ α 6 α 3 ↔ α 5 \alpha_3 \leftrightarrow\alpha_5 α 3 ↔ α 5 A 2 A_2 A 2 z z z V ′ V' V ′ z ( h ( − n ) ) = ( g h ) ( − n ) z(h(-n))=(gh)(-n) z ( h ( − n )) = ( g h ) ( − n ) z ( e λ ) = c ( λ ) e g λ z(e^\lambda)=c(\lambda)e^{g\lambda} z ( e λ ) = c ( λ ) e g λ c ( λ ) = ∏ i c i λ i ∏ i < j δ i j λ i λ j c(\lambda)=\prod_i c_i^{\lambda_i}\prod_{i<j}\delta_{ij}^{\lambda_i\lambda_j} c ( λ ) = ∏ i c i λ i ∏ i < j δ ij λ i λ j λ = ∑ i λ i α i \lambda=\sum_i\lambda_i\alpha_i λ = ∑ i λ i α i α 1 , … , α 6 \alpha_1,\ldots,\alpha_6 α 1 , … , α 6 E 6 E_6 E 6 α 7 , α 8 \alpha_7,\alpha_8 α 7 , α 8 A 2 A_2 A 2 δ i j = ε ( α i , α j ) ε ( g α i , g α j ) \delta_{ij}=\varepsilon(\alpha_i,\alpha_j)\varepsilon(g\alpha_i,g\alpha_j) δ ij = ε ( α i , α j ) ε ( g α i , g α j ) δ i i = 1 \delta_{ii}=1 δ ii = 1 c ( α i ) = c i c(\alpha_i)=c_i c ( α i ) = c i ( c 1 , … , c 8 ) = ( 4 5 , 1 , − 5 3 , − 1 , − 3 5 , 5 4 , 1 , 1 ) (c_1,\ldots,c_8)=\left(\frac{4}{5},1,-\frac{5}{3},-1,-\frac{3}{5},\frac{5}{4},1,1\right) ( c 1 , … , c 8 ) = ( 5 4 , 1 , − 3 5 , − 1 , − 5 3 , 4 5 , 1 , 1 ) z ∈ Z z\in Z z ∈ Z z z z U 2 U_2 U 2 S ( 9 , c ) S^{(9,c)} S ( 9 , c ) S ( 10 , c ) S^{(10,c)} S ( 10 , c ) c = 0 , 1 c=0,1 c = 0 , 1
In particular the four algebras S ( 9 , 0 ) S^{(9,0)} S ( 9 , 0 ) S ( 9 , 1 ) S^{(9,1)} S ( 9 , 1 ) S ( 10 , 0 ) S^{(10,0)} S ( 10 , 0 ) S ( 10 , 1 ) S^{(10,1)} S ( 10 , 1 ) S ( 9 , 0 ) = ψ ( V ) S^{(9,0)}=\psi(V) S ( 9 , 0 ) = ψ ( V )
Proof: First we prove part 1 without solving polynomial equations. Let z z z z ′ z' z ′ A 2 A_2 A 2 c i = 1 c_i=1 c i = 1 ρ \rho ρ Lemma 10.26 . The formula for c ( λ ) c(\lambda) c ( λ ) A 2 A_2 A 2 c i = 1 c_i=1 c i = 1 z ′ z' z ′ z z z z ′ z' z ′ δ i j = δ j i \delta_{ij}=\delta_{ji} δ ij = δ j i δ i i = 1 \delta_{ii}=1 δ ii = 1 c ( λ + μ ) ε ( λ , μ ) = c ( λ ) c ( μ ) ε ( g λ , g μ ) c(\lambda+\mu)\varepsilon(\lambda,\mu)=c(\lambda)c(\mu)\varepsilon(g\lambda,g\mu) c ( λ + μ ) ε ( λ , μ ) = c ( λ ) c ( μ ) ε ( g λ , g μ ) (27) and the Cartan elements gives z , z ′ ∈ Z z,z'\in Z z , z ′ ∈ Z h \mathfrak{h} h
z 2 = z ′ 2 = ρ 2 = 1 , z z ′ = z ′ z , z ρ = ρ z , z ′ ρ = ρ z ′ . z^2=z'^2=\rho^2=1,\qquad zz'=z'z,\qquad z\rho=\rho z,\qquad z'\rho=\rho z'. z 2 = z ′2 = ρ 2 = 1 , z z ′ = z ′ z , z ρ = ρ z , z ′ ρ = ρ z ′ . They generate eight distinct automorphisms: the isometry parts of z z z z ′ z' z ′ ρ \rho ρ Lemma 10.26 proves that these are all the elements of Z Z Z
The action on B B B b 0 = e , b 1 , … , b 4 b_0=e,b_1,\ldots,b_4 b 0 = e , b 1 , … , b 4 z ( b i ) = z ′ ( b i ) = b i z(b_i)=z'(b_i)=b_i z ( b i ) = z ′ ( b i ) = b i 0 ≤ i ≤ 3 0\leq i\leq3 0 ≤ i ≤ 3
z ( b 4 ) = z ′ ( b 4 ) = − b 4 − 1 12 b 1 + 5 24 b 2 + 1 16 b 3 , ρ ( b i ) = b i ( 0 ≤ i ≤ 4 ) . z(b_4)=z'(b_4)=-b_4-\frac{1}{12}b_1+\frac{5}{24}b_2+\frac{1}{16}b_3,\qquad\rho(b_i)=b_i\quad(0\leq i\leq4). z ( b 4 ) = z ′ ( b 4 ) = − b 4 − 12 1 b 1 + 24 5 b 2 + 16 1 b 3 , ρ ( b i ) = b i ( 0 ≤ i ≤ 4 ) . Thus z ∣ B = z ′ ∣ B = s z|_B=z'|_B=s z ∣ B = z ′ ∣ B = s ρ ∣ B = 1 \rho|_B=1 ρ ∣ B = 1 Lemma 10.26 .
An independent polynomial computation gives the same centralizer. An automorphism of V ′ V' V ′ V 1 ′ = e 6 ⊕ s l 3 V'_1=\mathfrak{e}_6\oplus\mathfrak{sl}_3 V 1 ′ = e 6 ⊕ sl 3 V ′ ≅ L E 6 ( 1 , 0 ) ⊗ L s l 3 ( 1 , 0 ) V'\cong L_{E_6}(1,0)\otimes L_{\mathfrak{sl}_3}(1,0) V ′ ≅ L E 6 ( 1 , 0 ) ⊗ L sl 3 ( 1 , 0 ) [54 ] . Under U 1 U_1 U 1 V 1 ′ V'_1 V 1 ′ ⨁ τ V ( τ ) ⊗ M τ \bigoplus_\tau V(\tau)\otimes M_\tau ⨁ τ V ( τ ) ⊗ M τ M τ M_\tau M τ τ = ( 0 , 2 ) \tau=(0,2) τ = ( 0 , 2 ) τ = ( 0 , 4 ) \tau=(0,4) τ = ( 0 , 4 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 2 , 0 ) (2,0) ( 2 , 0 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 5 ) (1,5) ( 1 , 5 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) ( 2 , 4 ) (2,4) ( 2 , 4 ) ( 3 , 1 ) (3,1) ( 3 , 1 ) Z Z Z id ⊗ z τ \operatorname{id}\otimes z_\tau id ⊗ z τ z τ z_\tau z τ e ′ e' e ′ e ′ ′ e'' e ′′ z ad ( m k ) = ∑ l ( z τ ) l k ad ( m l ) z z\operatorname{ad}(m_k)=\sum_l(z_\tau)_{lk}\operatorname{ad}(m_l)z z ad ( m k ) = ∑ l ( z τ ) l k ad ( m l ) z ( m k ) (m_k) ( m k ) Q \mathbb{Q} Q ( Z 2 ) 3 (\mathbb{Z}_2)^3 ( Z 2 ) 3 B ⊂ V 2 ′ B\subset V'_2 B ⊂ V 2 ′ b b b x ↦ x ( 1 ) b x\mapsto x(1)b x ↦ x ( 1 ) b V 1 ′ V'_1 V 1 ′ x ( 1 ) b = 0 x(1)b=0 x ( 1 ) b = 0 x x x [ x , y ] ( 2 ) b = x ( 1 ) y ( 1 ) b − y ( 1 ) x ( 1 ) b = 0 [x,y]_{(2)}b=x(1)y(1)b-y(1)x(1)b=0 [ x , y ] ( 2 ) b = x ( 1 ) y ( 1 ) b − y ( 1 ) x ( 1 ) b = 0 V 1 ′ = [ V 1 ′ , V 1 ′ ] V'_1=[V'_1,V'_1] V 1 ′ = [ V 1 ′ , V 1 ′ ] x ( n ) x_{(n)} x ( n ) n > 2 n>2 n > 2 b b b V 2 ′ V'_2 V 2 ′ V 1 ′ V'_1 V 1 ′ V ′ V' V ′ b = 0 b=0 b = 0 z ( b ) z(b) z ( b ) x ( 1 ) z ( b ) = z ( ( z − 1 x ) ( 1 ) b ) x(1)z(b)=z((z^{-1}x)(1)b) x ( 1 ) z ( b ) = z (( z − 1 x ) ( 1 ) b ) B B B s s s c c c c ( λ + μ ) = c ( λ ) c ( μ ) ε ( λ , μ ) ε ( g λ , g μ ) c(\lambda+\mu)=c(\lambda)c(\mu)\varepsilon(\lambda,\mu)\varepsilon(g\lambda,g\mu) c ( λ + μ ) = c ( λ ) c ( μ ) ε ( λ , μ ) ε ( g λ , g μ ) U 1 U_1 U 1 z z z U 2 U_2 U 2 K K K V ′ V' V ′ U 1 U_1 U 1 U 1 ⊕ U 2 U_1\oplus U_2 U 1 ⊕ U 2 U 1 ⊕ z ( U 2 ) U_1\oplus z(U_2) U 1 ⊕ z ( U 2 )
Lemma 10.28. Let S ⊂ U S\subset U S ⊂ U dim Rep ( S ) = dim Rep ( U ) \dim\operatorname{Rep}(S)=\dim\operatorname{Rep}(U) dim Rep ( S ) = dim Rep ( U ) S = U S=U S = U
Proof : Let M 0 = S M_0=S M 0 = S M 1 , … , M r M_1,\ldots,M_r M 1 , … , M r S S S h i h_i h i d i d_i d i U = ⨁ i m i M i U=\bigoplus_i m_iM_i U = ⨁ i m i M i S S S M i M_i M i dim U h i \dim U_{h_i} dim U h i h i h_i h i m 0 = 1 m_0=1 m 0 = 1 U 0 = C 1 U_0=\mathbb{C}1 U 0 = C 1 d i > 0 d_i>0 d i > 0
Write S S S^S S S S U S^U S U S S S τ = i / y \tau=i/y τ = i / y y ↓ 0 y\downarrow0 y ↓ 0
∑ i = 0 r m i d i = lim y ↓ 0 ch U ( i y ) ch S ( i y ) = S 00 U S 00 S = dim Rep ( S ) dim Rep ( U ) = 1. \sum_{i=0}^{r}m_i d_i=\lim_{y\downarrow0}\frac{\operatorname{ch} U(iy)}{\operatorname{ch} S(iy)}=\frac{S^U_{00}}{S^S_{00}}=\sqrt{\frac{\dim\operatorname{Rep}(S)}{\dim\operatorname{Rep}(U)}}=1. i = 0 ∑ r m i d i = y ↓ 0 lim ch S ( i y ) ch U ( i y ) = S 00 S S 00 U = dim Rep ( U ) dim Rep ( S ) = 1. Here d i = S i 0 S / S 00 S d_i=S^S_{i0}/S^S_{00} d i = S i 0 S / S 00 S S 00 S = ( dim Rep ( S ) ) − 1 / 2 S^S_{00}=(\dim\operatorname{Rep}(S))^{-1/2} S 00 S = ( dim Rep ( S ) ) − 1/2 U U U [49 , 19 ] . Since m 0 d 0 = 1 m_0d_0=1 m 0 d 0 = 1 d i > 0 d_i>0 d i > 0 m i m_i m i i > 0 i>0 i > 0 S ⊂ U S\subset U S ⊂ U
Theorem 10.29 (uniqueness in case (II)). Let U U U Rep ( U ) ≃ Z ( H 3 ) \operatorname{Rep}(U)\simeq Z(\mathcal{H}_3) Rep ( U ) ≃ Z ( H 3 ) Theorem 8.5 . Assuming ( H J ) (H_J) ( H J ) U U U V V V
Proof : By Proposition 4.1 , U ⊂ V ′ U\subset V' U ⊂ V ′ Theorem 8.5 , an inner automorphism h h h V ′ V' V ′ U U U g ′ ⊕ g ′ ′ \mathfrak{g}'\oplus\mathfrak{g}'' g ′ ⊕ g ′′ ω \omega ω V E 8 V_{E_8} V E 8 V 1 ′ ⊂ ( V E 8 ) 1 V'_1 \subset(V_{E_8})_1 V 1 ′ ⊂ ( V E 8 ) 1 V ′ = U ⊕ M b V' = U \oplus M_b V ′ = U ⊕ M b V ′ = h ( U ) ⊕ h ( M b ) V' = h(U) \oplus h(M_b) V ′ = h ( U ) ⊕ h ( M b ) Theorem 10.5 hold for h ( U ) h(U) h ( U ) U U U h ( U ) h(U) h ( U ) U 2 U_2 U 2 U U U S S S U 1 ⊕ U 2 U_1 \oplus U_2 U 1 ⊕ U 2 U 2 ⊂ S 2 ⊂ U 2 U_2 \subset S_2 \subset U_2 U 2 ⊂ S 2 ⊂ U 2 S 2 = U 2 S_2 = U_2 S 2 = U 2 ω ∈ U 2 \omega\in U_2 ω ∈ U 2 Theorem 7.1 , the vacuum character of U U U X 0 ( 0 ) X^{(0)}_0 X 0 ( 0 ) γ = 0 \gamma= 0 γ = 0 Proposition 10.27 , S ≅ S ( 9 , 0 ) = ψ ( V ) S \cong S^{(9,0)} = \psi(V) S ≅ S ( 9 , 0 ) = ψ ( V ) Theorem 10.24 ), and, assuming (HJ), S ( 9 , 0 ) S^{(9,0)} S ( 9 , 0 ) Rep ( S ( 9 , 0 ) ) ≃ Z ( H 3 ) \operatorname{Rep}(S^{(9,0)}) \simeq Z(\mathcal{H}_3) Rep ( S ( 9 , 0 ) ) ≃ Z ( H 3 ) X 0 ( 0 ) X^{(0)}_0 X 0 ( 0 ) Theorem 7.1 . Hence ch S = ch U \operatorname{ch} S = \operatorname{ch} U ch S = ch U S ⊂ U S \subset U S ⊂ U U = S ≅ V U = S \cong V U = S ≅ V
Remark 10.30 By Proposition 10.27 , the Galois conjugate is not a new algebra. The isomorphism is the lift of the outer automorphism of e 6 e_6 e 6 A 2 A_2 A 2 S ( 9 , c ) S^{(9,c)} S ( 9 , c ) S ( 10 , 1 − c ) S^{(10,1-c)} S ( 10 , 1 − c ) ψ τ ( U 2 , K ) = U 2 , K \psi_\tau(U_{2,K}) = U_{2,K} ψ τ ( U 2 , K ) = U 2 , K 10.27 ). The isometry part of an element of Z Z Z E 6 E_6 E 6 A 2 A_2 A 2 B B B s s s U 1 U_1 U 1 B B B s s s θ χ \theta_\chi θ χ S ( 9 , 0 ) S^{(9,0)} S ( 9 , 0 ) Gal ( K / Q ) \operatorname{Gal}(K/\mathbb{Q}) Gal ( K / Q ) U = S U = S U = S Theorem 10.29 . Once the inclusion and the positive strong rationality of S S S Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) Theorem 10.38 .
Proposition 10.31 Let S = S ( 9 , 0 ) S = S^{(9,0)} S = S ( 9 , 0 ) C = Com S ( A ) C = \operatorname{Com}_S(A) C = Com S ( A ) f 1 f_1 f 1 f 2 f_2 f 2 9 9 9 f 1 + f 2 = e f_1 + f_2 = e f 1 + f 2 = e ω 1 = 2 f 1 \omega_1 = 2f_1 ω 1 = 2 f 1 ω 2 = 2 f 2 \omega_2 = 2f_2 ω 2 = 2 f 2 182 95 \frac{182}{95} 95 182 32 19 \frac{32}{19} 19 32 Theorem 10.5 . In the basis b 0 = e , b 1 , … , b 4 b_0 = e, b_1, \ldots, b_4 b 0 = e , b 1 , … , b 4 B B B
f 2 = 80 171 b 0 + 13 − 13 114 b 1 − 25 + 5 13 228 b 2 + 7 − 13 152 b 3 + 4 13 19 b 4 , f 1 = e − f 2 . f_2 = \frac{80}{171}b_0 + \frac{\sqrt{13}-13}{114}b_1 - \frac{25+5\sqrt{13}}{228}b_2 + \frac{7-\sqrt{13}}{152}b_3 + \frac{4\sqrt{13}}{19}b_4,\qquad f_1 = e - f_2. f 2 = 171 80 b 0 + 114 13 − 13 b 1 − 228 25 + 5 13 b 2 + 152 7 − 13 b 3 + 19 4 13 b 4 , f 1 = e − f 2 . Write L i ( n ) = ( ω i ) ( n + 1 ) L_i(n) = (\omega_i)_{(n+1)} L i ( n ) = ( ω i ) ( n + 1 )
dim C 1 = 0 \dim C_1 = 0 dim C 1 = 0 dim C 2 = dim C 3 = 2 \dim C_2 = \dim C_3 = 2 dim C 2 = dim C 3 = 2 dim C 4 = 7 \dim C_4 = 7 dim C 4 = 7
On C 4 C_4 C 4 L 1 ( 0 ) L_1(0) L 1 ( 0 ) L 2 ( 0 ) L_2(0) L 2 ( 0 ) L 1 ( 0 ) + L 2 ( 0 ) = 4 L_1(0) + L_2(0) = 4 L 1 ( 0 ) + L 2 ( 0 ) = 4 C 4 C_4 C 4 ( h 1 , h 2 ) (h_1,h_2) ( h 1 , h 2 )
( 4 , 0 ) 2 , ( 0 , 4 ) 2 , ( 2 , 2 ) , ( 51 19 , 25 19 ) , ( 26 19 , 50 19 ) . (4,0)^2,\quad(0,4)^2,\quad(2,2),\quad\left(\frac{51}{19},\frac{25}{19}\right),\quad\left(\frac{26}{19},\frac{50}{19}\right). ( 4 , 0 ) 2 , ( 0 , 4 ) 2 , ( 2 , 2 ) , ( 19 51 , 19 25 ) , ( 19 26 , 19 50 ) . The first five eigenvectors can be taken to be L 1 ( − 4 ) 1 L_1(-4)1 L 1 ( − 4 ) 1 L 1 ( − 2 ) 2 1 L_1(-2)^2 1 L 1 ( − 2 ) 2 1 L 2 ( − 4 ) 1 L_2(-4)1 L 2 ( − 4 ) 1 L 2 ( − 2 ) 2 1 L_2(-2)^2 1 L 2 ( − 2 ) 2 1 L 1 ( − 2 ) L 2 ( − 2 ) 1 L_1(-2)L_2(-2)1 L 1 ( − 2 ) L 2 ( − 2 ) 1
The two eigenvectors of non-integral bi-weights are primary for Vir 1 ⊗ Vir 2 \operatorname{Vir}_1 \otimes\operatorname{Vir}_2 Vir 1 ⊗ Vir 2 L i ( 1 ) L_i(1) L i ( 1 ) L i ( 2 ) L_i(2) L i ( 2 ) i = 1 , 2 i = 1, 2 i = 1 , 2
Proof: We have C ^ 1 = 0 \widehat{C}_1 = 0 C 1 = 0 Lemma 10.1 ) and C 2 C_2 C 2 9 9 9 U 2 U_2 U 2 Theorem 10.5 . Proposition 10.17 establishes the finite certificate condition, so Theorem 10.21 gives N 4 ′ = S 4 N'_4 = S_4 N 4 ′ = S 4 K \mathbb{K} K p 3 p_3 p 3 S 4 S_4 S 4 Theorem 10.21 . Since C C C U 1 U_1 U 1 C 4 C_4 C 4 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ S 4 S_4 S 4 y ( k ) y_{(k)} y ( k ) y ∈ U 1 y \in U_1 y ∈ U 1 0 ≤ k ≤ 4 0 \leq k \leq4 0 ≤ k ≤ 4 dim C 4 ≤ 280 − 273 = 7 \dim C_4 \leq280 - 273 = 7 dim C 4 ≤ 280 − 273 = 7 w 1 , … , w 7 w_1,\ldots,w_7 w 1 , … , w 7 y ( k ) w j = 0 y_{(k)}w_j = 0 y ( k ) w j = 0 w j w_j w j dim C 4 = 7 \dim C_4 = 7 dim C 4 = 7 p 3 p_3 p 3 dim C 3 ≤ 2 \dim C_3 \leq2 dim C 3 ≤ 2 L 1 ( − 3 ) 1 L_1(-3)1 L 1 ( − 3 ) 1 L 2 ( − 3 ) 1 L_2(-3)1 L 2 ( − 3 ) 1 C 3 C_3 C 3 dim C 3 = 2 \dim C_3 = 2 dim C 3 = 2
The images L i ( 0 ) w j L_i(0)w_j L i ( 0 ) w j ( w j ) (w_j) ( w j ) 7 × 7 7 \times7 7 × 7 K \mathbb{K} K L 1 ( 0 ) L 2 ( 0 ) = L 2 ( 0 ) L 1 ( 0 ) L_1(0)L_2(0) = L_2(0)L_1(0) L 1 ( 0 ) L 2 ( 0 ) = L 2 ( 0 ) L 1 ( 0 ) L 1 ( 0 ) + L 2 ( 0 ) = 4 L_1(0) + L_2(0) = 4 L 1 ( 0 ) + L 2 ( 0 ) = 4 ∏ λ ( L 2 ( 0 ) − λ ) = 0 \prod_{\lambda}(L_2(0)-\lambda)=0 ∏ λ ( L 2 ( 0 ) − λ ) = 0 λ ∈ { 0 , 2 , 4 , 25 19 , 50 19 } \lambda\in\{0,2,4,\frac{25}{19},\frac{50}{19}\} λ ∈ { 0 , 2 , 4 , 19 25 , 19 50 } L 2 ( 0 ) L_2(0) L 2 ( 0 ) 25 19 \frac{25}{19} 19 25 50 19 \frac{50}{19} 19 50 C C C L ( − 4 ) 1 L(-4)1 L ( − 4 ) 1 L ( − 2 ) 2 1 L(-2)^2 1 L ( − 2 ) 2 1 c ∉ { 0 , − 22 5 } c \notin\{0,-\frac{22}{5}\} c ∈ / { 0 , − 5 22 } c 2 ( 5 c + 22 ) 2 \frac{c^2(5c+22)}{2} 2 c 2 ( 5 c + 22 ) L 1 ( − 2 ) L 2 ( − 2 ) 1 L_1(-2)L_2(-2)1 L 1 ( − 2 ) L 2 ( − 2 ) 1 ( 2 , 2 ) (2,2) ( 2 , 2 ) C 4 C_4 C 4 L i ( n ) L_i(n) L i ( n ) n ≥ 1 n \geq1 n ≥ 1 C 4 − n C_{4-n} C 4 − n
The 280 independent words in the Cartan block ( 0 , 0 ) (0,0) ( 0 , 0 ) N 4 ′ N'_4 N 4 ′ Proposition 10.17 . The supplementary material contains the seven kernel vectors, all 210 current-mode annihilations, the two Virasoro matrices and both primary vectors. A separate rational-matrix verification checks the matrix identities, spectral multiplicities and independence of the seven vectors without using the vertex-algebra implementation. The vertex-product identities are verified from the lattice formulas. The computational data order the Virasoro vectors by increasing central charge, so their indices are opposite to those of this proposition.
Corollary 10.32 There are no two nontrivial vertex operator algebras V 1 V^1 V 1 V 2 V^2 V 2 C ≅ V 1 ⊗ V 2 C \cong V^1 \otimes V^2 C ≅ V 1 ⊗ V 2 ω = ω 1 ⊗ 1 + 1 ⊗ ω 2 \omega= \omega^1 \otimes1 + 1 \otimes\omega^2 ω = ω 1 ⊗ 1 + 1 ⊗ ω 2 C C C K ( s l 2 , 17 ) Z 2 ⊗ Com ( L s l 2 ( 17 , 0 ) , L s l 2 ( 4 , 0 ) ⊗ L s l 2 ( 13 , 0 ) ) K(\mathfrak{sl}_2,17)_{\mathbb{Z}_2} \otimes\operatorname{Com}(L_{\mathfrak{sl}_2}(17,0), L_{\mathfrak{sl}_2}(4,0) \otimes L_{\mathfrak{sl}_2}(13,0)) K ( sl 2 , 17 ) Z 2 ⊗ Com ( L sl 2 ( 17 , 0 ) , L sl 2 ( 4 , 0 ) ⊗ L sl 2 ( 13 , 0 )) Com C ( ⟨ ω 2 ⟩ ) \operatorname{Com}_C(\langle\omega_2\rangle) Com C (⟨ ω 2 ⟩) Com C ( ⟨ ω 1 ⟩ ) \operatorname{Com}_C(\langle\omega_1\rangle) Com C (⟨ ω 1 ⟩) ω 1 \omega_1 ω 1 ω 2 \omega_2 ω 2 1 , 0 , 1 , 1 , 2 1,0,1,1,2 1 , 0 , 1 , 1 , 2
Proof : Suppose C ≅ V 1 ⊗ V 2 C \cong V^1 \otimes V^2 C ≅ V 1 ⊗ V 2 C 1 = 0 C_1 = 0 C 1 = 0 V 1 1 = V 1 2 = 0 V^1_1 = V^2_1 = 0 V 1 1 = V 1 2 = 0 C 2 = C ω 1 ⊗ 1 ⊕ C 1 ⊗ ω 2 C_2 = \mathbb{C}\omega^1 \otimes1 \oplus\mathbb{C}1 \otimes\omega^2 C 2 = C ω 1 ⊗ 1 ⊕ C 1 ⊗ ω 2 ω 1 , ω 2 \omega^1,\omega^2 ω 1 , ω 2 L ( 0 ) = 0 L(0)=0 L ( 0 ) = 0 1 2 ω 1 ⊗ 1 \frac{1}{2}\omega^1 \otimes1 2 1 ω 1 ⊗ 1 1 2 1 ⊗ ω 2 \frac{1}{2}1 \otimes\omega^2 2 1 1 ⊗ ω 2 C 2 C_2 C 2 e e e C 2 = C f 1 ⊕ C f 2 C_2 = \mathbb{C}f_1 \oplus\mathbb{C}f_2 C 2 = C f 1 ⊕ C f 2 0 0 0 f 1 f_1 f 1 f 2 f_2 f 2 e e e { ω 1 ⊗ 1 , 1 ⊗ ω 2 } = { ω 1 , ω 2 } \{\omega^1 \otimes1,1 \otimes\omega^2\} = \{\omega_1,\omega_2\} { ω 1 ⊗ 1 , 1 ⊗ ω 2 } = { ω 1 , ω 2 } L V 1 ( 0 ) ⊗ 1 L_{V^1}(0)\otimes1 L V 1 ( 0 ) ⊗ 1 V 1 ⊗ V 2 V^1 \otimes V^2 V 1 ⊗ V 2 L 1 ( 0 ) L_1(0) L 1 ( 0 ) 51 19 \frac{51}{19} 19 51 C 4 C_4 C 4 C n C_n C n ω 2 \omega_2 ω 2 L 2 ( 0 ) L_2(0) L 2 ( 0 ) Proposition 10.31 and C 3 = C L 1 ( − 3 ) 1 ⊕ C L 2 ( − 3 ) 1 C_3 = \mathbb{C}L_1(-3)1 \oplus\mathbb{C}L_2(-3)1 C 3 = C L 1 ( − 3 ) 1 ⊕ C L 2 ( − 3 ) 1 L 2 ( 0 ) L_2(0) L 2 ( 0 ) C n C_n C n n ≤ 4 n \leq4 n ≤ 4 1 1 1 ω 1 \omega_1 ω 1
:::
Remark 10.33 The central charges 182 45 \frac{182}{45} 45 182 32 19 \frac{32}{19} 19 32
D = Com ( L s l 2 ( 17 , 0 ) , L s l 2 ( 4 , 0 ) ⊗ L s l 2 ( 13 , 0 ) ) D = \operatorname{Com}(L_{\mathfrak{sl}_2}(17,0),L_{\mathfrak{sl}_2}(4,0) \otimes L_{\mathfrak{sl}_2}(13,0)) D = Com ( L sl 2 ( 17 , 0 ) , L sl 2 ( 4 , 0 ) ⊗ L sl 2 ( 13 , 0 )) and of the parafermion algebra K ( s l 2 , 17 ) K(\mathfrak{sl}_2,17) K ( sl 2 , 17 ) 1 , 0 , 2 , 2 , 7 1,0,2,2,7 1 , 0 , 2 , 2 , 7 C C C K ( s l 2 , 17 ) Z 2 ⊗ D K(\mathfrak{sl}_2,17)_{\mathbb{Z}_2} \otimes D K ( sl 2 , 17 ) Z 2 ⊗ D ( 4 , 0 ) (4,0) ( 4 , 0 ) ( 0 , 4 ) (0,4) ( 0 , 4 ) Corollary 10.32 shows that this agreement does not come from an isomorphism of the two algebras. Modulo Z \mathbb{Z} Z 25 19 \frac{25}{19} 19 25 50 19 \frac{50}{19} 19 50 h λ = λ ( λ + 2 ) 76 h_\lambda=\frac{\lambda(\lambda+2)}{76} h λ = 76 λ ( λ + 2 ) SU ( 2 ) 17 \operatorname{SU}(2)_{17} SU ( 2 ) 17 λ = 4 , 6 \lambda=4,6 λ = 4 , 6 x 19 \frac{x}{19} 19 x Z \mathbb{Z} Z 8 19 \frac{8}{19} 19 8 68 57 \frac{68}{57} 57 68 Vir 2 \operatorname{Vir}_2 Vir 2 Theorem 10.5 are not, so we do not draw any conclusion from it.
Case (I): uniqueness
Proposition 10.34. Assume the hypotheses of Theorem 8.5 in case (I). Then C 2 C_2 C 2 2400 1271 \frac{2400}{1271} 1271 2400 70 31 \frac{70}{31} 31 70 [67 Section 1] .
Proof: There is a direct exact exclusion that avoids choosing eigenvectors. For candidates 0, 3 and 4, use the primary space of type ( 1 , 13 ) (1,13) ( 1 , 13 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) X X X ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) P P P Y j Y_j Y j B ∩ C 2 ⊥ B\cap C_2^\perp B ∩ C 2 ⊥ u u u
( X u ) i u j − ( X u ) j u i = 0 , u T P Y j u = 0 , e ( 2 ) u = ( x C ) ( 2 ) u = 0 , u T P u ≠ 0. (Xu)_i u_j-(Xu)_j u_i=0,\qquad u^{\mathsf{T}}PY_j u=0,\qquad e_{(2)}u=(x_C)_{(2)}u=0,\qquad u^{\mathsf{T}}Pu\ne0. ( X u ) i u j − ( X u ) j u i = 0 , u T P Y j u = 0 , e ( 2 ) u = ( x C ) ( 2 ) u = 0 , u T P u = 0. In each projective chart set the first nonzero coordinate of u u u z u T P u − 1 = 0 zu^{\mathsf{T}}Pu-1=0 z u T P u − 1 = 0 Q \mathbb{Q} Q
A separate verification follows from the method of Steps 1 and 2 of the proof of Theorem 10.5 , applied in case (I): the subalgebras 0 and 3 are excluded by certified enclosures, 1 by the type ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 0 , 12 ) (0,12) ( 0 , 12 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 7 ) (1,7) ( 1 , 7 ) ( 1 , 13 ) (1,13) ( 1 , 13 )
Proposition 10.35. In case (I), C 2 C_2 C 2
Proof: Suppose that C 2 C_2 C 2 4 5 \frac{4}{5} 5 4 686 205 \frac{686}{205} 205 686 Π 1 , 13 \Pi_{1,13} Π 1 , 13 ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) Π 1 , 13 ∩ Ker e ( 2 ) ∩ Ker ( x C ) ( 2 ) \Pi_{1,13}\cap\operatorname{Ker}e_{(2)}\cap\operatorname{Ker}(x_C)_{(2)} Π 1 , 13 ∩ Ker e ( 2 ) ∩ Ker ( x C ) ( 2 ) 496 425 \frac{496}{425} 425 496 P 1 , 13 ( u , y ( 1 ) u ) P_{1,13}(u,y_{(1)}u) P 1 , 13 ( u , y ( 1 ) u ) − 184 425 -\frac{184}{425} − 425 184 E 1 , 13 E_{1,13} E 1 , 13 h 1 = 0 h_1=0 h 1 = 0 4 5 \frac{4}{5} 5 4 Π 1 , 1 \Pi_{1,1} Π 1 , 1 − 568 425 -\frac{568}{425} − 425 568 E 1 , 1 E_{1,1} E 1 , 1 h 1 = 2 5 h_1=\frac{2}{5} h 1 = 5 2 p 1 , 13 p_{1,13} p 1 , 13 p 1 , 1 p_{1,1} p 1 , 1 E 1 , 13 E_{1,13} E 1 , 13 E 1 , 1 E_{1,1} E 1 , 1 Q ( − 19758 ) \mathbb{Q}(\sqrt{-19758}) Q ( − 19758 ) Q ( − 4403 ) \mathbb{Q}(\sqrt{-4403}) Q ( − 4403 ) u u u p 1 , 13 p_{1,13} p 1 , 13 w w w p 1 , 1 p_{1,1} p 1 , 1 u ( 1 ) w ′ ∈ U 2 u_{(1)}w'\in U_2 u ( 1 ) w ′ ∈ U 2 w ′ w' w ′ g ′ ⊕ g ′ ′ \mathfrak{g}'\oplus\mathfrak{g}'' g ′ ⊕ g ′′ w w w x ⊗ y ↦ x ( 1 ) y x\otimes y\mapsto x_{(1)}y x ⊗ y ↦ x ( 1 ) y U 1 U_1 U 1 U 1 U_1 U 1 V ( 1 , 13 ) ⊗ V ( 1 , 1 ) = V ( 0 , 12 ) ⊕ V ( 0 , 14 ) ⊕ V ( 2 , 12 ) ⊕ V ( 2 , 14 ) V(1,13)\otimes V(1,1)=V(0,12)\oplus V(0,14)\oplus V(2,12)\oplus V(2,14) V ( 1 , 13 ) ⊗ V ( 1 , 1 ) = V ( 0 , 12 ) ⊕ V ( 0 , 14 ) ⊕ V ( 2 , 12 ) ⊕ V ( 2 , 14 ) ( 0 , 12 ) (0,12) ( 0 , 12 ) U 2 U_2 U 2 Theorem 8.5 ). Hence C ′ z = 0 = C ′ ′ z = 84 z C' z=0=C''z=84z C ′ z = 0 = C ′′ z = 84 z z = u ( 1 ) w ′ z=u_{(1)}w' z = u ( 1 ) w ′ C ′ , C ′ ′ C',C'' C ′ , C ′′ g ′ \mathfrak{g}' g ′ g ′ ′ \mathfrak{g}'' g ′′ C = h 2 2 + h + 2 f e C=\frac{h^2}{2}+h+2fe C = 2 h 2 + h + 2 f e a ( a + 2 ) 2 \frac{a(a+2)}{2} 2 a ( a + 2 ) V ( a ) V(a) V ( a ) ± 3 − 19758 / 148 \pm3\sqrt{-19758}/148 ± 3 − 19758 /148 ± 2 − 4403 / 37 \pm2\sqrt{-4403}/37 ± 2 − 4403 /37 w ′ = f ′ ( 0 ) w w'=f'(0)w w ′ = f ′ ( 0 ) w w w w f ′ ′ ( 0 ) w f''(0)w f ′′ ( 0 ) w f ′ ( 0 ) w f'(0)w f ′ ( 0 ) w f ′ ( 0 ) f ′ ′ ( 0 ) w f'(0)f''(0)w f ′ ( 0 ) f ′′ ( 0 ) w z z z ( 0 , 14 ) (0,14) ( 0 , 14 ) U 2 U_2 U 2 z = 0 z=0 z = 0 e ′ ′ ( 0 ) z = 0 e''(0)z=0 e ′′ ( 0 ) z = 0 C ′ ′ z = 112 z C''z=112z C ′′ z = 112 z ( C ′ ′ − 84 ) z = 28 z (C''-84)z=28z ( C ′′ − 84 ) z = 28 z ( C ′ ′ − 84 ) z ≠ 0 (C''-84)z\ne0 ( C ′′ − 84 ) z = 0 ( C ′ ′ − 84 ) u ( 1 ) w ′ (C''-84)u_{(1)}w' ( C ′′ − 84 ) u ( 1 ) w ′ e α e^\alpha e α α = ( 1 , 2 , 3 , 4 , 3 , 2 , 0 , 0 ) \alpha=(1,2,3,4,3,2,0,0) α = ( 1 , 2 , 3 , 4 , 3 , 2 , 0 , 0 ) ± 224 i 925 ( 5 4403 − 19758 ) \pm\frac{224i}{925}(5\sqrt{4403}-\sqrt{19758}) ± 925 224 i ( 5 4403 − 19758 ) ± 224 i 925 ( 5 4403 + 19758 ) \pm\frac{224i}{925}(5\sqrt{4403}+\sqrt{19758}) ± 925 224 i ( 5 4403 + 19758 ) 25 ⋅ 4403 − 19758 = 90317 > 0 25\cdot4403-19758=90317>0 25 ⋅ 4403 − 19758 = 90317 > 0 5 4403 − 19758 > 0 5\sqrt{4403}-\sqrt{19758}>0 5 4403 − 19758 > 0
The same test applied to the subalgebra 5 5 5 ( 1 , 13 ) (1,13) ( 1 , 13 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) U 2 U_2 U 2
We now compare with the data of [66 ] . There, U R = ⟨ A , e , f , v ⟩ ⊂ V E 6 ⊕ A 2 U^{\mathrm R}=\langle A,e,f,v\rangle\subset V_{E_6\oplus A_2} U R = ⟨ A , e , f , v ⟩ ⊂ V E 6 ⊕ A 2 e , f ∈ Com ( A ) 2 e,f\in\operatorname{Com}(A)_2 e , f ∈ Com ( A ) 2 70 31 , 2400 1271 \frac{70}{31},\frac{2400}{1271} 31 70 , 1271 2400 v v v ( 1 , 13 ) (1,13) ( 1 , 13 ) [67 Section 1] . The coordinates are specified in the certificates accompanying [67 ] , and the same vectors are given in [66 Section “The explicit candidate”] , in the bases fixed in [67 ] .
Proposition 10.36 There is an isomorphism of vertex operator algebras ψ R \psi_{\mathrm R} ψ R [66 ] onto V ′ V' V ′ [66 ] onto e ′ , f ′ , H ′ , e ′ ′ , f ′ ′ , H ′ ′ e',f',H',e'',f'',H'' e ′ , f ′ , H ′ , e ′′ , f ′′ , H ′′ e e e f f f 5 5 5 Table 3 , which they span, and v v v ( 1 , 13 ) (1,13) ( 1 , 13 ) 5 5 5
Proof : Exact computation, as for Theorem 10.24 . In [67 ] the coordinates 1, 2 are those of A 2 A_2 A 2 E 6 E_6 E 6 ϵ ( α i , α j ) \epsilon(\alpha_i,\alpha_j) ϵ ( α i , α j ) i > j i>j i > j K K K w = s 3 s 4 s 1 s 3 ∈ W ( E 6 ) w=s_3s_4s_1s_3\in W(E_6) w = s 3 s 4 s 1 s 3 ∈ W ( E 6 ) [67 ] onto H ′ , H ′ ′ H',H'' H ′ , H ′′ Theorem 10.24 , and the torus character with values 5 , − 3 2 , − 2 3 , 1 , − 2 3 3 , 4 , 1 , 1 5,-\frac{\sqrt{3}}{2},-2\sqrt{3},1,-\frac{2\sqrt{3}}{3},4,1,1 5 , − 2 3 , − 2 3 , 1 , − 3 2 3 , 4 , 1 , 1 ω V ′ \omega_{V'} ω V ′ A A A B B B ψ R ( e ) \psi_{\mathrm R}(e) ψ R ( e ) ψ R ( f ) \psi_{\mathrm R}(f) ψ R ( f ) Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 ) 5 5 5 Theorem 10.5 , the type ( 1 , 13 ) (1,13) ( 1 , 13 ) 5 5 5 E E E − 254 1445 + 508 1445 13 -\frac{254}{1445}+\frac{508}{1445}\sqrt{13} − 1445 254 + 1445 508 13 ( x C ) ( 1 ) (x_C)_{(1)} ( x C ) ( 1 ) Π 1 , 13 \Pi_{1,13} Π 1 , 13 E E E 1 30 ∓ − 39 130 \frac{1}{30}\mp\frac{\sqrt{-39}}{130} 30 1 ∓ 130 − 39 E E E Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 ) ψ R ( v ) \psi_{\mathrm R}(v) ψ R ( v ) E E E
Proposition 10.37 In case (I), the group Z Z Z V ′ V' V ′ U 1 U_1 U 1 ( Z 2 ) 3 (\mathbb{Z}_2)^3 ( Z 2 ) 3 B B B 5 5 5 z A z_A z A A 2 A_2 A 2 z 1 z_1 z 1 E 6 E_6 E 6 A 2 A_2 A 2 ( c 1 , … , c 8 ) = ( 4 5 , − 1 3 , 1 3 , 3 , 5 4 , 1 , 1 ) (c_1,\ldots,c_8)=(\frac45,-\frac13,\frac13,3,\frac54,1,1) ( c 1 , … , c 8 ) = ( 5 4 , − 3 1 , 3 1 , 3 , 4 5 , 1 , 1 ) Proposition 10.27 ), acts trivially on B B B ( 1 , 13 ) (1,13) ( 1 , 13 ) 5 5 5
Proof : The upper bound ∣ Z ∣ ≤ 8 |Z|\le8 ∣ Z ∣ ≤ 8 Lemma 10.26 . The explicitly given lifts z 1 , z A z_1,z_A z 1 , z A ρ \rho ρ ρ \rho ρ (26) . The supplementary material includes the exact verification. Thus Z ≅ ( Z 2 ) 3 Z \cong(\mathbb{Z}_2)^3 Z ≅ ( Z 2 ) 3
An independent polynomial computation gives the same centralizer, and its action on B B B ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 0 , 4 ) (0,4) ( 0 , 4 ) ψ R ( v ) \psi_R(v) ψ R ( v ) E E E
Theorem 10.38 (uniqueness in case (I)) Let U U U Theorem 8.5 . Assuming ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U U U U R U^{\mathrm{R}} U R
Proof : As in the proof of Theorem 10.29, we may assume U ⊂ V ′ U \subset V' U ⊂ V ′ U 1 = g ′ ⊕ g ′ ′ U_1=\mathfrak{g}'\oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ C 2 = U 2 ∩ Com ( A ) C_2=U_2\cap\operatorname{Com}(A) C 2 = U 2 ∩ Com ( A ) U U U z A ( U ) z_A(U) z A ( U ) ψ R ( e ) , ψ R ( f ) ∈ C 2 ⊂ U \psi_R(e),\psi_R(f)\in C_2\subset U ψ R ( e ) , ψ R ( f ) ∈ C 2 ⊂ U p 1 , 13 p_{1,13} p 1 , 13 U 2 U_2 U 2 E E E U U U z 1 ( U ) z_1(U) z 1 ( U ) U 1 U_1 U 1 C 2 C_2 C 2 ψ R ( v ) \psi_R(v) ψ R ( v ) U ⊃ ψ R ( U R ) U\supset\psi_R(U^{\mathrm{R}}) U ⊃ ψ R ( U R ) U U U ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U R U^{\mathrm{R}} U R X 0 ( 0 ) X_0^{(0)} X 0 ( 0 ) U = ψ R ( U R ) U=\psi_R(U^{\mathrm{R}}) U = ψ R ( U R )
Once C 2 C_2 C 2 ( 1 , 13 ) (1,13) ( 1 , 13 ) U R U^{\mathrm{R}} U R A A A e e e f f f v v v ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 0 , 12 ) (0,12) ( 0 , 12 ) ( 1 , 7 ) (1,7) ( 1 , 7 )
Proposition 10.39 The algebra ψ R ( U R ) ⊂ V ′ \psi_R(U^{\mathrm{R}})\subset V' ψ R ( U R ) ⊂ V ′
Proof : The coordinates of ψ R ( e ) \psi_R(e) ψ R ( e ) ψ R ( f ) \psi_R(f) ψ R ( f ) ψ R ( v ) \psi_R(v) ψ R ( v ) V ′ V' V ′ Q ( 13 , 3 , i ) \mathbb{Q}(\sqrt{13},\sqrt{3},i) Q ( 13 , 3 , i ) V ′ V' V ′ p p p − 1 -1 − 1 p p p p \mathfrak{p} p Q ( 13 , 3 , i ) \mathbb{Q}(\sqrt{13},\sqrt{3},i) Q ( 13 , 3 , i ) p p p p \mathfrak{p} p F p \mathbb{F}_p F p p \mathfrak{p} p p p p p \mathfrak{p} p p \mathfrak{p} p p p p e e e f f f v v v U 1 U_1 U 1 W 2 ⊂ V 2 ′ W_2\subset V'_2 W 2 ⊂ V 2 ′ ( A ) 2 (A)_2 ( A ) 2 e e e f f f v v v b ( 1 ) w b_{(1)}w b ( 1 ) w U 1 U_1 U 1 U 2 U_2 U 2 Table 2 , and W 2 ∩ C ^ = span ( e , f ) W_2\cap\widehat{C}=\operatorname{span}(e,f) W 2 ∩ C = span ( e , f ) W 2 W_2 W 2 ( A ) 2 (A)_2 ( A ) 2 y ( 1 ) S 4 ⊂ S 3 y_{(1)}S_4\subset S_3 y ( 1 ) S 4 ⊂ S 3 y ( 2 ) S 4 ⊂ W 2 y_{(2)}S_4\subset W_2 y ( 2 ) S 4 ⊂ W 2 y ∈ U 1 y\in U_1 y ∈ U 1 z 1 z_1 z 1 ( 1 , 13 ) (1,13) ( 1 , 13 ) V 2 ′ V'_2 V 2 ′ V 3 ′ V'_3 V 3 ′ ψ R ( U R ) \psi_R(U^{\mathrm{R}}) ψ R ( U R ) X 0 ( 0 ) X_0^{(0)} X 0 ( 0 ) [66 ] . This is an independent check of the graded dimensions in [66 ] up to q 11 / 3 q^{11/3} q 11/3 ch ψ R ( U R ) = X 0 ( 0 ) \operatorname{ch}\psi_R(U^{\mathrm{R}})=X_0^{(0)} ch ψ R ( U R ) = X 0 ( 0 ) Theorem 12.4 ).
Classification
Theorem 10.40 Let U U U R ) a n d ( H _{\mathrm{R}}) and (H R ) an d ( H U U U U R U^{\mathrm{R}} U R U 1 U_1 U 1 ( 1 , 39 ) (1,39) ( 1 , 39 ) V V V U 1 U_1 U 1 ( 3 , 13 ) (3,13) ( 3 , 13 )
Proof. By Theorem 8.5 , U U U Theorems 10.38 and 10.29 . The two algebras are not isomorphic, since the levels of the simple ideals of the weight-one Lie algebra are invariants.
The algebra announced by Huang [50 ] has currents of levels ( 1 , 39 ) (1,39) ( 1 , 39 ) [5 ] , Theorem 7.9, Corollary 7.10 and Theorem 7.13, using [20 ] , Proposition 4.1 and Theorem 7.3, this algebra and U R U^{\mathrm{R}} U R S ♮ S^{\natural} S ♮ S S S Theorem 12.3 ). In particular (HR _{\mathrm{R}} R Theorems 10.38 and 10.40 hold unconditionally.
Besides (HR _{\mathrm{R}} R J _{\mathrm{J}} J Theorem 10.40 uses the following results of this paper, with the computational dependencies recorded in Appendix § 13 : Theorems 7.1 and 8.5 , the determination of C 2 C_2 C 2 Theorem 10.5 , Propositions 10.34 and 10.35 ), and the explicit automorphisms and identifications (Theorem 10.24 , Propositions 10.12, 10.27 , 10.36 and 10.37 ). In the conditional uniqueness proofs, (HR _{\mathrm{R}} R J _{\mathrm{J}} J ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) ψ ( V ) \psi(V) ψ ( V ) V ′ V' V ′ X 0 ( 0 ) X_0^{(0)} X 0 ( 0 ) Theorem 12.4 .
Proposition 10.41 The algebras S ( 9 , 0 ) S^{(9,0)} S ( 9 , 0 ) ψ R ( U R ) = ⟨ A , e , f , v ⟩ \psi_{\mathrm{R}}(U^{\mathrm{R}})=\langle A,e,f,v\rangle ψ R ( U R ) = ⟨ A , e , f , v ⟩ U Q U_{\mathbb{Q}} U Q Q \mathbb{Q} Q 1 , ω ∈ U Q 1,\omega\in U_{\mathbb{Q}} 1 , ω ∈ U Q U Q ⊗ Q C = U U_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{C}=U U Q ⊗ Q C = U e e e f f f v v v ψ R ( e ) \psi_{\mathrm{R}}(e) ψ R ( e ) ψ R ( f ) \psi_{\mathrm{R}}(f) ψ R ( f ) ψ R ( v ) \psi_{\mathrm{R}}(v) ψ R ( v ) V ′ V' V ′ Proposition 10.36 ). The coordinates of v v v Q ( 13 , 3 , i ) \mathbb{Q}(\sqrt{13},\sqrt{3},i) Q ( 13 , 3 , i ) G = Gal ( K / Q ) = { 1 , σ , τ , σ τ } G=\operatorname{Gal}(K/\mathbb{Q})=\{1,\sigma,\tau,\sigma\tau\} G = Gal ( K / Q ) = { 1 , σ , τ , σ τ } τ \tau τ g ~ = id ⊗ g \widetilde{g}=\operatorname{id}\otimes g g = id ⊗ g V K ′ = V Q ′ ⊗ K V'_K=V'_{\mathbb{Q}}\otimes K V K ′ = V Q ′ ⊗ K
Case (II). Let z z z Proposition 10.27 , part 3, and z ′ z' z ′ A 2 A_2 A 2 z , z ′ ∈ Z z,z'\in Z z , z ′ ∈ Z Q \mathbb{Q} Q z 2 = z ′ 2 = 1 z^2=z'^2=1 z 2 = z ′2 = 1 z z ′ = z ′ z zz'=z'z z z ′ = z ′ z ψ σ = z σ ~ \psi_\sigma=z\widetilde{\sigma} ψ σ = z σ ψ τ = z z ′ τ ~ \psi_\tau=zz'\widetilde{\tau} ψ τ = z z ′ τ U 2 , K = U 2 ∩ V K ′ U_{2,K}=U_2\cap V'_K U 2 , K = U 2 ∩ V K ′ f 1 , f 2 f_1,f_2 f 1 , f 2 C 2 C_2 C 2 ψ σ \psi_\sigma ψ σ ψ τ \psi_\tau ψ τ
Case (I). Let z 1 z_1 z 1 z A z_A z A U 1 U_1 U 1 Proposition 10.37 . Their product z 1 z A z_1z_A z 1 z A E 6 E_6 E 6 Q \mathbb{Q} Q z 1 2 = z A 2 = 1 z_1^2=z_A^2=1 z 1 2 = z A 2 = 1 ψ τ = z 1 τ ~ \psi_\tau=z_1\widetilde{\tau} ψ τ = z 1 τ ψ σ = z 1 z A σ ~ \psi_\sigma=z_1z_A\widetilde{\sigma} ψ σ = z 1 z A σ e e e f f f [66 ] ). In addition, ψ σ ( v ′ ) = i v \psi_\sigma(v')=iv ψ σ ( v ′ ) = i v K \mathbb{K} K ψ τ ( v ′ ) = ψ σ ( v ′ ) = − v ′ \psi_\tau(v')=\psi_\sigma(v')=-v' ψ τ ( v ′ ) = ψ σ ( v ′ ) = − v ′ z A z_A z A z 1 z A z_1z_A z 1 z A ψ σ ( v ′ ) \psi_\sigma(v') ψ σ ( v ′ ) v ′ v' v ′
Put U 1 , K = U 1 ∩ V K ′ U_{1,K}=U_1\cap V'_K U 1 , K = U 1 ∩ V K ′ U K ⊂ V K ′ U_K\subset V'_K U K ⊂ V K ′ K \mathbb{K} K a ( n 1 ) 1 ⋯ a ( n r ) r 1 a^{1}_{(n_1)}\cdots a^{r}_{(n_r)}1 a ( n 1 ) 1 ⋯ a ( n r ) r 1 K \mathbb{K} K U 1 , K ⊕ U 2 , K U_{1,K}\oplus U_{2,K} U 1 , K ⊕ U 2 , K e , f , v ′ e,f,v' e , f , v ′ U = U K ⊗ K C U=U_K\otimes_{\mathbb{K}}\mathbb{C} U = U K ⊗ K C g ↦ ψ g g\mapsto\psi_g g ↦ ψ g ψ σ τ = ψ σ ψ τ \psi_{\sigma\tau}=\psi_\sigma\psi_\tau ψ σ τ = ψ σ ψ τ G G G U K U_K U K U Q = { u ∈ U K : ψ g ( u ) = u , g ∈ G } U_{\mathbb{Q}}=\{u\in U_K:\psi_g(u)=u,\ g\in G\} U Q = { u ∈ U K : ψ g ( u ) = u , g ∈ G } U Q U_{\mathbb{Q}} U Q e e e f f f − 39 v ′ = − 39 v \sqrt{-39}v'=-\sqrt{39}v − 39 v ′ = − 39 v
Proof. All the identities are exact: the rationality and the group relations of z z z z ′ z' z ′ z 1 z_1 z 1 z A z_A z A Z Z Z V 1 ′ V'_1 V 1 ′ V ′ V' V ′ K \mathbb{K} K U 2 , K = U 2 ( 9 , 0 ) ∩ V K ′ U_{2,K}=U_2(9,0)\cap V'_K U 2 , K = U 2 ( 9 , 0 ) ∩ V K ′ ψ σ \psi_\sigma ψ σ ψ τ \psi_\tau ψ τ ψ σ ψ τ \psi_\sigma\psi_\tau ψ σ ψ τ U 2 , K U_{2,K} U 2 , K K \mathbb{K} K ψ σ \psi_\sigma ψ σ 10.12 10.12 10.12 10.27 10.27 10.27 e e e f f f v ′ v' v ′ z z z z ′ z' z ′ z 1 z_1 z 1 z A z_A z A Q \mathbb{Q} Q σ ~ \widetilde{\sigma} σ τ ~ \widetilde{\tau} τ ψ σ 2 = ψ τ 2 = 1 \psi_{\sigma}^{2}=\psi_{\tau}^{2}=1 ψ σ 2 = ψ τ 2 = 1 ψ σ ψ τ = ψ τ ψ σ \psi_{\sigma}\psi_{\tau}=\psi_{\tau}\psi_{\sigma} ψ σ ψ τ = ψ τ ψ σ g ↦ ψ g g\mapsto\psi_g g ↦ ψ g ψ g \psi_g ψ g g g g u ( n ) v u_{(n)}v u ( n ) v V K ′ V'_K V K ′ U 2 , K U_{2,K} U 2 , K e e e f f f v ′ v' v ′ U K U_K U K H 1 ( G , GL n ( K ) ) = 1 H^1(G,\operatorname{GL}_n(K))=1 H 1 ( G , GL n ( K )) = 1 [70 Chapter X, Proposition 3] , each U K , n U_{K,n} U K , n K K K U K , n = ( U K , n ) G ⊗ Q K U_{K,n}=(U_{K,n})^G\otimes_{\mathbb{Q}}K U K , n = ( U K , n ) G ⊗ Q K Q \mathbb{Q} Q 1 1 1 ω \omega ω − 39 v ′ \sqrt{-39}v' − 39 v ′ σ ( − 39 ) = τ ( − 39 ) = − − 39 \sigma(\sqrt{-39})=\tau(\sqrt{-39})=-\sqrt{-39} σ ( − 39 ) = τ ( − 39 ) = − − 39 Q \mathbb{Q} Q U K U_K U K K K K U Q U_{\mathbb{Q}} U Q
In case (II) we have also checked that the traces ∑ g ψ g ( a u ) \sum_g\psi_g(au) ∑ g ψ g ( a u ) a ∈ { 1 , 13 , − 3 , − 39 } a\in\{1,\sqrt{13},\sqrt{-3},\sqrt{-39}\} a ∈ { 1 , 13 , − 3 , − 39 } u u u U 2 , K U_{2,K} U 2 , K Q \mathbb{Q} Q U 2 U_2 U 2 V Q ′ V'_{\mathbb{Q}} V Q ′ f 1 f_1 f 1 V ′ V' V ′ Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
Zhu-algebra computations and rationality
In this section, S ⊂ V ′ = V E 6 ⊕ A 2 S\subset V'=V_{E_6\oplus A_2} S ⊂ V ′ = V E 6 ⊕ A 2 U 1 = g ′ ⊕ g ′ ′ U_1=\mathfrak{g}'\oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ U 2 U_2 U 2 Theorem 10.5 . For coordinate computations we fix S = S ( 9 , 0 ) S=S^{(9,0)} S = S ( 9 , 0 ) Proposition 10.27 . We write L = E 6 ⊕ A 2 L=E_6\oplus A_2 L = E 6 ⊕ A 2 Theorem 10.10 , S S S c = 8 c=8 c = 8
Proposition 10.17 verifies the finite condition (F4) of Definition 10.16 . Theorem 10.21 therefore gives strong generation in weights at most three and C 2 C_2 C 2 [5 Appendix A] . The results of this section do not use (HJ). Theorem 11.1 , Propositions 11.3 and 11.4 and Theorem 11.6 are used in [5 Appendix A] .
We compare the algebraic steps with the proof of [67 ] for case (I). Remark 11.2 separates the branching calculation from categorical recognition. Section and page numbers refer to the working manuscript [67 ] and its package of computational certificates. Evans [Fya26, p. 3] cites the three constructions [50 , 52 , 66 ] . That of Jiao is the case (II) algebra (Theorem 10.24).
The structure of the proof in case (I)
The proof strategy of [67 ] separates the construction, strong generation, C 2 C_2 C 2 C 2 C_2 C 2 S S S Proposition 10.17 . Positivity follows from the determination of the simple modules and of their lowest weights in [5 Theorem 2.2(F1)–(F2)] .
Construction and positivity [67 Sections 2–3, pp. 16–19] . The algebra is generated by the currents and three weight-two vectors. A positive rescaling of the standard antilinear involution of the lattice algebra preserves the generators, which gives unitarity, simplicity and self-contragredience. For S S S Theorem 10.10 .
Strong generation [67 Section 4, pp. 20–24] . The finite mode closure in weights 2 and 3 [67 Proposition 4.1] , two obstruction calculations, in weights four and five, showing that at most three new generators appear in weight four and none in weight five [67 Proposition 4.2] , and a general lemma, triangular strong generation [67 Lemma 4.3] (Lemma 10.20). Proposition 10.17 supplies the finite weight-four condition (F4), so the lemma applies to S S S Theorem 10.21 . No generator of weight four is needed, hence no weight-five calculation either (Remark 10.23).
C 2 C_2 C 2 [67 Section 5, pp. 24–25] . For S S S Proposition 10.17 supplies the square identities in (F4), and part 3 of Theorem 10.21 gives the conclusion.
Rationality [67 Part III, pp. 25–55] . Semisimplicity of the Zhu algebra is proved through an extension of the affine subalgebra by its simple current, a Morita corner, a finite presentation by a quiver with relations, and an integral descent. The rationality criterion for C 2 C_2 C 2 S S S [5 Theorems A.10 and A.13] . For S S S Theorems 10.10 and 10.21 show that S S S C 2 C_2 C 2 [58 Main Theorem 3 (Corollary 5.19)] , S S S A ( S ) A(S) A ( S ) [72 ] . For S S S L s l 2 ( 3 , 0 ) ⊗ L s l 2 ( 13 , 0 ) L_{\mathfrak{sl}_2}(3,0) \otimes L_{\mathfrak{sl}_2}(13,0) L sl 2 ( 3 , 0 ) ⊗ L sl 2 ( 13 , 0 ) J = L ( 3 , 3 ) ⊗ L ( 13 , 13 ) J = L(3,3) \otimes L(13,13) J = L ( 3 , 3 ) ⊗ L ( 13 , 13 ) 3 4 + 13 4 = 4 \frac{3}{4} + \frac{13}{4} = 4 4 3 + 4 13 = 4 L ( 1 , 1 ) ⊗ L ( 39 , 39 ) L(1,1) \otimes L(39,39) L ( 1 , 1 ) ⊗ L ( 39 , 39 ) 1 4 + 39 4 = 10 \frac{1}{4} + \frac{39}{4} = 10 4 1 + 4 39 = 10 S S S V ( 3 , 13 ) V(3,13) V ( 3 , 13 ) J J J Proposition 11.4 ), so that S S S A ⊕ J A \oplus J A ⊕ J A + A^+ A + [67 Section 7] . In case (I) the current appears only in weight ten.
Modules and the categorical identification [67 Parts IV–V, pp. 56–70] . Six simple modules come from the nine V ′ V' V ′ W e , a W_{e,a} W e , a ( e , a ) ∈ L ∗ / L ≅ Z 3 2 (e,a) \in L^*/L \cong\mathbb{Z}_3^2 ( e , a ) ∈ L ∗ / L ≅ Z 3 2 [67 Section 11] . Then the two even unimodular overlattices of L L L V E 8 V_{E_8} V E 8 A ± = 1 ⊕ M b ⊕ 2 X ± A_{\pm} = 1 \oplus M_b \oplus2X_{\pm} A ± = 1 ⊕ M b ⊕ 2 X ± Rep ( U ) \operatorname{Rep}(U) Rep ( U ) Rep ( U ) ≃ Z ( D ± ) \operatorname{Rep}(U) \simeq\mathcal{Z}(\mathcal{D}_{\pm}) Rep ( U ) ≃ Z ( D ± ) D ± = Rep ( U ) A ± \mathcal{D}_{\pm} = \operatorname{Rep}(U)_{A_{\pm}} D ± = Rep ( U ) A ± [67 Lemma 12.2] [23 ] . The categorical argument uses the full branching of the lattice modules, dimension constraints and an exhaustion statement for fusion categories with the required fusion rules and untwisted pointed part. Here that exhaustion statement is hypothesis (C) of Remark 11.2 . The classification within the family of [34 Theorems 2 and 3] is distinct from proving this exhaustion. For S S S [5 Theorem 2.2(F2)–(F3)] . The recognition of the braided category in [5 Theorem 6.1] proceeds through the conformal net.
The changes in the explicit calculations include the affine levels, the weight of the simple current and the commutant, whose two commuting Virasoro vectors have central charges 182 95 \frac{182}{95} 95 182 32 19 \frac{32}{19} 19 32 2400 1271 \frac{2400}{1271} 1271 2400 70 31 \frac{70}{31} 31 70
The categorical route
The two isotropic lines in L ∗ / L L^*/L L ∗ / L H ± H_{\pm} H ± S S S [5 Theorem 2.2(F3)] give the branching input for categorical recognition. Remark 11.2 describes the additional categorical steps.
Part 2 of Proposition 10.13 gives a finite-degree check modulo p 3 p_3 p 3 W 0 , 0 ∣ S = S ⊕ M b W_{0,0}|_S = S \oplus M_b W 0 , 0 ∣ S = S ⊕ M b [5 Theorem 2.2(F3)] in weights at most three: modulo the prime p 3 p_3 p 3 R b R_b R b M b M_b M b S S S V ′ = S ⊕ S ⊥ V' = S \oplus S^\perp V ′ = S ⊕ S ⊥ S S S S ⊥ S^\perp S ⊥ S S S R b R_b R b S 1 = U 1 S_1 = U_1 S 1 = U 1 Theorem 10.21 ).
A conditional theorem
Theorem 11.1 Assume the hypothesis (Z) that the Zhu algebra A ( S ) A(S) A ( S ) S S S S S S [66 , 50 ] .
Proof. Proposition 10.17 and Theorem 10.21 give C 2 C_2 C 2 Theorem 10.10 gives simplicity, self-contragredience and CFT type. Thus [58 Main Theorem 3 (Corollary 5.19)] applies under (Z) and proves strong rationality. For the last assertion, recall that S 1 = U 1 S_1 = U_1 S 1 = U 1 Theorem 10.21 ). An isomorphism with an algebra W W W [66 , 50 ] would force dim W 1 = 6 \dim W_1 = 6 dim W 1 = 6 W 1 W_1 W 1 W W W S 1 S_1 S 1 W 1 W_1 W 1
Theorem 2.2(F1)–(F3) of [ArhIII] proves (Z) and (M). Theorem 11.1 therefore gives strong rationality, and Rep ( S ) \operatorname{Rep}(S) Rep ( S ) [49 ] . The nonvacuum lowest weights are positive, so the quantum dimensions are positive [19 Lemma 4.2] and equal the categorical dimensions [49 ] . Unitarity of S S S 1 13 Z ∖ Z \frac{1}{13}\mathbb{Z} \setminus\mathbb{Z} 13 1 Z ∖ Z ( W e , a ) ′ ≅ W − e , − a (W_{e,a})' \cong W_{-e,-a} ( W e , a ) ′ ≅ W − e , − a
The two even unimodular overlattices of L L L S S S 1 ⊕ M b ⊕ 2 X ± 1 \oplus M_b \oplus2X_\pm 1 ⊕ M b ⊕ 2 X ± A ± A_\pm A ± Rep ( S ) ≃ Z ( D ± ) \operatorname{Rep}(S) \simeq\mathcal{Z}(D_\pm) Rep ( S ) ≃ Z ( D ± ) D ± = Rep ( S ) A ± D_\pm= \operatorname{Rep}(S)_{A_\pm} D ± = Rep ( S ) A ± [23 ] . The fusion rules and the pointed associators follow from [5 Theorem 9.1] . These are the inputs to the recognition argument of [67 Sections 11.3 and 12–13] under (C).
Hypothesis (C) is the exhaustion statement formulated in [67 Proposition 12.5] . The classification in [34 Theorem 2] concerns the listed family. The proof of ( H J ) (H_J) ( H J ) [5 Theorem 6.1] uses the conformal-net comparison independently of (C).
The Zhu algebra of S S S
We now adapt to S S S [67 Part III] for the hypothesis (Z). We recall that the Zhu algebra is A ( V ) = V / O ( V ) A(V) = V/O(V) A ( V ) = V / O ( V ) a ∗ b = ∑ j ≥ 0 ( wt a j ) a ( j − 1 ) b a*b = \sum_{j\geq0} \binom{\operatorname{wt}a}{j}a_{(j-1)}b a ∗ b = ∑ j ≥ 0 ( j wt a ) a ( j − 1 ) b [72 Definition] 2.1.2 and Theorem 2.1.1. For h h h a a a x x x [ L ( − 1 ) x ] = − wt ( x ) [ x ] [L(-1)x] = -\operatorname{wt}(x)[x] [ L ( − 1 ) x ] = − wt ( x ) [ x ] [72 Lemma 2.1.1] and, by [72 Lemma 2.1.3] ,
[ h ( − 1 ) x ] = [ x ] ∗ [ h ] , [ h ( 0 ) x ] = [ h ] ∗ [ x ] − [ x ] ∗ [ h ] , [ a ( 0 ) x ] = [ a ] ∗ [ x ] − [ x ] ∗ [ a ] − [ a ( 1 ) x ] . (28) \begin{aligned}
[h(-1)x] &= [x] * [h], \qquad[h(0)x] = [h] * [x] - [x] * [h], \qquad[a(0)x] = [a] * [x] - [x] * [a] - [a(1)x]. \tag*{(28)}
\end{aligned} [ h ( − 1 ) x ] = [ x ] ∗ [ h ] , [ h ( 0 ) x ] = [ h ] ∗ [ x ] − [ x ] ∗ [ h ] , [ a ( 0 ) x ] = [ a ] ∗ [ x ] − [ x ] ∗ [ a ] − [ a ( 1 ) x ] . (28) Proposition 11.3 The algebra A ( S ) A(S) A ( S ) U 2 U_2 U 2 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ e ( − 2 ) ′ 1 e'_{(-2)}1 e ( − 2 ) ′ 1 e ( − 1 ) ′ e ′ e'_{(-1)}e' e ( − 1 ) ′ e ′ e ( − 2 ) ′ ′ 1 e''_{(-2)}1 e ( − 2 ) ′′ 1 e ( − 1 ) ′ ′ e ′ ′ e''_{(-1)}e'' e ( − 1 ) ′′ e ′′ e ( − 1 ) ′ e ′ ′ e'_{(-1)}e'' e ( − 1 ) ′ e ′′ ( A ) 2 (A)_2 ( A ) 2 e e e x C x_C x C ( 0 , 4 ) (0,4) ( 0 , 4 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 1 , 5 ) (1,5) ( 1 , 5 ) ( 1 , 7 ) (1,7) ( 1 , 7 ) ( 2 , 2 ) (2,2) ( 2 , 2 ) ( 2 , 6 ) (2,6) ( 2 , 6 ) ( 3 , 3 ) (3,3) ( 3 , 3 )
Proof : By (28) , the classes of L ( − 1 ) U 2 L(-1)U_2 L ( − 1 ) U 2 U 1 ( − 1 ) U 2 U_1(-1)U_2 U 1 ( − 1 ) U 2 ( U 2 ) ( 0 ) U 2 (U_2)(0)U_2 ( U 2 ) ( 0 ) U 2 [ U 1 ] [U_1] [ U 1 ] [ U 2 ] [U_2] [ U 2 ] ( U 2 ) ( 1 ) U 2 ⊂ U 2 (U_2)(1)U_2 \subset U_2 ( U 2 ) ( 1 ) U 2 ⊂ U 2 [ S 3 ] [S_3] [ S 3 ] S 3 = N 3 + ( U 2 ) ( 0 ) U 2 S_3 = N_3 + (U_2)(0)U_2 S 3 = N 3 + ( U 2 ) ( 0 ) U 2 Theorem 10.21 , S S S U 1 ⊕ U 2 ⊕ D 3 U_1 \oplus U_2 \oplus D_3 U 1 ⊕ U 2 ⊕ D 3 A ( S ) A(S) A ( S ) [ L ( − 1 ) x ] = − wt ( x ) [ x ] [L(-1)x] = -\operatorname{wt}(x)[x] [ L ( − 1 ) x ] = − wt ( x ) [ x ] g ( − 1 ) y g_{(-1)}y g ( − 1 ) y [ g ] ∗ [ y ] [g] * [y] [ g ] ∗ [ y ] f = f ′ f=f' f = f ′ f = f ′ ′ f=f'' f = f ′′ [ f ( 0 ) z ] = [ f ] ∗ [ z ] − [ z ] ∗ [ f ] [f(0)z] = [f] * [z] - [z] * [f] [ f ( 0 ) z ] = [ f ] ∗ [ z ] − [ z ] ∗ [ f ] [ U 2 ] [U_2] [ U 2 ] Table 2 ). The first seven are L ( − 1 ) e ( − 1 ) e L(-1)e_{(-1)}e L ( − 1 ) e ( − 1 ) e e ( − 1 ) ′ e ′ e'_{(-1)}e' e ( − 1 ) ′ e ′ L ( − 1 ) e ′ L(-1)e' L ( − 1 ) e ′ e ( − 1 ) ′ ′ e ′ ′ e''_{(-1)}e'' e ( − 1 ) ′′ e ′′ e ( − 1 ) ′ e ′ ′ e'_{(-1)}e'' e ( − 1 ) ′ e ′′ (28) .
Unlike in case (I), no strong generator of weight four is involved.
Proposition 11.4 Let A = L s l 2 ( 3 , 0 ) ⊗ L s l 2 ( 13 , 0 ) ⊂ S A = L_{\mathfrak{sl}_2}(3,0) \otimes L_{\mathfrak{sl}_2}(13,0) \subset S A = L sl 2 ( 3 , 0 ) ⊗ L sl 2 ( 13 , 0 ) ⊂ S C = Com S ( A ) C = \operatorname{Com}_S(A) C = Com S ( A )
The Cartan weight space of weight ( 3 , 13 ) (3,13) ( 3 , 13 ) V 4 ′ V'_4 V 4 ′ e w e^w e w w = ( 2 , 2 , 4 , 5 , 4 , 2 , 1 , 1 ) w=(2,2,4,5,4,2,1,1) w = ( 2 , 2 , 4 , 5 , 4 , 2 , 1 , 1 ) ( w , w ) = 8 (w,w)=8 ( w , w ) = 8 e w ∈ S e^w \in S e w ∈ S J = L s l 2 ( 3 , 3 ) ⊗ L s l 2 ( 13 , 13 ) J=L_{\mathfrak{sl}_2}(3,3)\otimes L_{\mathfrak{sl}_2}(13,13) J = L sl 2 ( 3 , 3 ) ⊗ L sl 2 ( 13 , 13 ) A + = A ⊕ J A^+=A\oplus J A + = A ⊕ J S S S
Com S ( C ) = A ⊕ J \operatorname{Com}_S(C)=A\oplus J Com S ( C ) = A ⊕ J
The inclusion A + ⊂ S A^+\subset S A + ⊂ S A ( A + ) → A ( S ) A(A^+)\to A(S) A ( A + ) → A ( S )
Proof : We first identify the kernel of the coset grading. Write ω = ω A + ω C \omega=\omega_A+\omega_C ω = ω A + ω C ω C = 2 e ∈ C \omega_C=2e\in C ω C = 2 e ∈ C ( L n C ) ∗ = L − n C (L_n^C)^*=L_{-n}^C ( L n C ) ∗ = L − n C H χ H_\chi H χ Theorem 10.10 , the Virasoro relation gives
∥ L − 1 C x ∥ χ 2 = ∥ L 1 C x ∥ χ 2 + 2 H χ ( L 0 C x , x ) . (29) \lVert L_{-1}^C x\rVert_\chi^2 = \lVert L_1^C x\rVert_\chi^2 + 2H_\chi(L_0^C x,x). \tag*{(29)} ∥ L − 1 C x ∥ χ 2 = ∥ L 1 C x ∥ χ 2 + 2 H χ ( L 0 C x , x ) . (29) The regularity of A A A S S S A A A ( i , j ) (i,j) ( i , j ) L 0 C = L ( 0 ) − h A ( i , j ) L_0^C=L(0)-h_A(i,j) L 0 C = L ( 0 ) − h A ( i , j ) h A ( i , j ) = i ( i + 2 ) 20 + j ( j + 2 ) 60 h_A(i,j)=\frac{i(i+2)}{20}+\frac{j(j+2)}{60} h A ( i , j ) = 20 i ( i + 2 ) + 60 j ( j + 2 ) L n C L_n^C L n C x x x λ \lambda λ L 1 C x = 0 L_1^C x=0 L 1 C x = 0 (29) gives ∥ L − 1 C x ∥ χ 2 = 2 λ ∥ x ∥ χ 2 \lVert L_{-1}^C x\rVert_\chi^2=2\lambda\lVert x\rVert_\chi^2 ∥ L − 1 C x ∥ χ 2 = 2 λ ∥ x ∥ χ 2 λ ≥ 0 \lambda\geq0 λ ≥ 0 L 0 C L_0^C L 0 C A A A S S S
For x ∈ Ker L 1 C x\in\operatorname{Ker}L_1^C x ∈ Ker L 1 C L − 1 C x = 0 L_{-1}^C x=0 L − 1 C x = 0 (29) gives L 0 C x = 0 L_0^C x=0 L 0 C x = 0 c ∈ C c\in C c ∈ C c ( n ) x = 0 c_{(n)}x=0 c ( n ) x = 0 n n n [ L − 1 C , c ( n + 1 ) ] = − ( n + 1 ) c ( n ) [L_{-1}^C,c_{(n+1)}]=-(n+1)c_{(n)} [ L − 1 C , c ( n + 1 ) ] = − ( n + 1 ) c ( n ) c ( n ) x = 0 c_{(n)}x=0 c ( n ) x = 0 n ≥ 0 n\geq0 n ≥ 0 ω C \omega_C ω C L 0 C x = 0 L_0^C x=0 L 0 C x = 0 Ker L 0 C = Com S ( C ) \operatorname{Ker}L_0^C=\operatorname{Com}_S(C) Ker L 0 C = Com S ( C )
The lattice realization gives the following lower bound: if the Cartan weight space of weight ( a , b ) (a,b) ( a , b ) V m ′ V'_m V m ′
m ≥ a 2 12 + b 2 52 . (30) m \ge\frac{a^2}{12}+\frac{b^2}{52}. \tag*{(30)} m ≥ 12 a 2 + 52 b 2 . (30) Indeed, the two Cartan vectors are orthogonal, of squared norms 6 and 26. A lattice vector of weight ( a , b ) (a,b) ( a , b ) a 2 / 6 + b 2 / 26 a^2/6+b^2/26 a 2 /6 + b 2 /26 ( 3 , 13 ) (3,13) ( 3 , 13 ) 1 2 ( H ′ + H ′ ′ ) \frac{1}{2}(H'+H'') 2 1 ( H ′ + H ′′ ) 3 2 6 + 13 2 26 = 8 \frac{3^2}{6}+\frac{13^2}{26}=8 6 3 2 + 26 1 3 2 = 8 w w w V 1 ′ V'_1 V 1 ′
Let u ( 1 , 7 ) u_{(1,7)} u ( 1 , 7 ) u ( 2 , 6 ) u_{(2,6)} u ( 2 , 6 ) Table 4 . A direct exact product gives
Table 4. The seven pairs in the product (31) , in the simple-root coordinates of E 6 ⊕ A 2 E_6\oplus A_2 E 6 ⊕ A 2
( u ( 1 , 7 ) ) ( − 1 ) u ( 2 , 6 ) = ( 507 5 13 + 3549 5 − 3 ) e w . (31) (u_{(1,7)})_{(-1)}u_{(2,6)}=\left(\frac{507}{5}\sqrt{13}+\frac{3549}{5}\sqrt{-3}\right)e^w. \tag*{(31)} ( u ( 1 , 7 ) ) ( − 1 ) u ( 2 , 6 ) = ( 5 507 13 + 5 3549 − 3 ) e w . (31) Here is also a direct verification of this product. Put r = 13 r=\sqrt{13} r = 13 s = − 3 s=\sqrt{-3} s = − 3 u ( 1 , 7 ) = ∑ i = 1 7 a i e α i u_{(1,7)}=\sum_{i=1}^{7}a_i e^{\alpha_i} u ( 1 , 7 ) = ∑ i = 1 7 a i e α i u ( 2 , 6 ) = ∑ i = 1 7 b i e w − α i u_{(2,6)}=\sum_{i=1}^{7}b_i e^{w-\alpha_i} u ( 2 , 6 ) = ∑ i = 1 7 b i e w − α i Table 4 . Each of α i \alpha_i α i w − α i w-\alpha_i w − α i ⟨ α i , w − α i ⟩ L = 0 \langle\alpha_i,w-\alpha_i\rangle_L=0 ⟨ α i , w − α i ⟩ L = 0 α i + w − α j ≠ w \alpha_i+w-\alpha_j\ne w α i + w − α j = w (23) therefore gives
( u ( 1 , 7 ) ) ( − 1 ) u ( 2 , 6 ) = ∑ i = 1 7 ε ( α i , w − α i ) a i b i e w = 26 + 104 + 26 + 104 + 65 + 65 + 117 5 ( r + 7 s ) e w . (32) \begin{aligned}
(u_{(1,7)})_{(-1)}u_{(2,6)}
&=\sum_{i=1}^{7}\varepsilon(\alpha_i,w-\alpha_i)a_i b_i e^w\\
&=\frac{26+104+26+104+65+65+117}{5}(r+7s)e^w. \tag*{(32)}
\end{aligned} ( u ( 1 , 7 ) ) ( − 1 ) u ( 2 , 6 ) = i = 1 ∑ 7 ε ( α i , w − α i ) a i b i e w = 5 26 + 104 + 26 + 104 + 65 + 65 + 117 ( r + 7 s ) e w . (32) This proves (31) by arithmetic in Q ( r , s ) \mathbb{Q}(r,s) Q ( r , s ) e w ∈ S e^w\in S e w ∈ S
The highest weight conditions also follow from (30) . The vectors e ( 0 ) ′ e w e'_{(0)}e^w e ( 0 ) ′ e w e ( 0 ) ′ ′ e w e''_{(0)}e^w e ( 0 ) ′′ e w ( 5 , 13 ) (5,13) ( 5 , 13 ) ( 3 , 15 ) (3,15) ( 3 , 15 ) 16 / 3 16/3 16/3 66 / 13 66/13 66/13 y y y ( 0 , 0 ) (0,0) ( 0 , 0 ) ( ± 2 , 0 ) (\pm2,0) ( ± 2 , 0 ) ( 0 , ± 2 ) (0,\pm2) ( 0 , ± 2 ) y ( n ) e w y_{(n)}e^w y ( n ) e w n ≥ 1 n \ge1 n ≥ 1 40 / 13 > 3 40/13 > 3 40/13 > 3 e w e^w e w A A A J J J
An independent exact calculation reconstructs the product and verifies the 26 annihilation identities for e ( 0 ) ′ e'_{(0)} e ( 0 ) ′ e ( 0 ) ′ ′ e''_{(0)} e ( 0 ) ′′ Table 4 . These computational checks supplement the preceding analytic calculation.
Its affine conformal weight is 4, so L 0 C e w = 0 L^C_0 e^w = 0 L 0 C e w = 0 L 0 C L^C_0 L 0 C A A A A A A J J J Ker L 0 C = Com S ( C ) \operatorname{Ker} L^C_0 = \operatorname{Com}_S(C) Ker L 0 C = Com S ( C ) A A A J J J S 0 = C 1 S_0 = \mathbb{C}1 S 0 = C 1 ( 3 , 13 ) (3,13) ( 3 , 13 ) A A A A ⊠ A = A A \boxtimes A = A A ⊠ A = A A ⊠ J = J A \boxtimes J = J A ⊠ J = J J ⊠ J = A J \boxtimes J = A J ⊠ J = A A ⊕ J A \oplus J A ⊕ J A + A^+ A + ω A \omega_A ω A L 0 C L^C_0 L 0 C L − 1 C L^C_{-1} L − 1 C A + A^+ A + S S S [21 Theorem 4.5] .
By Proposition 10.17 , Theorem 10.21 gives S 1 = U 1 S_1 = U_1 S 1 = U 1 S 2 = U 2 S_2 = U_2 S 2 = U 2 A A A ( i , j ) (i,j) ( i , j ) h A ( i , j ) h_A(i,j) h A ( i , j ) 0 ≤ i ≤ 3 0 \le i \le3 0 ≤ i ≤ 3 0 ≤ j ≤ 13 0 \le j \le13 0 ≤ j ≤ 13 h A ( i , j ) ∈ Z h_A(i,j) \in\mathbb{Z} h A ( i , j ) ∈ Z ( 0 , 0 ) (0,0) ( 0 , 0 ) ( 3 , 3 ) (3,3) ( 3 , 3 ) ( 0 , 10 ) (0,10) ( 0 , 10 ) ( 3 , 13 ) (3,13) ( 3 , 13 ) S 0 = C 1 S_0 = \mathbb{C}1 S 0 = C 1 S 1 = U 1 S_1 = U_1 S 1 = U 1 ( 3 , 3 ) (3,3) ( 3 , 3 ) U 2 U_2 U 2 ( 0 , 10 ) (0,10) ( 0 , 10 ) Table 2 ), and the Cartan weight space ( 3 , 13 ) (3,13) ( 3 , 13 ) S 4 S_4 S 4 Com S ( C ) = A ⊕ J \operatorname{Com}_S(C) = A \oplus J Com S ( C ) = A ⊕ J
On A + A^+ A + L 0 C L^C_0 L 0 C L − 1 C L^C_{-1} L − 1 C A + A^+ A + S S S ∗ \ast ∗ ∘ \circ ∘ A + A^+ A + S S S O ( A + ) ⊂ O ( S ) O(A^+) \subset O(S) O ( A + ) ⊂ O ( S )
The simple A A A L ( i , j ) = L s l 2 ( 3 , i ) ⊗ L s l 2 ( 13 , j ) L(i,j) = L_{\mathfrak{sl}_2}(3,i) \otimes L_{\mathfrak{sl}_2}(13,j) L ( i , j ) = L sl 2 ( 3 , i ) ⊗ L sl 2 ( 13 , j ) V ( i , j ) V(i,j) V ( i , j ) L ( i , j ) L(i,j) L ( i , j ) J J J ( − 1 ) i + j (-1)^{i+j} ( − 1 ) i + j J ⊠ L ( i , j ) ≅ L ( 3 − i , 13 − j ) J \boxtimes L(i,j) \cong L(3-i,13-j) J ⊠ L ( i , j ) ≅ L ( 3 − i , 13 − j ) A + A^+ A + [21 , 45 ] to the fourteen orbits { ( i , j ) , ( 3 − i , 13 − j ) } \{(i,j),(3-i,13-j)\} {( i , j ) , ( 3 − i , 13 − j )} i + j i+j i + j h A ( 3 − i , 13 − j ) − h A ( i , j ) = 4 − i + j 2 h_A(3-i,13-j)-h_A(i,j)=4-\frac{i+j}{2} h A ( 3 − i , 13 − j ) − h A ( i , j ) = 4 − 2 i + j T t T_t T t M t M_t M t V ( i , j ) V(i,j) V ( i , j ) i + j < 8 i+j<8 i + j < 8 i + j = 8 i+j=8 i + j = 8
T 0 , … , T 5 = V ( 0 , 0 ) , V ( 0 , 2 ) , V ( 1 , 1 ) , V ( 2 , 0 ) , V ( 0 , 4 ) , V ( 1 , 3 ) , T 6 , … , T 11 = V ( 2 , 2 ) , V ( 3 , 1 ) , V ( 0 , 6 ) , V ( 1 , 5 ) , V ( 2 , 4 ) , V ( 3 , 3 ) , T 12 = V ( 1 , 7 ) ⊕ V ( 2 , 6 ) ( h = 5 6 ) , T 13 = V ( 0 , 8 ) ⊕ V ( 3 , 5 ) ( h = 4 3 ) . (33) \begin{aligned}
T_0,\ldots,T_5 &= V(0,0),\ V(0,2),\ V(1,1),\ V(2,0),\ V(0,4),\ V(1,3),\\
T_6,\ldots,T_{11} &= V(2,2),\ V(3,1),\ V(0,6),\ V(1,5),\ V(2,4),\ V(3,3),\\
T_{12} &= V(1,7)\oplus V(2,6) \quad(h=\frac{5}{6}),\qquad T_{13}=V(0,8)\oplus V(3,5) \quad(h=\frac{4}{3}). \tag*{(33)}
\end{aligned} T 0 , … , T 5 T 6 , … , T 11 T 12 = V ( 0 , 0 ) , V ( 0 , 2 ) , V ( 1 , 1 ) , V ( 2 , 0 ) , V ( 0 , 4 ) , V ( 1 , 3 ) , = V ( 2 , 2 ) , V ( 3 , 1 ) , V ( 0 , 6 ) , V ( 1 , 5 ) , V ( 2 , 4 ) , V ( 3 , 3 ) , = V ( 1 , 7 ) ⊕ V ( 2 , 6 ) ( h = 6 5 ) , T 13 = V ( 0 , 8 ) ⊕ V ( 3 , 5 ) ( h = 3 4 ) . (33) Thus A ( A + ) ≅ ⨁ t = 0 13 End ( T t ) A(A^+) \cong\bigoplus_{t=0}^{13}\operatorname{End}(T_t) A ( A + ) ≅ ⨁ t = 0 13 End ( T t ) e t ∈ A ( S ) e_t \in A(S) e t ∈ A ( S ) End ( T t ) \operatorname{End}(T_t) End ( T t ) e Z = ∑ t e t e_Z=\sum_t e_t e Z = ∑ t e t A ( A + ) A(A^+) A ( A + ) E i j ( t ) E^{(t)}_{ij} E ij ( t ) E 11 ( t ) E^{(t)}_{11} E 11 ( t ) 1 = ∑ t , i E i 1 ( t ) E 11 ( t ) E 1 i ( t ) 1=\sum_{t,i}E^{(t)}_{i1}E^{(t)}_{11}E^{(t)}_{1i} 1 = ∑ t , i E i 1 ( t ) E 11 ( t ) E 1 i ( t ) Proposition 11.4 proves that e Z e_Z e Z A ( S ) A(S) A ( S ) A ( S ) A(S) A ( S ) D = e Z A ( S ) e Z D=e_ZA(S)e_Z D = e Z A ( S ) e Z [67 Lemma 8.2] . In particular A ( S ) A(S) A ( S ) D D D
Remark 11.5 Every g ′ ⊕ g ′ ′ \mathfrak{g}'\oplus\mathfrak{g}'' g ′ ⊕ g ′′ Table 2 is local and of the form ( i , j ) (i,j) ( i , j ) i + j < 8 i+j<8 i + j < 8 ( 3 − i , 13 − j ) (3-i,13-j) ( 3 − i , 13 − j ) S S S M M M J J J M M M A ( S ) A(S) A ( S ) Table 2 , then D ≅ ⨁ z M n z ( C ) D\cong\bigoplus_z M_{n_z}(\mathbb{C}) D ≅ ⨁ z M n z ( C ) n z n_z n z z z z 1 , 1 , 2 , 2 , 2 , 3 , 5 , 7 , 11 , 12 , 12 , 16 1,1,2,2,2,3,5,7,11,12,12,16 1 , 1 , 2 , 2 , 2 , 3 , 5 , 7 , 11 , 12 , 12 , 16 0 , c 1 , d 1 , d 4 , d 5 , d 6 , c 2 , d 2 , b , a , c 0 , d 3 0,c_1,d_1,d_4,d_5,d_6,c_2,d_2,b,a,c_0,d_3 0 , c 1 , d 1 , d 4 , d 5 , d 6 , c 2 , d 2 , b , a , c 0 , d 3 dim D = 762 \dim D=762 dim D = 762 T 12 , T 13 T_{12},T_{13} T 12 , T 13 [67 Proposition 9.5] .
To obtain an upper bound for D D D [67 Section 9] . Put g = g ′ ⊕ g ′ ′ \mathfrak{g}=\mathfrak{g}'\oplus\mathfrak{g}'' g = g ′ ⊕ g ′′ Γ Z \Gamma_Z Γ Z 0 , … , 13 0,\ldots,13 0 , … , 13 z z z Proposition 11.3 and each pair of constituents λ ⊂ T t \lambda\subset T_t λ ⊂ T t μ ⊂ T v \mu\subset T_v μ ⊂ T v Hom ( V ( λ ) , V ( μ ) ) \operatorname{Hom}(V(\lambda),V(\mu)) Hom ( V ( λ ) , V ( μ )) z z z t → v t\to v t → v [ z ] [z] [ z ] A ( S ) A(S) A ( S ) A ( A + ) A(A^+) A ( A + ) C Γ Z → D \mathbb{C}\Gamma_Z\to D C Γ Z → D [67 Propositions 8.3 and 8.6] . The relations are the projections of the Zhu images of the three families of identities in S S S
the kernel of U 2 ⊕ ( g ⊗ U 2 ) ⊕ Λ 2 U 2 → S 3 U_2\oplus(\mathfrak{g}\otimes U_2)\oplus\Lambda^2U_2\to S_3 U 2 ⊕ ( g ⊗ U 2 ) ⊕ Λ 2 U 2 → S 3 ( v , h ⊗ w , a ∧ b ) ↦ L ( − 1 ) v + h ( − 1 ) w + 1 2 ( a ( 0 ) b − b ( 0 ) a ) (v,h\otimes w,a\wedge b)\mapsto L(-1)v+h(-1)w+\frac{1}{2}(a(0)b-b(0)a) ( v , h ⊗ w , a ∧ b ) ↦ L ( − 1 ) v + h ( − 1 ) w + 2 1 ( a ( 0 ) b − b ( 0 ) a )
the kernel of S 2 U 2 → S 4 S^2U_2\to S_4 S 2 U 2 → S 4 a ⋅ b ↦ 1 2 ( a ( − 1 ) b + b ( − 1 ) a ) a\cdot b\mapsto\frac{1}{2}(a(-1)b+b(-1)a) a ⋅ b ↦ 2 1 ( a ( − 1 ) b + b ( − 1 ) a )
the kernel of S 3 ⊕ ( g ⊗ S 3 ) ⊕ S 2 U 2 → S 4 S_3\oplus(\mathfrak{g}\otimes S_3)\oplus S^2U_2\to S_4 S 3 ⊕ ( g ⊗ S 3 ) ⊕ S 2 U 2 → S 4 ( u , h ⊗ v , a ⋅ b ) ↦ L ( − 1 ) u + h ( − 1 ) v + 1 2 ( a ( − 1 ) b + b ( − 1 ) a ) (u,h\otimes v,a\cdot b)\mapsto L(-1)u+h(-1)v+\frac{1}{2}(a(-1)b+b(-1)a) ( u , h ⊗ v , a ⋅ b ) ↦ L ( − 1 ) u + h ( − 1 ) v + 2 1 ( a ( − 1 ) b + b ( − 1 ) a )
together with the centrality of [ ω ] = [ ω A ] + 2 [ e ] [\omega]=[\omega_A]+2[e] [ ω ] = [ ω A ] + 2 [ e ] g \mathfrak{g} g (28) and from
[ 1 2 ( a ( − 1 ) b + b ( − 1 ) a ) ] = 1 2 ( [ a ] ∗ [ b ] + [ b ] ∗ [ a ] ) + [ c + ( a , b ) ] , \left[\frac{1}{2}(a(-1)b+b(-1)a)\right]=\frac{1}{2}([a]*[b]+[b]*[a])+[c_+(a,b)], [ 2 1 ( a ( − 1 ) b + b ( − 1 ) a ) ] = 2 1 ([ a ] ∗ [ b ] + [ b ] ∗ [ a ]) + [ c + ( a , b )] , [ 1 2 ( a ( 0 ) b − b ( 0 ) a ) ] = [ a ] ∗ [ b ] − [ b ] ∗ [ a ] − [ c − ( a , b ) ] , \left[\frac{1}{2}(a(0)b-b(0)a)\right]=[a]*[b]-[b]*[a]-[c_-(a,b)], [ 2 1 ( a ( 0 ) b − b ( 0 ) a ) ] = [ a ] ∗ [ b ] − [ b ] ∗ [ a ] − [ c − ( a , b )] , where c ± ( a , b ) = 1 2 ( a ( 1 ) b ± b ( 1 ) a ) ∈ U 2 c_\pm(a,b)=\frac{1}{2}(a(1)b\pm b(1)a)\in U_2 c ± ( a , b ) = 2 1 ( a ( 1 ) b ± b ( 1 ) a ) ∈ U 2
Theorem 11.6 (computation modulo a prime). Let p = p 3 = 8388571 p=p_3=8388571 p = p 3 = 8388571 ( O K ) p → F p (\mathcal{O}_{K})_{\mathfrak{p}}\to\mathbb{F}_p ( O K ) p → F p p \mathfrak{p} p p p p 13 \sqrt{13} 13 − 3 \sqrt{-3} − 3 p \mathfrak{p} p p p p
the three families have 458, 70 and 532 highest weight relations, and S 3 S_3 S 3
the projected relations, 19981 rows in the path algebra F p Γ Z \mathbb{F}_p\Gamma_Z F p Γ Z
the resulting algebra D p D_p D p D p D_p D p D p ≅ ⨁ z M n z ( F p ) D_p\cong\bigoplus_z M_{n_z}(\mathbb{F}_p) D p ≅ ⨁ z M n z ( F p )
for each block, the ranks of the vertex idempotents e t e_t e t T t T_t T t Table 2 , for all twelve modules.
Proof. All products a ( n ) b a_{(n)}b a ( n ) b a , b ∈ U 2 a,b\in U_2 a , b ∈ U 2 − 1 ≤ n ≤ 3 -1\le n\le3 − 1 ≤ n ≤ 3 h ( − 1 ) s h_{(-1)}s h ( − 1 ) s L ( − 1 ) s L(-1)s L ( − 1 ) s s ∈ S 3 s\in S_3 s ∈ S 3 K K K p p p F p \mathbb{F}_p F p [8 Theorem 1.2] .
The inclusion ambiguities are needed because the system has cubic as well as quadratic leading words. These checks concern the projected presentation.
For completeness, the algebraic consequences of the last checks require no further VOA computation. If x x x D p D_p D p x y xy x y y ∈ D p y \in D_p y ∈ D p x y xy x y x = 0 x = 0 x = 0 F p \mathbb{F}_p F p F p \mathbb{F}_p F p D p D_p D p
Lemma 11.8 The nine V ′ V' V ′ W e , a W_{e,a} W e , a S S S W e , a ∣ S W_{e,a}|_S W e , a ∣ S S S S S S S A ( S ) A(S) A ( S ) W e , a W_{e,a} W e , a
Proof: By [24 Theorem 4.12] , V L + λ V_{L+\lambda} V L + λ ( V L , ϕ 1 ) (V_L,\phi_1) ( V L , ϕ 1 ) H 1 H_1 H 1 g = exp ( h χ ( 0 ) ) g = \exp(h_\chi(0)) g = exp ( h χ ( 0 )) Theorem 10.10 , g 1 / 2 g^{1/2} g 1/2 V L + λ V_{L+\lambda} V L + λ e 1 2 ( h χ , μ ) e^{\frac{1}{2}(h_\chi,\mu)} e 2 1 ( h χ , μ ) M ( 1 ) ⊗ e μ M(1) \otimes e^\mu M ( 1 ) ⊗ e μ H χ ( u , v ) = H 1 ( g 1 / 2 u , g 1 / 2 v ) H_\chi(u,v) = H_1(g^{1/2}u,g^{1/2}v) H χ ( u , v ) = H 1 ( g 1/2 u , g 1/2 v ) ( V ′ , ϕ χ ) (V',\phi_\chi) ( V ′ , ϕ χ ) S S S ϕ χ ( S ) = S \phi_\chi(S) = S ϕ χ ( S ) = S X X X A ( S ) A(S) A ( S ) N N N ⟨ X ⟩ \langle X\rangle ⟨ X ⟩ ⟨ X ⟩ = N ⊕ N ⊥ \langle X\rangle= N \oplus N^\perp ⟨ X ⟩ = N ⊕ N ⊥ X = ( N ∩ X ) ⊕ ( N ⊥ ∩ X ) X = (N \cap X) \oplus(N^\perp\cap X) X = ( N ∩ X ) ⊕ ( N ⊥ ∩ X ) N ∩ X = 0 N \cap X = 0 N ∩ X = 0 X ⊂ N ⊥ X \subset N^\perp X ⊂ N ⊥ N = 0 N = 0 N = 0
Completion of the rationality argument
Appendix A of [5 ] proves (Z) and (M) for S S S [5 ] states (Z) and the resulting strong rationality. Parts (F2)–(F3) give (M), including the complete lattice branching and the lowest weight spaces of the twelve simple modules. The proof of (M) combines the restrictions of the lattice modules with Galois symmetry to show that all twelve generic blocks survive in the Zhu corner. The braided equivalence that completes the proof of (HJ _J J [5 Theorem 6.1] .
The associated lattice and the twin algebra
In this section we describe the Haagerup algebras through the decomposition of a strongly rational vertex operator algebra into its Heisenberg commutant and its associated lattice [57 , 48 , 62 ] , and we use the notions of genus of Höhn [47 ] (the bulk genus, given by the representation category and the central charge) and of Moriwaki [61 ] and Möller–Rayhaun [62 ] (the hyperbolic genus). We call a Haagerup algebra a positive strongly rational vertex operator algebra U U U Rep ( U ) ≃ Z ( H 3 ) \operatorname{Rep}(U) \simeq\mathcal{Z}(\mathcal{H}_3) Rep ( U ) ≃ Z ( H 3 ) Proposition 4.1 and Theorem 8.5 , after an automorphism of V ′ V' V ′ U ⊂ V ′ U \subset V' U ⊂ V ′ U 1 = g ′ ⊕ g ′ ′ U_1 = \mathfrak{g}' \oplus\mathfrak{g}'' U 1 = g ′ ⊕ g ′′ ( k ′ , k ′ ′ ) = ( 1 , 39 ) (k',k'') = (1,39) ( k ′ , k ′′ ) = ( 1 , 39 ) ( 3 , 13 ) (3,13) ( 3 , 13 ) h = C h ′ ⊕ C h ′ ′ \mathfrak{h} = \mathbb{C}h' \oplus\mathbb{C}h'' h = C h ′ ⊕ C h ′′ U 1 U_1 U 1 V 1 ′ V'_1 V 1 ′ h ′ ∈ g ′ h' \in\mathfrak{g}' h ′ ∈ g ′ h ′ ′ ∈ g ′ ′ h'' \in\mathfrak{g}'' h ′′ ∈ g ′′ ⟨ h ′ , h ′ ⟩ = 2 k ′ \langle h',h'\rangle= 2k' ⟨ h ′ , h ′ ⟩ = 2 k ′ ⟨ h ′ ′ , h ′ ′ ⟩ = 2 k ′ ′ \langle h'',h''\rangle= 2k'' ⟨ h ′′ , h ′′ ⟩ = 2 k ′′ ⟨ h ′ , h ′ ′ ⟩ = 0 \langle h',h''\rangle= 0 ⟨ h ′ , h ′′ ⟩ = 0 h \mathfrak{h} h ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ λ ∈ h \lambda\in\mathfrak{h} λ ∈ h ( m ′ , m ′ ′ ) = ( ⟨ h ′ , λ ⟩ , ⟨ h ′ ′ , λ ⟩ ) (m',m'') = (\langle h',\lambda\rangle,\langle h'',\lambda\rangle) ( m ′ , m ′′ ) = (⟨ h ′ , λ ⟩ , ⟨ h ′′ , λ ⟩) λ \lambda λ h \mathfrak{h} h v v v h ( 0 ) ′ v = m ′ v h'_{(0)}v=m'v h ( 0 ) ′ v = m ′ v h ( 0 ) ′ ′ v = m ′ ′ v h''_{(0)}v=m''v h ( 0 ) ′′ v = m ′′ v
∣ λ ∣ 2 = m ′ 2 2 k ′ + m ′ ′ 2 2 k ′ ′ . |\lambda|^2 = \frac{m'^2}{2k'} + \frac{m''^2}{2k''}. ∣ λ ∣ 2 = 2 k ′ m ′2 + 2 k ′′ m ′′2 . We denote by Γ ( U ) ⊂ h \Gamma(U) \subset\mathfrak{h} Γ ( U ) ⊂ h h \mathfrak{h} h U U U
Ω U : = Com U ( M ( h ) ) \Omega_U \mathrel{\mathop:}= \operatorname{Com}_U(M(\mathfrak{h})) Ω U : = Com U ( M ( h )) the Heisenberg commutant of U U U L L L associated lattice , defined by Com U ( Ω U ) ≅ V L \operatorname{Com}_U(\Omega_U) \cong V_L Com U ( Ω U ) ≅ V L [62 Definition 2.12] . Throughout this section L L L E 6 ⊕ A 2 E_6 \oplus A_2 E 6 ⊕ A 2 [62 Remark 2.13] . For an even lattice L L L C ( L ′ / L ) \mathcal{C}(L'/L) C ( L ′ / L ) ( L ′ / L , q L ) (L'/L,q_L) ( L ′ / L , q L ) q L ( λ + L ) = ∣ λ ∣ 2 2 m o d 1 q_L(\lambda+L)=\frac{|\lambda|^2}{2}\bmod1 q L ( λ + L ) = 2 ∣ λ ∣ 2 mod 1 Rep ( V L ) ≅ C ( L ′ / L ) \operatorname{Rep}(V_L)\cong\mathcal{C}(L'/L) Rep ( V L ) ≅ C ( L ′ / L ) B ‾ \overline{\mathcal{B}} B B \mathcal{B} B
Two binary lattices
Put Γ 0 = { ( m ′ , m ′ ′ ) ∈ Z 2 : m ′ ≡ m ′ ′ m o d 2 } \Gamma_0 = \{(m',m'') \in\mathbb{Z}^2 : m' \equiv m'' \bmod2\} Γ 0 = {( m ′ , m ′′ ) ∈ Z 2 : m ′ ≡ m ′′ mod 2 } ( k ′ , k ′ ′ ) ∈ { ( 1 , 39 ) , ( 3 , 13 ) } (k',k'') \in\{(1,39),(3,13)\} ( k ′ , k ′′ ) ∈ {( 1 , 39 ) , ( 3 , 13 )}
L k ′ , k ′ ′ : = Γ 0 ∗ = { ( k ′ a , k ′ ′ c ) : a , c ∈ Z , a ≡ c m o d 2 } , L_{k',k''} \mathrel{\mathop:}= \Gamma_0^* = \{(k'a,k''c) : a,c \in\mathbb{Z},\ a \equiv c \bmod2\}, L k ′ , k ′′ : = Γ 0 ∗ = {( k ′ a , k ′′ c ) : a , c ∈ Z , a ≡ c mod 2 } , the dual being taken for the form above. It has the basis ( 2 k ′ , 0 ) (2k',0) ( 2 k ′ , 0 ) ( k ′ , k ′ ′ ) (k',k'') ( k ′ , k ′′ ) L I = L 1 , 39 L_{\mathrm{I}}=L_{1,39} L I = L 1 , 39 L I I = L 3 , 13 L_{\mathrm{II}}=L_{3,13} L II = L 3 , 13
Lemma 12.1.
The Gram matrices of L I L_{\mathrm{I}} L I L I I L_{\mathrm{II}} L II ( 2 1 1 20 ) \begin{pmatrix}2 & 1 \\ 1 & 20\end{pmatrix} ( 2 1 1 20 ) ( 6 3 3 8 ) \begin{pmatrix}6 & 3 \\ 3 & 8\end{pmatrix} ( 6 3 3 8 ) 39 39 39 2 2 2 6 6 6 x 2 + x y + 10 y 2 x^2+xy+10y^2 x 2 + x y + 10 y 2 3 x 2 + 3 x y + 4 y 2 3x^2+3xy+4y^2 3 x 2 + 3 x y + 4 y 2 L ′ = Γ 0 L'=\Gamma_0 L ′ = Γ 0 L ′ / L ≅ Z / 39 L'/L\cong\mathbb{Z}/39 L ′ / L ≅ Z /39 ( 1 , 1 ) (1,1) ( 1 , 1 ) q L I ( 1 , 1 ) = 10 39 q_{L_{\mathrm{I}}}(1,1)=\frac{10}{39} q L I ( 1 , 1 ) = 39 10 q L I I ( 1 , 1 ) = 4 39 q_{L_{\mathrm{II}}}(1,1)=\frac{4}{39} q L II ( 1 , 1 ) = 39 4
The even positive-definite lattices of rank two whose discriminant form is isometric to that of L I L_{\mathrm{I}} L I L I L_{\mathrm{I}} L I L I I L_{\mathrm{II}} L II − 39 -39 − 39 ( 1 , 1 , 10 ) (1,1,10) ( 1 , 1 , 10 ) ( 2 , ± 1 , 5 ) (2,\pm1,5) ( 2 , ± 1 , 5 ) ( 3 , 3 , 4 ) (3,3,4) ( 3 , 3 , 4 ) − 39 -39 − 39 { ( 1 , 1 , 10 ) , ( 3 , 3 , 4 ) } \{(1,1,10),(3,3,4)\} {( 1 , 1 , 10 ) , ( 3 , 3 , 4 )} { ( 2 , 1 , 5 ) , ( 2 , − 1 , 5 ) } \{(2,1,5),(2,-1,5)\} {( 2 , 1 , 5 ) , ( 2 , − 1 , 5 )} L I L_{\mathrm{I}} L I
The isometries L I I ′ / L I I → L I ′ / L I L'_{\mathrm{II}}/L_{\mathrm{II}}\to L'_{\mathrm{I}}/L_{\mathrm{I}} L II ′ / L II → L I ′ / L I ( 1 , 1 ) ↦ u ( 1 , 1 ) (1,1)\mapsto u(1,1) ( 1 , 1 ) ↦ u ( 1 , 1 ) u ∈ { 4 , 17 , 22 , 35 } u\in\{4,17,22,35\} u ∈ { 4 , 17 , 22 , 35 } O ( L ′ / L ) = { ± 1 , ± 14 } O(L'/L)=\{\pm1,\pm14\} O ( L ′ / L ) = { ± 1 , ± 14 }
The image of the orthogonal group O ( L I I ) O(L_{\mathrm{II}}) O ( L II ) O ( L I I ′ / L I I ) O(L'_{\mathrm{II}}/L_{\mathrm{II}}) O ( L II ′ / L II ) w ′ = diag ( − 1 , 1 ) w'=\operatorname{diag}(-1,1) w ′ = diag ( − 1 , 1 ) w ′ ′ = diag ( 1 , − 1 ) w''=\operatorname{diag}(1,-1) w ′′ = diag ( 1 , − 1 ) h \mathfrak{h} h g ′ \mathfrak{g}' g ′ g ′ ′ \mathfrak{g}'' g ′′ 14 14 14 25 25 25 O ( L I ) O(L_{\mathrm{I}}) O ( L I ) O ( L I ′ / L I ) O(L'_{\mathrm{I}}/L_{\mathrm{I}}) O ( L I ′ / L I ) { ± 1 } \{\pm1\} { ± 1 } w ′ w' w ′ w ′ ′ w'' w ′′ − 1 -1 − 1
Proof. We give a proof with integer arithmetic, independently of the computational verification. For part 1, the form m ′ 2 2 k ′ + m ′ ′ 2 2 k ′ ′ \frac{m^{\prime2}}{2k^{\prime}}+\frac{m^{\prime\prime2}}{2k^{\prime\prime}} 2 k ′ m ′2 + 2 k ′′ m ′′2 ( 2 k ′ , 0 ) (2k^{\prime},0) ( 2 k ′ , 0 ) ( k ′ , k ′ ′ ) (k^{\prime},k^{\prime\prime}) ( k ′ , k ′′ ) M k ′ , k ′ ′ M_{k^{\prime},k^{\prime\prime}} M k ′ , k ′′ M 11 = 2 k ′ M_{11}=2k^{\prime} M 11 = 2 k ′ M 12 = M 21 = k ′ M_{12}=M_{21}=k^{\prime} M 12 = M 21 = k ′ M 22 = k ′ + k ′ ′ 2 M_{22}=\frac{k^{\prime}+k^{\prime\prime}}{2} M 22 = 2 k ′ + k ′′ k ′ k ′ ′ = 39 k^{\prime}k^{\prime\prime}=39 k ′ k ′′ = 39 n ( 1 , 1 ) n(1,1) n ( 1 , 1 ) L k ′ , k ′ ′ L_{k^{\prime},k^{\prime\prime}} L k ′ , k ′′ 39 39 39 n n n ( 1 , 1 ) (1,1) ( 1 , 1 ) 1 4 k ′ + 1 4 k ′ ′ \frac{1}{4k^{\prime}}+\frac{1}{4k^{\prime\prime}} 4 k ′ 1 + 4 k ′′ 1
x 2 + x y + 10 y 2 = ( x + y 2 ) 2 + 39 4 y 2 , 3 x 2 + 3 x y + 4 y 2 = 3 ( x + y 2 ) 2 + 13 4 y 2 x^2+xy+10y^2=\left(x+\frac{y}{2}\right)^2+\frac{39}{4}y^2,\qquad3x^2+3xy+4y^2=3\left(x+\frac{y}{2}\right)^2+\frac{13}{4}y^2 x 2 + x y + 10 y 2 = ( x + 2 y ) 2 + 4 39 y 2 , 3 x 2 + 3 x y + 4 y 2 = 3 ( x + 2 y ) 2 + 4 13 y 2 show that the shortest vectors are precisely the two vectors ± ( 2 k ′ , 0 ) \pm(2k^{\prime},0) ± ( 2 k ′ , 0 )
An even positive-definite binary lattice of determinant 39 corresponds to a primitive form a x 2 + b x y + c y 2 ax^2+bxy+cy^2 a x 2 + b x y + c y 2 − 39 -39 − 39 ∣ b ∣ ≤ a ≤ c |b|\leq a\leq c ∣ b ∣ ≤ a ≤ c b ≥ 0 b\geq0 b ≥ 0 ∣ b ∣ = a |b|=a ∣ b ∣ = a a = c a=c a = c 39 = 4 a c − b 2 ≥ 3 a 2 39=4ac-b^2\geq3a^2 39 = 4 a c − b 2 ≥ 3 a 2 1 ≤ a ≤ 3 1\leq a\leq3 1 ≤ a ≤ 3 b b b c = 39 + b 2 4 a c=\frac{39+b^2}{4a} c = 4 a 39 + b 2 a = 1 , 2 , 3 a=1,2,3 a = 1 , 2 , 3 ( 1 , 1 , 10 ) (1,1,10) ( 1 , 1 , 10 ) ( 2 , ± 1 , 5 ) (2,\pm1,5) ( 2 , ± 1 , 5 ) ( 3 , 3 , 4 ) (3,3,4) ( 3 , 3 , 4 ) [ 4 1 1 10 ] \begin{bmatrix}4&1\\1&10\end{bmatrix} [ 4 1 1 10 ] 5 39 \frac{5}{39} 39 5 L I L_{\mathrm{I}} L I 5 u 2 ≡ 10 ( m o d 39 ) 5u^2\equiv10\pmod{39} 5 u 2 ≡ 10 ( mod 39 ) u u u 10 ⋅ 4 2 ≡ 4 ( m o d 39 ) 10\cdot4^2\equiv4\pmod{39} 10 ⋅ 4 2 ≡ 4 ( mod 39 ) L I L_{\mathrm{I}} L I L I I L_{\mathrm{II}} L II [65 Corollary 1.9.4] . The four reduced forms are the four proper classes. Inversion interchanges the two distinct classes ( 2 , 1 , 5 ) (2,1,5) ( 2 , 1 , 5 ) ( 2 , − 1 , 5 ) (2,-1,5) ( 2 , − 1 , 5 ) ( 3 , 3 , 4 ) (3,3,4) ( 3 , 3 , 4 ) [18 ] .
An isometry between the discriminant forms is multiplication by a unit u u u 10 u 2 ≡ 4 ( m o d 39 ) 10u^2\equiv4\pmod{39} 10 u 2 ≡ 4 ( mod 39 ) u 2 ≡ 16 ( m o d 39 ) u^2\equiv16\pmod{39} u 2 ≡ 16 ( mod 39 ) 4 , 17 , 22 , 35 4,17,22,35 4 , 17 , 22 , 35 u 2 ≡ 1 ( m o d 39 ) u^2\equiv1\pmod{39} u 2 ≡ 1 ( mod 39 ) O ( L ′ / L ) = { 1 , 14 , 25 , 38 } O(L^{\prime}/L)=\{1,14,25,38\} O ( L ′ / L ) = { 1 , 14 , 25 , 38 }
An orthogonal transformation preserving either lattice preserves its unique shortest line, so its orthogonal group has at most four elements. The four sign changes in the two coordinates ( m ′ , m ′ ′ ) (m^{\prime},m^{\prime\prime}) ( m ′ , m ′′ ) L k ′ , k ′ ′ L_{k^{\prime},k^{\prime\prime}} L k ′ , k ′′ ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) ( 1 , − 1 ) (1,-1) ( 1 , − 1 ) L k ′ , k ′ ′ L_{k^{\prime},k^{\prime\prime}} L k ′ , k ′′
( − 1 , 1 ) − 14 ( 1 , 1 ) = ( − 15 , − 13 ) ∈ L I I , ( 1 , − 1 ) − 25 ( 1 , 1 ) = ( − 24 , − 26 ) ∈ L I I . (-1,1)-14(1,1)=(-15,-13)\in L_{\mathrm{II}},\qquad(1,-1)-25(1,1)=(-24,-26)\in L_{\mathrm{II}}. ( − 1 , 1 ) − 14 ( 1 , 1 ) = ( − 15 , − 13 ) ∈ L II , ( 1 , − 1 ) − 25 ( 1 , 1 ) = ( − 24 , − 26 ) ∈ L II . In case (I), ( − 1 , 1 ) − ( 1 , 1 ) = ( − 2 , 0 ) ∈ L I (-1,1)-(1,1)=(-2,0)\in L_{\mathrm{I}} ( − 1 , 1 ) − ( 1 , 1 ) = ( − 2 , 0 ) ∈ L I ( 1 , − 1 ) + ( 1 , 1 ) = ( 2 , 0 ) ∈ L I (1,-1)+(1,1)=(2,0)\in L_{\mathrm{I}} ( 1 , − 1 ) + ( 1 , 1 ) = ( 2 , 0 ) ∈ L I
The supplementary material includes independent integer-arithmetic checks of these lattice calculations. They are not needed for the preceding proof.
The Heisenberg commutant and the associated lattice
For a simple vertex operator algebra Ω \Omega Ω Ω \Omega Ω M M M
Vac Ω ( M ) : = { v ∈ M : a ( n ) v = 0 for every a ∈ Ω and n ≥ 0 } . \operatorname{Vac}_{\Omega}(M)\mathrel{\mathop:}=\{v\in M:a_{(n)}v=0\text{ for every }a\in\Omega\text{ and }n\geq0\}. Vac Ω ( M ) : = { v ∈ M : a ( n ) v = 0 for every a ∈ Ω and n ≥ 0 } . A nonzero vector in this space is called vacuum-like. If M M M v v v a ↦ a ( − 1 ) v a\mapsto a_{(-1)}v a ↦ a ( − 1 ) v Ω \Omega Ω Ω → M \Omega\to M Ω → M M ≅ Ω M\cong\Omega M ≅ Ω [56 Section 4.7] .
Proposition 12.2 Let U U U
The Heisenberg commutant Ω U \Omega_U Ω U is a strongly rational vertex operator algebra of central charge six with ( Ω U ) 1 = 0 (\Omega_U)_1=0 ( Ω U ) 1 = 0 There is a ribbon-reversing equivalence τ \tau τ from C ( L ′ / L ) \mathcal{C}(L^{\prime}/L) C ( L ′ / L ) onto the full pointed fusion subcategory generated by the invertible objects of Rep ( Ω U ) \operatorname{Rep}(\Omega_U) Rep ( Ω U ) Writing τ \tau τ also for the induced map on isomorphism classes of simple objects, we have
U = ⨁ α ∈ L ′ / L Ω U τ ( α ) ⊗ V α + L U=\bigoplus_{\alpha\in L^{\prime}/L}\Omega_U^{\tau(\alpha)}\otimes V_{\alpha+L} U = α ∈ L ′ / L ⨁ Ω U τ ( α ) ⊗ V α + L as an Ω U ⊗ V L \Omega_U\otimes V_L Ω U ⊗ V L module, and Rep ( Ω U ) ≅ Z ( H 3 ) ⊠ C ( L ′ / L ) \operatorname{Rep}(\Omega_U)\cong Z(H_3)\boxtimes\mathcal{C}(L^{\prime}/L) Rep ( Ω U ) ≅ Z ( H 3 ) ⊠ C ( L ′ / L )
Every irreducible Ω U \Omega_U Ω U module has a nonnegative conformal weight, and for α ≠ 0 \alpha\ne0 α = 0 the module Ω U τ ( α ) \Omega_U^{\tau(\alpha)} Ω U τ ( α ) contains no vacuum-like vector. Consequently, a vector of U U U of h \mathfrak{h} h weight λ \lambda λ has conformal weight at least ∣ λ ∣ 2 2 \frac{|\lambda|^2}{2} 2 ∣ λ ∣ 2
Γ ( U ) = Γ 0 \Gamma(U)=\Gamma_0 Γ ( U ) = Γ 0 and the associated lattice is L = L k ′ , k ′ ′ L=L_{k',k''} L = L k ′ , k ′′ it is L I L_{\mathrm{I}} L I in case ( I ) (I) ( I ) and L I I L_{\mathrm{II}} L II in case ( I I ) (II) ( I I )
For λ = ( k ′ , k ′ ′ ) \lambda=(k',k'') λ = ( k ′ , k ′′ ) the vector e λ e^\lambda e λ of V L ⊂ U V_L\subset U V L ⊂ U is a highest weight vector of weight ( k ′ , k ′ ′ ) (k',k'') ( k ′ , k ′′ ) for the affine subalgebra A = L g ′ ( k ′ , 0 ) ⊗ L g ′ ′ ( k ′ ′ , 0 ) A=L_{\mathfrak{g}'}(k',0)\otimes L_{\mathfrak{g}''}(k'',0) A = L g ′ ( k ′ , 0 ) ⊗ L g ′′ ( k ′′ , 0 ) of conformal weight k ′ + k ′ ′ 4 \frac{k'+k''}{4} 4 k ′ + k ′′ and it generates an A A A submodule of U U U isomorphic to the simple current J = L s l 2 ( k ′ , k ′ ) ⊗ L s l 2 ( k ′ ′ , k ′ ′ ) J=L_{\mathfrak{sl}_2}(k',k')\otimes L_{\mathfrak{sl}_2}(k'',k'') J = L sl 2 ( k ′ , k ′ ) ⊗ L sl 2 ( k ′′ , k ′′ ) of conformal weight 10 10 10 in case ( I ) (I) ( I ) and 4 4 4 in case ( I I ) (II) ( I I )
Proof. 1. By [62 Definition 2.12, Proposition 2.14] , which rest on [57 ] , Ω U \Omega_U Ω U V L V_L V L U U U Ω U \Omega_U Ω U [13 ] , and U U U Ω U ⊗ V L \Omega_U\otimes V_L Ω U ⊗ V L A U ⊂ L ′ / L A_U\subset L^{\prime}/L A U ⊂ L ′ / L L ′ / L L^{\prime}/L L ′ / L Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U)\cong Z(H_3) Rep ( U ) ≅ Z ( H 3 ) (3) . It is pseudo-unitary for the ribbon structure of U U U U U U qdim U ( X ) = S X , U / S U , U \operatorname{qdim}_U(X)=S_{X,U}/S_{U,U} qdim U ( X ) = S X , U / S U , U [19 ] , and they are the categorical dimensions of Rep ( U ) \operatorname{Rep}(U) Rep ( U ) [49 ] . Alternatively, Z ( H 3 ) Z(H_3) Z ( H 3 ) [62 Corollary 4.10] , applied with V = W = U V=W=U V = W = U A U = L ′ / L A_U=L^{\prime}/L A U = L ′ / L τ \tau τ Rep ( Ω U ) ≅ Rep ( U ) ⊠ C ( L ′ / L ) ‾ \operatorname{Rep}(\Omega_U)\cong\operatorname{Rep}(U)\boxtimes\overline{\mathcal{C}(L^{\prime}/L)} Rep ( Ω U ) ≅ Rep ( U ) ⊠ C ( L ′ / L ) Ω U \Omega_U Ω U 8 − rank L = 6 8-\operatorname{rank}L=6 8 − rank L = 6 ( Ω U ) 1 (\Omega_U)_1 ( Ω U ) 1 h \mathfrak{h} h h \mathfrak{h} h U 1 = s l 2 ⊕ s l 2 U_1=\mathfrak{sl}_2\oplus\mathfrak{sl}_2 U 1 = sl 2 ⊕ sl 2 h \mathfrak{h} h h \mathfrak{h} h
Let μ \mu μ Ω U \Omega_U Ω U μ \mu μ τ ( α ) \tau(\alpha) τ ( α ) χ μ ( α ) \chi_\mu(\alpha) χ μ ( α ) α ↦ χ μ ( α ) \alpha\mapsto\chi_\mu(\alpha) α ↦ χ μ ( α ) L ′ / L L^{\prime}/L L ′ / L L ′ / L L^{\prime}/L L ′ / L β ∈ L ′ / L \beta\in L^{\prime}/L β ∈ L ′ / L μ ⊠ β \mu\boxtimes\beta μ ⊠ β τ ( α ) ⊠ α \tau(\alpha)\boxtimes\alpha τ ( α ) ⊠ α U U U X = ⨁ α ( τ ( α ) μ ) ⊗ V α + β + L X=\bigoplus_{\alpha}(\tau(\alpha)\mu)\otimes V_{\alpha+\beta+L} X = ⨁ α ( τ ( α ) μ ) ⊗ V α + β + L τ ( α ) μ \tau(\alpha)\mu τ ( α ) μ U U U [45 , 15 ] , a finite sum of irreducible U U U μ ⊗ V β + L \mu\otimes V_{\beta+L} μ ⊗ V β + L y > 0 y>0 y > 0
ch X ( i y ) ≥ ch μ ( i y ) θ β + L ( i y ) η ( i y ) 2 . \operatorname{ch}_X(iy)\geq\operatorname{ch}_\mu(iy)\frac{\theta_{\beta+L}(iy)}{\eta(iy)^2}. ch X ( i y ) ≥ ch μ ( i y ) η ( i y ) 2 θ β + L ( i y ) . As y → 0 + y\to0^+ y → 0 + θ β + L ( i y ) ∼ ( det L ) − 1 2 y − 1 \theta_{\beta+L}(iy)\sim(\det L)^{-\frac12}y^{-1} θ β + L ( i y ) ∼ ( det L ) − 2 1 y − 1 η ( i y ) − 2 = y η ( i / y ) − 2 ∼ y e 2 π 24 y \eta(iy)^{-2}=y\eta(i/y)^{-2}\sim ye^{\frac{2\pi}{24y}} η ( i y ) − 2 = y η ( i / y ) − 2 ∼ y e 24 y 2 π ( det L ) − 1 2 e 2 π 24 y (\det L)^{-\frac12}e^{\frac{2\pi}{24y}} ( det L ) − 2 1 e 24 y 2 π ch X ( i y ) = ∑ Y S X , Y ch Y ( i / y ) \operatorname{ch}_X(iy)=\sum_Y S_{X,Y}\operatorname{ch}_Y(i/y) ch X ( i y ) = ∑ Y S X , Y ch Y ( i / y ) U U U Y Y Y [72 ] at the point i / y i/y i / y X X X ch Y ( i / y ) = O ( e 2 π ⋅ 8 24 y ) \operatorname{ch}_Y(i/y)=O(e^{\frac{2\pi\cdot8}{24y}}) ch Y ( i / y ) = O ( e 24 y 2 π ⋅ 8 ) U U U ch μ ( i y ) = O ( e 2 π ⋅ 6 24 y ) \operatorname{ch}_\mu(iy)=O(e^{\frac{2\pi\cdot6}{24y}}) ch μ ( i y ) = O ( e 24 y 2 π ⋅ 6 ) Ω U \Omega_U Ω U μ \mu μ Ω U \Omega_U Ω U ν \nu ν h ν h_\nu h ν Ω U \Omega_U Ω U ch ν ( i / y ) = ∑ μ S ν , μ Ω ch μ ( i y ) = O ( e 2 π ⋅ 6 24 y ) \operatorname{ch}_\nu(i/y)=\sum_\mu S^\Omega_{\nu,\mu}\operatorname{ch}_\mu(iy)=O(e^{\frac{2\pi\cdot6}{24y}}) ch ν ( i / y ) = ∑ μ S ν , μ Ω ch μ ( i y ) = O ( e 24 y 2 π ⋅ 6 ) ch ν ( i / y ) ≥ dim ( ν h ) e 2 π y ( 6 24 − h ν ) \operatorname{ch}_\nu(i/y)\geq\dim(\nu_h)e^{\frac{2\pi}{y}(\frac{6}{24}-h_\nu)} ch ν ( i / y ) ≥ dim ( ν h ) e y 2 π ( 24 6 − h ν ) h ν ≥ 0 h_{\nu} \ge0 h ν ≥ 0 α ≠ 0 \alpha\ne0 α = 0 Ω U τ ( α ) \Omega_U^{\tau(\alpha)} Ω U τ ( α ) Ω U \Omega_U Ω U Vac Ω U ( Ω U τ ( α ) ) = 0 \operatorname{Vac}_{\Omega_U}(\Omega_U^{\tau(\alpha)}) = 0 Vac Ω U ( Ω U τ ( α ) ) = 0 h \mathfrak{h} h λ ∈ α + L \lambda\in\alpha+ L λ ∈ α + L Ω U τ ( α ) ⊗ M ( h , λ ) \Omega_U^{\tau(\alpha)} \otimes M(\mathfrak{h},\lambda) Ω U τ ( α ) ⊗ M ( h , λ ) h τ ( α ) + ∣ λ ∣ 2 2 ≥ ∣ λ ∣ 2 2 h_{\tau(\alpha)} + \frac{|\lambda|^2}{2} \ge\frac{|\lambda|^2}{2} h τ ( α ) + 2 ∣ λ ∣ 2 ≥ 2 ∣ λ ∣ 2
The h \mathfrak{h} h Ω U τ ( α ) ⊗ V α + L \Omega_U^{\tau(\alpha)} \otimes V_{\alpha+L} Ω U τ ( α ) ⊗ V α + L α + L \alpha+ L α + L Γ ( U ) = L ′ \Gamma(U) = L' Γ ( U ) = L ′ L = Γ ( U ) ∗ L = \Gamma(U)^* L = Γ ( U ) ∗ V 1 ′ = U 1 ⊕ R b V'_1 = U_1 \oplus R_b V 1 ′ = U 1 ⊕ R b Table 2 , every irreducible U 1 U_1 U 1 V 1 ′ V'_1 V 1 ′ ( a , b ) (a,b) ( a , b ) a ≡ b ( m o d 2 ) a \equiv b \pmod2 a ≡ b ( mod 2 ) Γ 0 \Gamma_0 Γ 0 V ′ V' V ′ V 1 ′ V'_1 V 1 ′ h \mathfrak{h} h V ′ V' V ′ Γ 0 \Gamma_0 Γ 0 Γ ( U ) ⊂ Γ 0 \Gamma(U) \subset\Gamma_0 Γ ( U ) ⊂ Γ 0 U 1 U_1 U 1 ( 2 , 0 ) (2,0) ( 2 , 0 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) V ( 1 , 1 ) ⊂ U 2 V(1,1) \subset U_2 V ( 1 , 1 ) ⊂ U 2 Table 2 gives ( 1 , 1 ) (1,1) ( 1 , 1 ) Γ 0 \Gamma_0 Γ 0 Γ ( U ) = L ′ \Gamma(U) = L' Γ ( U ) = L ′ Γ ( U ) = Γ 0 \Gamma(U) = \Gamma_0 Γ ( U ) = Γ 0 L = L k ′ , k ′ ′ L = L_{k',k''} L = L k ′ , k ′′
Since λ = ( k ′ , k ′ ′ ) ∈ L \lambda= (k',k'') \in L λ = ( k ′ , k ′′ ) ∈ L e λ ∈ V L ⊂ U e^\lambda\in V_L \subset U e λ ∈ V L ⊂ U ∣ λ ∣ 2 2 = k ′ + k ′ ′ 4 \frac{|\lambda|^2}{2} = \frac{k'+k''}{4} 2 ∣ λ ∣ 2 = 4 k ′ + k ′′ x x x U 1 U_1 U 1 μ ∈ { ( ± 2 , 0 ) , ( 0 , ± 2 ) } \mu\in\{(\pm2,0),(0,\pm2)\} μ ∈ {( ± 2 , 0 ) , ( 0 , ± 2 )} x ∈ h x \in\mathfrak{h} x ∈ h μ = 0 \mu= 0 μ = 0 n ≥ 0 n \ge0 n ≥ 0 x ( n ) e λ x_{(n)}e^\lambda x ( n ) e λ h \mathfrak{h} h λ + μ \lambda+\mu λ + μ k ′ + k ′ ′ 4 − n \frac{k'+k''}{4}-n 4 k ′ + k ′′ − n n = 0 n=0 n = 0 μ ∈ { ( 2 , 0 ) , ( 0 , 2 ) } \mu\in\{(2,0),(0,2)\} μ ∈ {( 2 , 0 ) , ( 0 , 2 )} ∣ λ + μ ∣ 2 2 − k ′ + k ′ ′ 4 \frac{|\lambda+\mu|^2}{2}-\frac{k'+k''}{4} 2 ∣ λ + μ ∣ 2 − 4 k ′ + k ′′ 1 + 1 k ′ 1+\frac{1}{k'} 1 + k ′ 1 1 + 1 k ′ ′ 1+\frac{1}{k''} 1 + k ′′ 1 n ≥ 1 n \ge1 n ≥ 1 ∣ λ + μ ∣ 2 2 − ( k ′ + k ′ ′ 4 − n ) ≥ min ( 0 , − 1 + 1 k ′ , − 1 + 1 k ′ ′ ) + n > 0 \frac{|\lambda+\mu|^2}{2}-\left(\frac{k'+k''}{4}-n\right) \ge\min(0,-1+\frac{1}{k'},-1+\frac{1}{k''})+n > 0 2 ∣ λ + μ ∣ 2 − ( 4 k ′ + k ′′ − n ) ≥ min ( 0 , − 1 + k ′ 1 , − 1 + k ′′ 1 ) + n > 0 x ( n ) e λ = 0 x_{(n)}e^\lambda=0 x ( n ) e λ = 0 e λ e^\lambda e λ e ( 0 ) e_{(0)} e ( 0 ) f ( 0 ) f_{(0)} f ( 0 ) h ( 0 ) h_{(0)} h ( 0 ) h ( 0 ) ′ h'_{(0)} h ( 0 ) ′ k ′ k' k ′ k ′ ′ k'' k ′′ ( k ′ , k ′ ′ ) (k',k'') ( k ′ , k ′′ ) A A A A A A U U U A A A e λ e^\lambda e λ ( k ′ , k ′ ′ ) (k',k'') ( k ′ , k ′′ ) J J J k ′ ( k ′ + 2 ) 4 ( k ′ + 2 ) + k ′ ′ ( k ′ ′ + 2 ) 4 ( k ′ ′ + 2 ) = k ′ + k ′ ′ 4 \frac{k'(k'+2)}{4(k'+2)}+\frac{k''(k''+2)}{4(k''+2)}=\frac{k'+k''}{4} 4 ( k ′ + 2 ) k ′ ( k ′ + 2 ) + 4 ( k ′′ + 2 ) k ′′ ( k ′′ + 2 ) = 4 k ′ + k ′′
In case (II), part 4 is Proposition 11.4, proved there for S S S A ⊕ J ( 1 , 39 ) ⊂ U R A \oplus J(1,39) \subset U^R A ⊕ J ( 1 , 39 ) ⊂ U R [67 Proposition 7.1] , obtained there from the explicit vector e w e^w e w ( w , w ) = 20 (w,w)=20 ( w , w ) = 20 V ′ V' V ′ V ( 1 , 1 ) V(1,1) V ( 1 , 1 ) U 2 U_2 U 2 L I L_I L I L I I L_{II} L I I 2 k ′ Z ⊕ 2 k ′ ′ Z \sqrt{2k'}\mathbb{Z} \oplus\sqrt{2k''}\mathbb{Z} 2 k ′ Z ⊕ 2 k ′′ Z J J J
The lattice exchange
Let U U U L L L L ♮ L^{\natural} L ♮ { L , L ♮ } = { L I , L I I } \{L,L^{\natural}\}=\{L_I,L_{II}\} { L , L ♮ } = { L I , L I I } ψ : L ′ / L → ( L ♮ ) ′ / L ♮ \psi:L'/L \to(L^{\natural})'/L^{\natural} ψ : L ′ / L → ( L ♮ ) ′ / L ♮ Rep ( Ω U ⊗ V L ) ≅ Rep ( Ω U ) ⊠ C ( L ′ / L ) \operatorname{Rep}(\Omega_U \otimes V_L) \cong\operatorname{Rep}(\Omega_U) \boxtimes\mathcal{C}(L'/L) Rep ( Ω U ⊗ V L ) ≅ Rep ( Ω U ) ⊠ C ( L ′ / L ) Rep ( Ω U ⊗ V L ♮ ) ≅ Rep ( Ω U ) ⊠ C ( ( L ♮ ) ′ / L ♮ ) \operatorname{Rep}(\Omega_U \otimes V_{L^{\natural}}) \cong\operatorname{Rep}(\Omega_U) \boxtimes\mathcal{C}((L^{\natural})'/L^{\natural}) Rep ( Ω U ⊗ V L ♮ ) ≅ Rep ( Ω U ) ⊠ C (( L ♮ ) ′ / L ♮ ) [36 , 15 ] . The isometry ψ \psi ψ Φ ψ : C ( L ′ / L ) → C ( ( L ♮ ) ′ / L ♮ ) \Phi_\psi:\mathcal{C}(L'/L) \to\mathcal{C}((L^{\natural})'/L^{\natural}) Φ ψ : C ( L ′ / L ) → C (( L ♮ ) ′ / L ♮ ) [30 Section 8.4] , and the twists e 2 π i q e^{2\pi i q} e 2 π i q Φ = id ⊠ Φ ψ \Phi=\operatorname{id}\boxtimes\Phi_\psi Φ = id ⊠ Φ ψ [45 ] , U U U A U = ⨁ α ∈ L ′ / L Ω U τ ( α ) ⊠ α A_U=\bigoplus_{\alpha\in L'/L}\Omega_U^{\tau(\alpha)}\boxtimes\alpha A U = ⨁ α ∈ L ′ / L Ω U τ ( α ) ⊠ α Rep ( Ω U ) ⊠ C ( L ′ / L ) \operatorname{Rep}(\Omega_U)\boxtimes\mathcal{C}(L'/L) Rep ( Ω U ) ⊠ C ( L ′ / L ) U ♮ = U ψ ♮ U^{\natural}=U^{\natural}_{\psi} U ♮ = U ψ ♮ U ⊗ V L ♮ U\otimes V_{L^{\natural}} U ⊗ V L ♮ Φ ( A U ) \Phi(A_U) Φ ( A U ) [45 , 15 ] . As an Ω U ⊗ V L ♮ \Omega_U\otimes V_{L^{\natural}} Ω U ⊗ V L ♮
U ♮ = ⨁ α ∈ L ′ / L Ω U τ ( α ) ⊗ V ψ ( α ) + L ♮ . U^{\natural}=\bigoplus_{\alpha\in L'/L}\Omega_U^{\tau(\alpha)}\otimes V_{\psi(\alpha)+L^{\natural}}. U ♮ = α ∈ L ′ / L ⨁ Ω U τ ( α ) ⊗ V ψ ( α ) + L ♮ . Theorem 12.3 Let U U U L ♮ L^{\natural} L ♮ ψ \psi ψ
U ♮ U^{\natural} U ♮ Theorem 8.5 . For the Cartan subalgebra L ♮ ⊗ C L^{\natural}\otimes\mathbb{C} L ♮ ⊗ C U 1 ♮ U^{\natural}_1 U 1 ♮ Ω U \Omega_U Ω U L ♮ L^{\natural} L ♮
Let P P P Rep ( Ω U ) \operatorname{Rep}(\Omega_U) Rep ( Ω U ) Aut ( Ω U ) \operatorname{Aut}(\Omega_U) Aut ( Ω U ) O ( P ) O(P) O ( P ) O ( P ) O(P) O ( P )
strongly rational vertex operator algebra Ω \Omega Ω N ∈ { L I , L I I } N \in\{L_{\mathrm{I}}, L_{\mathrm{II}}\} N ∈ { L I , L II } Ω \Omega Ω N N N
Up to isomorphism, U ♭ U^{\flat} U ♭ ψ \psi ψ ( U ♭ ) ♭ ≅ U (U^{\flat})^{\flat} \cong U ( U ♭ ) ♭ ≅ U U ↦ U ♭ U \mapsto U^{\flat} U ↦ U ♭ Ω U \Omega_U Ω U U U U U ♭ U^{\flat} U ♭
U U U U ♭ U^{\flat} U ♭ U U U U ♭ U^{\flat} U ♭
Proof: 1. Extension and representation category. Put B = Ω U ⊗ V L ♭ B = \Omega_U \otimes V_{L^{\flat}} B = Ω U ⊗ V L ♭ A = Φ ( A U ) A = \Phi(\mathcal{A}_U) A = Φ ( A U ) B B B [45 Theorem 3.2 and Remark 3.3] . Strict positivity of the nonvacuum B B B A A A W W W A A A End A ( A ) ≅ Hom ( 1 , A ) = C \operatorname{End}_A(A) \cong\operatorname{Hom}(\mathbf{1}, A) = \mathbb{C} End A ( A ) ≅ Hom ( 1 , A ) = C A A A W W W A A A W W W Rep ( B ) \operatorname{Rep}(B) Rep ( B ) [45 , 15 ] . Their category is ribbon equivalent, through Φ \Phi Φ A U \mathcal{A}_U A U Rep ( W ) ≅ Rep ( U ) \operatorname{Rep}(W) \cong\operatorname{Rep}(U) Rep ( W ) ≅ Rep ( U )
CFT type. For α ≠ 0 \alpha\ne0 α = 0 Ω U τ ( α ) ⊗ V ψ ( α ) + L ♭ \Omega_U^{\tau(\alpha)} \otimes V_{\psi(\alpha)+L^{\flat}} Ω U τ ( α ) ⊗ V ψ ( α ) + L ♭ h τ ( α ) + q L ♭ ( ψ ( α ) ) + Z = h τ ( α ) + q L ( α ) + Z = Z h_{\tau(\alpha)}+q_{L^{\flat}}(\psi(\alpha))+\mathbb{Z}=h_{\tau(\alpha)}+q_L(\alpha)+\mathbb{Z}=\mathbb{Z} h τ ( α ) + q L ♭ ( ψ ( α )) + Z = h τ ( α ) + q L ( α ) + Z = Z h τ ( α ) + min γ ∈ ψ ( α ) + L ♭ ∣ γ ∣ 2 2 > 0 h_{\tau(\alpha)}+\min_{\gamma\in\psi(\alpha)+L^{\flat}}\frac{\lvert\gamma\rvert^2}{2}>0 h τ ( α ) + min γ ∈ ψ ( α ) + L ♭ 2 ∣ γ ∣ 2 > 0 Proposition 12.2 , part 2. Hence they are at least 1. The summand α = 0 \alpha=0 α = 0 Ω U ⊗ V L ♭ \Omega_U \otimes V_{L^{\flat}} Ω U ⊗ V L ♭ W W W
Self-contragredience. Since ( Ω U ) 1 = 0 (\Omega_U)_1=0 ( Ω U ) 1 = 0 W 1 = 1 ⊗ ( V L ♭ ) 1 ⊕ ⨁ α ≠ 0 ( Ω U τ ( α ) ⊗ V ψ ( α ) + L ♭ ) 1 W_1=\mathbf{1}\otimes(V_{L^{\flat}})_1\oplus\bigoplus_{\alpha\ne0}(\Omega_U^{\tau(\alpha)}\otimes V_{\psi(\alpha)+L^{\flat}})_1 W 1 = 1 ⊗ ( V L ♭ ) 1 ⊕ ⨁ α = 0 ( Ω U τ ( α ) ⊗ V ψ ( α ) + L ♭ ) 1 L ( 1 ) L(1) L ( 1 ) W 1 W_1 W 1 W 0 = C 1 W_0=\mathbb{C}\mathbf{1} W 0 = C 1 α = 0 \alpha=0 α = 0 L ( 1 ) ( V L ♭ ) 1 = 0 L(1)(V_{L^{\flat}})_1=0 L ( 1 ) ( V L ♭ ) 1 = 0 L ( 1 ) W 1 = 0 L(1)W_1=0 L ( 1 ) W 1 = 0 W W W [55 ] .
C 2 C_2 C 2 W W W C 2 C_2 C 2 C 2 C_2 C 2 Ω U ⊗ V L ♭ \Omega_U\otimes V_{L^{\flat}} Ω U ⊗ V L ♭ C 2 C_2 C 2 [11 ] , and C 2 C_2 C 2 C 2 ( W ) C_2(W) C 2 ( W )
Rationality. Let M M M W W W [1 ] , M M M Ω U ⊗ V L ♭ \Omega_U\otimes V_{L^{\flat}} Ω U ⊗ V L ♭ B = Ω U ⊗ V L ♭ B=\Omega_U\otimes V_{L^{\flat}} B = Ω U ⊗ V L ♭ M ′ M' M ′ W W W B B B M 0 M_0 M 0 M ′ M' M ′ w ( n ) x w_{(n)}x w ( n ) x w ∈ W w\in W w ∈ W x ∈ N x\in N x ∈ N n ∈ Z n\in\mathbb{Z} n ∈ Z B B B K K K
Hom B ( M ′ , K ) ⟶ I B ( K W N ) . \operatorname{Hom}_B(M',K)\longrightarrow I_B\left(\begin{matrix} K \\ W\quad N \end{matrix}\right). Hom B ( M ′ , K ) ⟶ I B ( K W N ) . It is injective because a map in its kernel vanishes on this spanning set. The target is finite dimensional and is zero for all but finitely many K K K B B B M ′ M' M ′ K K K Hom B ( M ′ , K ) \operatorname{Hom}_B(M',K) Hom B ( M ′ , K ) M ′ M' M ′ Rep ( B ) \operatorname{Rep}(B) Rep ( B ) [45 , 15 ] , it is a local module for the connected étale algebra Φ ( A U ) \Phi(\mathcal{A}_U) Φ ( A U ) [23 ] . Hence this submodule is a finite direct sum of irreducible W W W M M M W W W [62 Proposition 5.3] , which is the same construction in the holomorphic case.
Positivity. Let X X X W W W Ω U ν ⊗ V γ + L ♮ \Omega^{\nu}_{U} \otimes V_{\gamma+L^{\natural}} Ω U ν ⊗ V γ + L ♮ h ν ≥ 0 h_{\nu} \ge0 h ν ≥ 0 Proposition 12.2 , part 2), so that h X ≥ 0 h_X \ge0 h X ≥ 0 h X = 0 h_X = 0 h X = 0 h ν = 0 h_{\nu} = 0 h ν = 0 γ ∈ L ♮ \gamma\in L^{\natural} γ ∈ L ♮ X X X Ω U ν ⊗ V L ♮ \Omega^{\nu}_{U} \otimes V_{L^{\natural}} Ω U ν ⊗ V L ♮ h ν = 0 h_{\nu} = 0 h ν = 0 A U \mathcal{A}_{U} A U X ♭ = Φ − 1 ( X ) X^{\flat} = \Phi^{-1}(X) X ♭ = Φ − 1 ( X ) U U U Φ − 1 ( ν ⊠ 0 ) = ν ⊠ 0 \Phi^{-1}(\nu\boxtimes0) = \nu\boxtimes0 Φ − 1 ( ν ⊠ 0 ) = ν ⊠ 0 Ω U ν ⊗ V L \Omega^{\nu}_{U} \otimes V_{L} Ω U ν ⊗ V L h X ♭ ≤ h ν = 0 h_{X^{\flat}} \le h_{\nu} = 0 h X ♭ ≤ h ν = 0 X ♭ ≅ U X^{\flat} \cong U X ♭ ≅ U U U U X ≅ Φ ( A U ) = W X \cong\Phi(\mathcal{A}_{U}) = W X ≅ Φ ( A U ) = W W W W Theorem 8.5 it is in case (I) or (II).
Heisenberg commutant. Put h ♮ = 1 ⊗ ( L ♮ ⊗ C ) ( − 1 ) 1 ⊂ W 1 \mathfrak{h}^{\natural} = 1 \otimes(L^{\natural} \otimes\mathbb{C})(-1)1 \subset W_{1} h ♮ = 1 ⊗ ( L ♮ ⊗ C ) ( − 1 ) 1 ⊂ W 1 W 1 W_{1} W 1 h ♮ \mathfrak{h}^{\natural} h ♮ ( Ω U ) 1 ⊗ 1 + 1 ⊗ h ♮ = h ♮ (\Omega_{U})_{1} \otimes1 + 1 \otimes\mathfrak{h}^{\natural} = \mathfrak{h}^{\natural} ( Ω U ) 1 ⊗ 1 + 1 ⊗ h ♮ = h ♮ ( Ω U ) 1 = 0 (\Omega_{U})_{1} = 0 ( Ω U ) 1 = 0 h ♮ \mathfrak{h}^{\natural} h ♮ W 1 W_{1} W 1 [25 ] acting semisimply on W W W M ( h ♮ ) M(\mathfrak{h}^{\natural}) M ( h ♮ ) V γ + L ♮ V_{\gamma+L^{\natural}} V γ + L ♮ C e γ \mathbb{C}e_{\gamma} C e γ γ ∈ L ♮ \gamma\in L^{\natural} γ ∈ L ♮ Com W ( M ( h ♮ ) ) = Ω U ⊗ 1 \operatorname{Com}_{W}(M(\mathfrak{h}^{\natural})) = \Omega_{U} \otimes1 Com W ( M ( h ♮ )) = Ω U ⊗ 1 Com W ( Ω U ) = ⨁ α V ψ ( α ) + L ♮ ⊗ Ω U τ ( α ) = V L ♮ \operatorname{Com}_{W}(\Omega_{U}) = \bigoplus_{\alpha} V_{\psi(\alpha)+L^{\natural}} \otimes\Omega^{\tau(\alpha)}_{U} = V_{L^{\natural}} Com W ( Ω U ) = ⨁ α V ψ ( α ) + L ♮ ⊗ Ω U τ ( α ) = V L ♮ Proposition 12.2 , part 2. So the Heisenberg commutant of W W W Ω U \Omega_{U} Ω U L ♮ L^{\natural} L ♮ Proposition 12.2 , part 3, applied to W W W L I L_{I} L I L I I L_{II} L I I L ♮ L^{\natural} L ♮ L L L Lemma 12.1 ), W W W
The map τ : L ′ / L → P \tau: L^{\prime}/L \to P τ : L ′ / L → P U I I U_{II} U I I U U U U ♮ U^{\natural} U ♮ L I I L_{II} L I I τ I I : L I I ′ / L I I → P \tau_{II} : L^{\prime}_{II}/L_{II} \to P τ I I : L I I ′ / L I I → P ( U I I ) 1 (U_{II})_{1} ( U I I ) 1 U I I U_{II} U I I Ω U = Com ( M ( h ) ) \Omega_{U} = \operatorname{Com}(M(\mathfrak{h})) Ω U = Com ( M ( h )) V L I I = Com ( Ω U ) V_{L_{II}} = \operatorname{Com}(\Omega_{U}) V L I I = Com ( Ω U ) θ ∈ Aut ( Ω U ) \theta\in\operatorname{Aut}(\Omega_{U}) θ ∈ Aut ( Ω U ) V L I I V_{L_{II}} V L I I w ′ w' w ′ w ′ ′ w'' w ′′ L I I ′ / L I I L'_{II}/L_{II} L I I ′ / L I I Ω U τ I I ( α ) ⊗ V α + L I I \Omega^{\tau_{II}(\alpha)}_{U} \otimes V_{\alpha+L_{II}} Ω U τ I I ( α ) ⊗ V α + L I I θ ˉ \bar{\theta} θ ˉ θ ˉ τ I I = τ I I w ˉ \bar{\theta}\tau_{II} = \tau_{II}\bar{w} θ ˉ τ I I = τ I I w ˉ Lemma 12.1 , part 4, w ˉ ′ = 14 \bar{w}' = 14 w ˉ ′ = 14 w ˉ ′ ′ = 25 \bar{w}'' = 25 w ˉ ′′ = 25 O ( L I I ′ / L I I ) O(L'_{II}/L_{II}) O ( L I I ′ / L I I ) θ ˉ ∣ P \bar{\theta}|_{P} θ ˉ ∣ P O ( P ) = τ I I O ( L I I ′ / L I I ) τ I I − 1 O(P) = \tau_{II}O(L'_{II}/L_{II})\tau_{II}^{-1} O ( P ) = τ I I O ( L I I ′ / L I I ) τ I I − 1 Y 1 , Y 2 Y_{1}, Y_{2} Y 1 , Y 2 Ω \Omega Ω N N N σ 1 , σ 2 : N ′ / N → P ( Ω ) \sigma_{1},\sigma_{2} : N'/N \to P(\Omega) σ 1 , σ 2 : N ′ / N → P ( Ω ) σ 2 = θ ˉ σ 1 \sigma_{2} = \bar{\theta}\sigma_{1} σ 2 = θ ˉ σ 1 θ ∈ Aut ( Ω ) \theta\in\operatorname{Aut}(\Omega) θ ∈ Aut ( Ω ) Y 1 Y_{1} Y 1 θ ⊗ id \theta\otimes\operatorname{id} θ ⊗ id Ω ⊗ V N \Omega\otimes V_{N} Ω ⊗ V N Y 1 Y_{1} Y 1 Ω σ 2 ( α ) ⊗ V α + N \Omega^{\sigma_{2}(\alpha)} \otimes V_{\alpha+N} Ω σ 2 ( α ) ⊗ V α + N Y 2 Y_{2} Y 2 Ω ⊗ V N \Omega\otimes V_{N} Ω ⊗ V N N ′ / N N'/N N ′ / N
M α ⊠ M β ≅ M α + β . M_{\alpha} \boxtimes M_{\beta} \cong M_{\alpha+\beta}. M α ⊠ M β ≅ M α + β . All summands are nonzero, both extensions are simple, and the fusion multiplicities are one. Thus [25 Proposition 5.3] applies to their common graded module decomposition. Hence Y 1 ≅ Y 2 Y_{1} \cong Y_{2} Y 1 ≅ Y 2
The algebra U ψ ♮ U^{\natural}_{\psi} U ψ ♮ Ω U \Omega_{U} Ω U L ♮ L^{\natural} L ♮ ψ \psi ψ U ψ ♮ U^{\natural}_{\psi} U ψ ♮ ψ − 1 \psi^{-1} ψ − 1 U U U ( U ψ ♮ ) ♮ ≅ U (U^{\natural}_{\psi})^{\natural} \cong U ( U ψ ♮ ) ♮ ≅ U [25 Proposition 5.3] . If Y ≅ Y ′ Y \cong Y' Y ≅ Y ′ Y ♮ ≅ ( Y ′ ) ♮ Y^{\natural} \cong(Y')^{\natural} Y ♮ ≅ ( Y ′ ) ♮ ♮ \natural ♮ W W W Ω W ≅ Ω U \Omega_{W} \cong\Omega_{U} Ω W ≅ Ω U L L L L ♮ L^{\natural} L ♮ W ≅ U W \cong U W ≅ U W ≅ U ♮ W \cong U^{\natural} W ≅ U ♮
By [62 Corollary 4.10] , two strongly rational vertex operator algebras with equivalent unpointed pseudo-unitary representation categories belong to the same hyperbolic genus if and only if they have the same central charge and isomorphic Heisenberg commutants. This applies to U U U U ♮ U^{\natural} U ♮ V V V
Theorems 4.7 and 4.8]. Rep ( V ) \operatorname{Rep}(V) Rep ( V ) Rep ( U ) \operatorname{Rep}(U) Rep ( U ) V V V Ω U \Omega_U Ω U L L L L I L_{\mathrm{I}} L I L I I L_{\mathrm{II}} L II Lemma 12.1 ). By the proof of [MoR24a, Corollary 4.10], V = ⨁ α ∈ P Ω U σ ( α ) ⊗ V α + N V=\bigoplus_{\alpha\in P}\Omega_U^{\sigma(\alpha)}\otimes V_{\alpha+N} V = ⨁ α ∈ P Ω U σ ( α ) ⊗ V α + N N N N σ \sigma σ P P P Y 1 Y_1 Y 1 U U U U ♮ U^\natural U ♮ N N N Y 2 = V Y_2=V Y 2 = V Y 2 Y_2 Y 2 V ≅ Y 1 V\cong Y_1 V ≅ Y 1
Theorem 12.4
Assuming ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) V ≅ S V\cong S V ≅ S ( I I ) (\mathrm{II}) ( II ) S ♮ S^\natural S ♮ ( I ) (\mathrm{I}) ( I ) S ♮ S^\natural S ♮ V ′ V' V ′ ψ R ( U R ) \psi_R(U^R) ψ R ( U R ) A A A e e e f f f v v v Proposition 10.36 ), with equality in weights at most four.
Assuming ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U R U^R U R ( I ) (\mathrm{I}) ( I ) ( U R ) ♮ (U^R)^\natural ( U R ) ♮ ( I I ) (\mathrm{II}) ( II ) ( U R ) ♮ (U^R)^\natural ( U R ) ♮ V ′ V' V ′ S S S
Assuming ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) S ♮ ≅ U R S^\natural\cong U^R S ♮ ≅ U R ( U R ) ♮ ≅ V (U^R)^\natural\cong V ( U R ) ♮ ≅ V
Assuming ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J )
Proof: 1. By Theorem 10.29 , every Haagerup algebra in case ( I I ) (\mathrm{II}) ( II ) V ≅ S V\cong S V ≅ S Theorem 12.3 , part 3, every Haagerup algebra in case ( I ) (\mathrm{I}) ( I ) S ♮ S^\natural S ♮ Theorem 10.38 shows, without using [66 ] , that a Haagerup algebra in case ( I ) (\mathrm{I}) ( I ) ψ R ( U R ) \psi_R(U^R) ψ R ( U R ) V ′ V' V ′ ψ R ( U R ) \psi_R(U^R) ψ R ( U R ) Theorem 7.1 ). 2. The same argument with Theorem 10.38 . The proof of Theorem 10.29 shows, without [52 ] , that a Haagerup algebra in case ( I I ) (\mathrm{II}) ( II ) S S S V ′ V' V ′ Theorem 10.21 , using the independent certificate of Proposition 10.17 , gives the dimensions 1, 6, 120, 956, 6010 of S S S Theorem 12.3 , parts 3 and 4.
Theorem 12.3 pairs the two cases within each hyperbolic genus. For the explicit algebra S S S ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) Theorem 12.4 then give the complete classification by the pair S S S S ♮ S^\natural S ♮ ( I ) (\mathrm{I}) ( I )
The common fixed-point algebra of [DJX26, Theorem 7.3] is used in [ArhIII, Theorem 7.9] to identify S ♮ S^\natural S ♮ V H V^{\mathrm{H}} V H S ♮ ≅ V H ≅ U R S^\natural\cong V^{\mathrm{H}}\cong U^R S ♮ ≅ V H ≅ U R ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) Theorem 12.4 apply, and ( U R ) ♮ ≅ S (U^R)^\natural\cong S ( U R ) ♮ ≅ S Theorem 12.3 (3). The ambient containment and the finite weight comparison in Theorem 12.4 (1) come from the calculations in this article.
Remark 12.5 The decomposition of Proposition 12.2 can be tested on the graded dimensions. For μ ∈ Γ 0 \mu\in\Gamma_0 μ ∈ Γ 0 h \mathfrak{h} h μ \mu μ U n U_n U n ∏ m ≥ 1 ( 1 − q m ) 2 \prod_{m\geq1}(1-q^m)^2 ∏ m ≥ 1 ( 1 − q m ) 2 q − ∣ μ ∣ 2 / 2 q^{-\lvert\mu\rvert^2/2} q − ∣ μ ∣ 2 /2 f α f_\alpha f α S U Ω S^\Omega_U S U Ω α = μ + L \alpha=\mu+L α = μ + L U 2 U_2 U 2 U 3 U_3 U 3 ( I I ) (\mathrm{II}) ( II ) ( I ) (\mathrm{I}) ( I ) μ \mu μ ( I I ) (\mathrm{II}) ( II ) ( I ) (\mathrm{I}) ( I ) ψ \psi ψ Lemma 12.1 , part 3, the 88 coefficients of f α f_\alpha f α ( I I ) (\mathrm{II}) ( II ) f ψ ( α ) f_{\psi(\alpha)} f ψ ( α ) ( I ) (\mathrm{I}) ( I ) Theorem 12.3 proves for U U U U ♭ U^\flat U ♭
Remark 12.6 Let V V V Rep ( V ) ≃ Z ( H 3 ) \operatorname{Rep}(V) \simeq Z(\mathcal{H}_3) Rep ( V ) ≃ Z ( H 3 ) Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [42 Theorem 6.3] . Since pivotal structures differ by tensor automorphisms of the identity functor, which are characters of this group, Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [31 ] . Since c ( V ) ≡ 0 ≡ c ( Z ( H 3 ) ) ( m o d 8 ) c(V) \equiv0 \equiv c(Z(\mathcal{H}_3)) \pmod8 c ( V ) ≡ 0 ≡ c ( Z ( H 3 )) ( mod 8 ) S S S V V V S S S s s s § 2 [62 Lemma 3.6] . Suppose that the minimal conformal weight h min h_{\min} h m i n V V V J J J h min h_{\min} h m i n 0 0 0 J J J { b , a , c 0 } \{b,a,c_0\} { b , a , c 0 } { c 1 } \{c_1\} { c 1 } { c 2 } \{c_2\} { c 2 } { d l } \{d_l\} { d l } i i i s i j ≤ 0 s_{ij}\leq0 s ij ≤ 0 j ∈ J j\in J j ∈ J j j j
Table 6.
The signs follow directly from the entries − y / 3 -y/3 − y /3 − 1 / 3 -1/3 − 1/3 0 0 0 S S S ch i ( i y ) ∼ ( ∑ j ∈ J s i j dim V h min j ) e 2 π y ( c 24 − h min ) \operatorname{ch}_i(iy)\sim\left(\sum_{j\in J}s_{ij}\dim V_{h_{\min}}^j\right)\mathrm{e}^{\frac{2\pi}{y}(\frac{c}{24}-h_{\min})} ch i ( i y ) ∼ ( ∑ j ∈ J s ij dim V h m i n j ) e y 2 π ( 24 c − h m i n ) y > 0 y>0 y > 0 [62 Proposition 3.10] with a weaker sign condition: Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [62 ] (for J = { b , a } J=\{b,a\} J = { b , a } i i i s i b < 0 s_{ib}<0 s ib < 0 s i a < 0 s_{ia}<0 s ia < 0 J J J 0 0 0 h min < 0 h_{\min}<0 h m i n < 0 V V V b b b a a a c 0 c_0 c 0
Computational methods and certificates
The proofs distinguish exact calculations, certified exclusions, calculations modulo primes and numerical checks. Exact calculations use rational arithmetic, arithmetic in K = Q ( 13 , − 3 ) K=\mathbb{Q}(\sqrt{13},\sqrt{-3}) K = Q ( 13 , − 3 ) 1 , 13 , − 3 , − 39 1,\sqrt{13},\sqrt{-3},\sqrt{-39} 1 , 13 , − 3 , − 39 Q \mathbb{Q} Q Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
For a matrix integral at a prime p p p K K K K K K K K K
The supplementary material indexes the source code, coordinates, exact witnesses and verification results by the numbered statements. It specifies the computational environments and commands. The lattice construction gives the vertex algebra identities in arbitrary weights, while § 10 justifies the momentum truncation used in the certificates.
The following table identifies the scripts associated with numbered statements and the nature of their computations. It distinguishes exact verification, certified exclusions, reductions modulo primes and auxiliary numerical tests. An entry does not assert more than the precise statement and its proof establish. In particular, modular calculations are not treated as characteristic-zero lifts.
Table 7.
Coverage of the certificates
The trace certificates give the dual forms, vanishing relations, matrix ranks and eighteen moments of U 2 U_2 U 2 Theorem 8.5 .
For Theorem 10.5 , the thirteen excluded commutant candidates give thirty-nine rational projective charts with unit Gröbner bases. The pairwise closure calculation gives 62 certified nonzero interval witnesses, which exclude 254 of the 256 choices. The coverage check uses only these certified witnesses. The primary-space computations retain all characteristic-polynomial factors and do not discard an inconclusive eigenspace.
The weight-four certificate of Proposition 10.17 checks the complete input domains, the identity of each pivot word, integrality at the chosen prime, full rank and every coordinate of each membership identity. It covers all 131 nonzero Cartan blocks and checks the 886 zero inputs separately. Missing or truncated data, duplicate labels and nonzero residuals are rejected. This establishes (F4) without an external existence theorem. Corollary 10.22 gives a separate dimension proof when an ambient Haagerup algebra is available.
For Proposition 10.31 , the supplementary data give the seven exact commutant vectors, the two Virasoro matrices and the primary vectors. Their matrix identities and ranks admit a separate verification in rational arithmetic. This verification is distinct from the lattice-mode checks of the vertex-product identities.
Independent mode comparisons. A second evaluator, using simultaneous normal ordering and direct exponential coefficients, agrees with the iterate-formula implementation on 24643 basis-state products in the stated weight window and 310 generator products. The coefficient-field multiplication and three Virasoro identities are checked separately. These comparisons use the supplied generator coordinates. They do not independently reconstruct those coordinates from the printed definitions, or regenerate every coordinate of (F4) by a second method.
The modular Zhu calculation. All 131698 overlap ambiguities and 151 inclusion ambiguities of the final presentation are checked. At 8388571, the algebra defined by the projected relations has dimension 762, nondegenerate trace form and the twelve stated blocks. These checks concern that presentation. They do not prove a characteristic-zero lift, recompute the vertex-algebra origin of every relation or show that every block survives in the Zhu corner. The agreement with the calculation at a second prime has the status specified in Remark 11.7 .
AI statement. The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions, the writing of the verification scripts and feedback during the development and preparation of this manuscript. The author takes full responsibility for the mathematical content of the manuscript.
Competing interests. The author declares that he has no competing interests.
Data availability. The source code, certificates and reproduction instructions are available at https://github.com/cedricarhancet/haagerup-c8 . The two complete weight-four coordinate archives are distributed as release assets. Table 5 relates the computational scripts to the numbered statements. Author’s GitHub profile. https://github.com/cedricarhancet
Table 5. Correspondence between numbered statements and computational scripts.
Bibliography 