Introduction
The Haagerup subfactor of index 5 + 13 2 \frac{5+\sqrt{13}}{2} 2 5 + 13 [5 ] 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 [41 ] . 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 ≔ 3 + 13 2 . (2) d \coloneqq\frac{3+\sqrt{13}}{2}. \tag*{(2)} d : = 2 3 + 13 . (2) Whether the Haagerup categories, which do not come from finite groups or quantum groups, are realized in conformal field theory is a long-standing question. Numerically, Huang, Lin, Ohmori, Tachikawa and Tezuka [44 ] and Vanhoove and collaborators [69 ] found evidence for a Haagerup symmetric conformal field theory with c ≈ 2 c \approx2 c ≈ 2 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 [2 ] suggested that the chiral ( e 8 ) 1 (\mathfrak{e}_8)_1 ( e 8 ) 1 [61 ] , and Gannon and Rayhaun proposed [39 ] , 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 [39 Section 4.3] . Recently, Jiao [50 ] 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 ) [48 ] subsequently each announced such an algebra, with currents of levels ( 1 , 39 ) (1,39) ( 1 , 39 ) [33 ] 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.
In this paper we determine all such algebras, up to the existence results of [50 Rie26a] , which we use only as hypotheses, and we explain why there are two of them. 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^{-c/24}(1+O(q)) q − c /24 ( 1 + O ( q )) [70 , 47 ] . Both vectors of [EvG11] 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 Haagerup algebra . Then the character vector of 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 [8 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 Lemma 6.1 ) gives relations of weights four and six, which, together with the integrality of the weights of s l 2 \mathfrak{sl}_2 sl 2 e 8 \mathfrak{e}_8 e 8
Theorem 1.2 Let U U U Haagerup algebra . Then 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 ) ( I ) (I) ( I ) ( 3 , 13 ) (3,13) ( 3 , 13 ) ( I I ) (II) ( I I ) 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 [65 ] of Riesen, which are therefore forced by the representation category once the levels are ( 1 , 39 ) (1,39) ( 1 , 39 ) 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 [50 ] .
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^{\mathrm{R}} = \langle A,e,f,v\rangle U R = ⟨ A , e , f , v ⟩ Proposition 10.36 ), and exchange the remaining choices by automorphisms of V ′ V' V ′ S S S ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) V ′ V' V ′ ψ R \psi_{\mathrm{R}} ψ R V ′ V' V ′ Theorems 10.38 and 10.29 give equality, granting the corresponding existence result.
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 [55 , 60 ] . This decomposition was used by Höhn [46 ] to study the genus of [45 ] , and it leads to the hyperbolic genus of Moriwaki [59 ] and Möller–Rayhaun [60 ] .
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 ) ≅ Z ( H 3 ) ⊠ C ( L ′ / L ) ‾ \operatorname{Rep}(\Omega_U) \cong\mathcal{Z}(\mathcal{H}_3) \boxtimes\overline{\mathcal{C}(L'/L)} Rep ( Ω U ) ≅ Z ( 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^{\natural} U ♮ Ω U \Omega_U Ω U U ↦ U ♮ U \mapsto U^{\natural} U ↦ U ♮ U U U U ♮ U^{\natural} U ♮ U U U
The precise statements are Proposition 12.2 and Theorem 12.3. Part 1 combines [60 Proposition 2.14] and the proof of [60 Corollary 4.10] with the weight decomposition in Table 2 and a growth estimate for characters. In part 2, U ♮ U^{\natural} 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^{\natural\prime}/L^{\natural}) Rep ( Ω U ) ⊠ C ( L ′ / L ) → Rep ( Ω U ) ⊠ C ( L ♮′ / L ♮ ) U ♮ U^{\natural} U ♮ U U U ± 1 \pm1 ± 1 [60 Proposition 5.3] for holomorphic algebras; what is specific here is its computation for Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 )
External existence hypotheses. We write ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) V V V [50 ] is positive and strongly rational, with a braided equivalence Rep ( V ) ≅ Z ( H 3 ) \operatorname{Rep}(V) \cong Z(\mathcal{H}_3) Rep ( V ) ≅ Z ( H 3 ) ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U R U^{\mathrm{R}} U R [50 ] asserts this positivity, and the working manuscript [65 Section 11.1] asserts it for U R U^{\mathrm{R}} U R ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R )
Theorem 1.5
Assuming ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) U ↦ U ♮ U \mapsto U^{\natural} U ↦ U ♮
Assuming ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) S S S S ♮ S^{\natural} S ♮ S ♮ S^{\natural} S ♮ V ′ V^{\prime} V ′ ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) ( 1 , 39 ) (1,39) ( 1 , 39 ) [50 ] .
Assuming ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) U R U^{\mathrm{R}} U R S ♮ ≅ U R S^{\natural} \cong U^{\mathrm{R}} S ♮ ≅ U R ( U R ) ♮ ≅ S (U^{\mathrm{R}})^{\natural} \cong S ( U R ) ♮ ≅ S [48 ] is a Haagerup algebra, it is isomorphic to U R U^{\mathrm{R}} U R
These are Theorems 10.40 and 12.4. Without [50 Rie26a] , the number of isomorphism classes of Haagerup algebras in each case equals the number of hyperbolic genera of Haagerup algebras, and uniqueness in one case implies uniqueness in the other. Table 1 summarizes what depends on which announcement.
Table 1. Dependence of the main statements on [50 Rie26a] and on computations.
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 S S S J J J S S S Table 2 . This is an upper presentation. The computation is Theorem 11.6. Identifying this presentation with the Zhu corner in characteristic zero and proving that all twelve blocks survive remain separate steps. Theorem 11.1 gives a criterion for strong rationality. The further categorical recognition is separated in Remark 11.2: the proposed branching data do not, by themselves, establish the fusion rules, the pointed associator or the required Morita class.
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 K K V ′ V' V ′ Q \mathbb{Q} Q Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
Relation to central charge two. The results of this paper are used in a companion paper [4 ] to exclude rational Haagerup symmetric theories of product type at central charge two, by gluing and through dual pairs inside Haagerup algebras; the twin algebra U ♯ U^\sharp U ♯ U ♯ U^\sharp U ♯ Ω U \Omega_U Ω U
Notation. Throughout, V ′ = V E 6 ⊕ A 2 ′ V' = V_{E_6 \oplus A_2}' V ′ = V E 6 ⊕ A 2 ′ U ⊂ V ′ U \subset V' 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 ) K = \mathbb{Q}(\sqrt{13}, \sqrt{-3}) K = Q ( 13 , − 3 ) S S S Theorem 1.3 and U R = ⟨ A , e , f , v ⟩ U^R = \langle A,e,f,v\rangle U R = ⟨ A , e , f , v ⟩ § 12 , h \mathfrak{h} h U 1 U_1 U 1 Ω U \Omega_U Ω U L L L
Structure of the paper. § 2 fixes the modular data of Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) § 3 treats the character vectors at central charge eight, and § 4 establishes 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 § 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 discusses the generation problem. § 10 determines the commutant of the currents and U 2 U_2 U 2 [50 ] and [Rie26a], and proves the uniqueness theorems and the rational forms. § 11 outlines an independent route to the rationality of S S S § 12 determines the associated lattices and constructs the twin algebras. Appendix § 13 describes the computations.
Preliminaries
The modular data of the Drinfeld center
We use the modular data of Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 )
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,\ 3d+1,\ 3d+2,\ 3d+2,\ 3d+2,\ 3d+2,\ 3d,\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 6 l 2 13 ( 1 ≤ l ≤ 6 ) . (4) 1,\ 1,\ 1,\ 1,\ e^{\frac{2\pi i}{3}},\ e^{-\frac{2\pi i}{3}},\ e^{\frac{2\pi i6l^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 6 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 = 1 3 ( 3 d + 2 ) S_{0,0}=\frac{1}{3(3d+2)} S 0 , 0 = 3 ( 3 d + 2 ) 1
The modular data of Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [49 ] ; the form above is that of [EvG11], and its identification with the Drinfeld center of the Haagerup category is established in [42 ] . 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 The labels 0 0 0 b b b a a a c 0 c_0 c 0 1 1 1 π 1 \pi_1 π 1 π 2 \pi_2 π 2 σ 1 \sigma_1 σ 1 [2 , 9 ] , as stated in [48 p. 2] . The induction of the unit of H 3 \mathcal{H}_3 H 3 I ( 1 ) ≅ 0 ⊕ b ⊕ 2 a I(1)\cong0\oplus b\oplus2a I ( 1 ) ≅ 0 ⊕ b ⊕ 2 a 0 0 0 b b b a a a ρ \rho ρ d d d d − = d e f 3 − 13 2 d_- \mathrel{\mathop{=}^{\mathrm{def}}} \frac{3-\sqrt{13}}{2} d − = def 2 3 − 13
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 [47 ] , 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 Z(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) [40 Theorem 6.3] . Pivotal structures differ by a tensor automorphism of the identity [29 ] ; 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 = d e f e − 2 π i c 24 T T_c\mathrel{\mathop{=}^{\mathrm{def}}}e^{-\frac{2\pi i c}{24}}T T c = def 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_cX(\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 [70 ] and the identification of the modular transformation of the characters with the categorical S S S [47 ] , the following holds.
Lemma 2.3 Let V V V c c c ( Tr M q L 0 − c 24 ) M ∈ Irr ( Rep ( V ) ) (\operatorname{Tr}_{M}q^{L_0-\frac{c}{24}})_{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 [70 ] , with the S S S [47 ] . 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 ′ [63 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)}=\overline{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 [EvG11, 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 e 2 π i Λ = T 8 e^{2\pi i\Lambda}=T_8 e 2 π i Λ = T 8 F = ( J − 240 ) Δ E 10 = ∑ k ≥ 0 f k q k F=(J-240)\frac{\Delta}{E_{10}}=\sum_{k\geq0}f_kq^k F = ( J − 240 ) E 10 Δ = ∑ k ≥ 0 f k q k G = Δ E 10 = ∑ k ≥ 1 g k q k G=\frac{\Delta}{E_{10}}=\sum_{k\geq1}g_kq^k G = E 10 Δ = ∑ 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 = I d \Xi_{-1}=\mathrm{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 − I d ) + 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-\mathrm{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 I d + ∑ n ≥ 0 Ξ n q n ) . (10) \Xi(\tau)\mathrel{\mathop:}=q^\Lambda\left(q^{-1}\mathrm{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 − I d ) F + ( χ + [ Λ , χ ] ) G ) q\frac{\mathrm{d}\Xi}{\mathrm{d}q}=\Xi\left((\Lambda-\mathrm{Id})F+(\chi+[\Lambda,\chi])G\right) q d q d Ξ = Ξ ( ( Λ − Id ) F + ( χ + [ Λ , χ ]) G ) coefficientwise. Indeed, the coefficient of 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 − I d ) \Xi_n(\Lambda-\mathrm{Id}) Ξ n ( Λ − Id ) f 0 = 1 f_0=1 f 0 = 1 g 0 = 0 g_0=0 g 0 = 0 [6 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^{-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 − 1/3 ( 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 [EvG11, 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 = I d \Xi_{-1}=\mathrm{Id} Ξ − 1 = Id C ( J ) \mathbb{C}(J) C ( J ) [6 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 5 5
The modularity of the columns is the content of [EvG11, 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_mq^{\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 2 − 1 = − 1 2 \Lambda_{00}-1=\frac{1}{2}-1=-\frac{1}{2} Λ 00 − 1 = 2 1 − 1 = − 2 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 = χ d 4 , 0 + γ χ d 4 , d 3 = 5 − 3 γ , (\Xi_0)_{d_4,0}+\gamma(\Xi_0)_{d_4,d_3}=\chi_{d_4,0}+\gamma\chi_{d_4,d_3}=5-3\gamma, ( Ξ 0 ) d 4 , 0 + γ ( Ξ 0 ) d 4 , d 3 = χ d 4 , 0 + γ χ 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 [EvG11, 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 U U U 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 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
Table 4.1.
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 ( 1 ) ≅ 0 ⊕ b ⊕ 2 a I(1)\cong0\oplus b\oplus2a I ( 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' \mathrel{\mathop:}= U\oplus M_b V ′ : = U ⊕ M b H H H g g g H H H H g = V ′ H^g=V' 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'\cong V_{E_6\oplus A_2} V ′ ≅ V E 6 ⊕ A 2 V 1 ′ V'_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 ( 1 ) I(1) I ( 1 ) Rep ( U ) \operatorname{Rep}(U) Rep ( U ) [43 , 14 ] , it defines a strongly rational extension H H H U U U Rep ( H ) ≅ Vec \operatorname{Rep}(H)\cong\operatorname{Vec} Rep ( H ) ≅ Vec H H H V E 8 V_{E_8} V E 8
By [21 Theorem 4.10 p. 18] , the subalgebras of I ( 1 ) I(1) I ( 1 ) H 3 \mathcal{H}_3 H 3 Vec Z 3 \operatorname{Vec}_{\mathbb{Z}_3} Vec Z 3 H 3 \mathcal{H}_3 H 3
corresponds to a connected étale subalgebra 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 [43 ] , 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 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 [21 Corollary 3.32 p. 15] . It is Witt equivalent to Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [21 Proposition 5.4 p. 22] , hence of the form Z ( D ) Z(\mathcal{D}) Z ( D ) D \mathcal{D} D [21 Proposition 5.8 p. 23] , with FPdim ( D ) = 3 \operatorname{FPdim}(\mathcal{D})=3 FPdim ( D ) = 3 [29 Corollary 8.30] . Here D ≅ Vec Z 3 ω \mathcal{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^{\frac{2\pi i k}{3}} e 3 2 π ik N k N_k N k g g g H g = V ′ H^g=V' H g = V ′
3. The automorphism group of V E 8 V_{E_8} V E 8 E 8 E_8 E 8 [26 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 [52 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 C D 7 ⊕ C E 7 ⊕ C E_7\oplus 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 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 Z 3 \mathbb{Z}_3 Z 3 [39 Section 4.3 pp. 83–88] , and the announced algebra of V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 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}\pmod{1} 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 )} E 8 / ( E 6 ⊕ A 2 ) = E_8/(E_6\oplus A_2)= E 8 / ( E 6 ⊕ A 2 ) = { ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 ) } \{(0,0),(1,1),(2,2)\} {( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 )} A 2 A_2 A 2 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. Let ( 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(\mathcal{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 [70 Section 5] , [47 ] . 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 [23 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{s o}_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 [23 Theorem 1 p. 2] . By Lemma 4.3, this applies in particular to 3 3 3 27 27 27 p 2 ( s ) p_2(\mathfrak{s}) p 2 ( s ) p 1 ( s ) p_1(\mathfrak{s}) p 1 ( s )
reductive subalgebras of 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 ) \mathfrak{p}_1(\mathfrak{s}) p 1 ( s ) Ω \Omega Ω 27 \mathbf{27} 27 27 27 27 ω 1 \omega_1 ω 1 t \mathfrak{t} t p 1 ( s ) \mathfrak{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 Qv Q v
Remark 4.5 The announced algebra of [Rie26a, Hua26] 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},\mathbf{2})\oplus(\mathbf{15},\mathbf{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 [67 ] and Dong–Mason [DofM04b]. In the context of the Haagerup modular data, weight-two one-point functions of currents were introduced by Gannon [Gan19, 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
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 , (14) 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*{(14)} 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 , (14) 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 SL 2 ( Z ) \operatorname{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\geq2 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\geq1}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\geq1 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\left(A_m+m\kappa\operatorname{Tr}_M Q\right). 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\geq1}\frac{mq^m}{1-q^m}=(1-E_2)/24 ∑ m ≥ 1 1 − q m m q m = ( 1 − E 2 ) /24 κ = 0 \kappa=0 κ = 0 h [ − 1 ] h h[-1]h h [ − 1 ] h L [ 0 ] L[0] L [ 0 ] [70 ] together with the identification of the representation [47 ] .
The obstruction principle
We denote by ϱ ∗ d e f = ( ϱ T ) − 1 \varrho^*\mathrel{\mathop{\rm def}}=(\varrho^{\mathsf T})^{-1} ϱ ∗ def = ( ϱ T ) − 1 § 2 , we have S T = S S^{\mathsf T}=S S 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 ] ∈ SL 2 ( Z ) \gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix}\in\operatorname{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))^{\mathsf T}\varrho(\gamma)^{-\mathsf 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 SL 2 ( Z ) \operatorname{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 s 2 = t s^2=t s 2 = t 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 [8 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 ) . (15) \pi=(c_1\ c_2)(d_1\ d_5)(d_2\ d_3)(d_4\ d_6). \tag*{(15)} π = ( c 1 c 2 ) ( d 1 d 5 ) ( d 2 d 3 ) ( d 4 d 6 ) . (15) 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^{-\frac{2\pi i}{3}}\eta(\tau)^{-8} η ( τ + 1 ) − 8 = e − 3 2 π i 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 ϱ G ( τ ) ) z G_z(\tau+1)=e^{2\pi i(\Lambda_{\pi(z)}-\frac{1}{3})}G_z(\tau)=e^{-2\pi i\Lambda_z}G_z(\tau)=(T\varrho G(\tau))_z G z ( τ + 1 ) = e 2 π i ( Λ π ( z ) − 3 1 ) G z ( τ ) = e − 2 π i Λ z G z ( τ ) = ( T ϱG ( τ ) ) z η ( − 1 τ ) − 8 = τ − 4 η ( τ ) − 8 \eta(-\frac{1}{\tau})^{-8}=\tau^{-4}\eta(\tau)^{-8} η ( − τ 1 ) − 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 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 ) . (16) w_0=[q^{-\frac{1}{3}}]W_0,\qquad w'_0=[q^{\frac{2}{3}}]W_0,\qquad w_z=[q^{\mu_z}]W_z\quad(z\ne0). \tag*{(16)} w 0 = [ q − 3 1 ] W 0 , w 0 ′ = [ q 3 2 ] W 0 , w z = [ q μ z ] W z ( z = 0 ) . (16) 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 (16) 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 + 17 819 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,\qquad
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}{819}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 + 819 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. (17) 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,\qquad
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*{(17)} 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. (17) 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
Step 1: the relations. Consider the 4 ⋅ 12 ⋅ 2 = 96 4\cdot12\cdot2=96 4 ⋅ 12 ⋅ 2 = 96 E 4 Ξ ∙ , j J m , ϑ 2 D ( Ξ ∙ , j J m ) , D J ⋅ D ( Ξ ∙ , j J m ) , ( D J ) 2 Ξ ∙ , j J m , 0 ≤ j ≤ 11 , m ∈ { 0 , 1 } , E_4\Xi_{\bullet,j}J^m,\quad\vartheta_2D(\Xi_{\bullet,j}J^m),\quad DJ\cdot D(\Xi_{\bullet,j}J^m),\quad(DJ)^2\Xi_{\bullet,j}J^m,\quad0\leq j\leq11,\ m\in\{0,1\}, E 4 Ξ ∙ , j J m , ϑ 2 D ( Ξ ∙ , j J m ) , D J ⋅ D ( Ξ ∙ , j J m ) , ( D J ) 2 Ξ ∙ , j J m , 0 ≤ j ≤ 11 , m ∈ { 0 , 1 } , Y Y Y G G G ϱ ∗ \varrho^* ϱ ∗ 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 5.3); since G G G 5.3) of every W ∈ M 2 ( μ ) W \in 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 5.4) and in the files above. Using m ≤ 2 m \le2 m ≤ 2
Step 2: exact elements. The forms D ( Ξ ∙ , j J m ) D(\Xi_{\bullet,j}^{Jm}) D ( Ξ ∙ , j J m ) D J ⋅ Ξ ∙ , j J m D_J \cdot\Xi_{\bullet,j}^{Jm} 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 ) M_2(\mu) M 2 ( μ ) 5.3) on this subspace span a space of dimension five for 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 ) M_2(\mu) M 2 ( μ )
Step 3: injectivity. Let W ∈ M 2 ( μ ) W \in M_2(\mu) W ∈ M 2 ( μ ) 5.3). As in the proof of 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 3 C [ [ q ] ] W_0 \in q^3\mathbb{C}[[q]] W 0 ∈ q 3 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 k z = Λ z + Λ π ( z ) − 1 3 ∈ { 0 , 1 } k_z=\Lambda_z+\Lambda_{\pi(z)}-\frac{1}{3}\in\{0,1\} k z = Λ z + Λ π ( z ) − 3 1 ∈ { 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 M 2 ( μ ) M_2(\mu) M 2 ( μ ) dim M 2 ( μ ) \dim 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 s \mathfrak{s} s h ( 0 ) h(0) h ( 0 ) U U U U U U [55 Theorem 1, p. 3] . Normalize ⟨ 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{s}\mathfrak{l}_3 h = ( h 1 , h 2 ) ∈ e 6 ⊕ s l 3 h 1 h_1 h 1 27 27 27 h 2 h_2 h 2 3 3 3 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 5 ( ad h ) 2 , τ j = d e f Tr R d j h ( 0 ) 2 ≥ 0. u \mathrel{\mathop{=}^{\mathrm{def}}} \langle h_1,h_1\rangle\in[0,1], \qquad T \mathrel{\mathop{=}^{\mathrm{def}}} \operatorname{Tr}_{\mathfrak{s}_5}(\operatorname{ad} h)^2,\qquad\tau_j \mathrel{\mathop{=}^{\mathrm{def}}} \operatorname{Tr}_{R_{d_j}} h(0)^2 \ge0. u = def ⟨ h 1 , h 1 ⟩ ∈ [ 0 , 1 ] , T = def Tr s 5 ( 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{s}\mathfrak{l}_3)\ominus\mathfrak{s} ( e 6 ⊕ s l 3 ) ⊖ s ( 27 , 3 ) (27,3) ( 27 , 3 ) ( 27 , 3 ‾ ) (27,\overline{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{s}\mathfrak{l}_3}(\operatorname{ad}h_2)^2=6(1-u) Tr s l 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 Lemma 5.1 , 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 ) 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 Proposition 5.4 are equivalent to
τ 1 = T 15 − 7 τ 5 15 − 3 τ 6 5 − 3 u 5 + 17 5 , τ 2 = 31 T 270 − 209 τ 5 270 − 7 τ 6 6 − 31 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{31T}{270}-\frac{209\tau_5}{270}-\frac{7\tau_6}{6}-\frac{31u}{30}+\frac{2009}{90}, τ 1 = 15 T − 15 7 τ 5 − 5 3 τ 6 − 5 3 u + 5 17 , τ 2 = 270 31 T − 270 209 τ 5 − 6 7 τ 6 − 30 31 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 [65 ] 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 ≔ h [ − 1 ] n 1 v_n\coloneqq h[-1]^n\mathbf{1} v n : = 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 (18) 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*{(18)} 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 (18) 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 [70 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 ) = Z_M(v_n,\tau)= Z M ( v n , τ ) = ∑ λ Z F λ ( v n , τ ) Tr Ω λ q L ′ ( 0 ) − c − 1 24 \sum_\lambda Z_{F_\lambda}(v_n,\tau)\operatorname{Tr}_{\Omega_\lambda}q^{L'(0)-\frac{c-1}{24}} ∑ λ Z F λ ( v n , τ ) Tr Ω λ q L ′ ( 0 ) − 24 c − 1 F λ F_\lambda F λ [70 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+n e P_{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 n ( n − 1 2 k ) + ( n − 2 k + 2 ) ( n − 1 2 k − 2 ) = ( n + 1 ) ( n 2 k ) n\binom{n-1}{2k}+(n-2k+2)\binom{n-1}{2k-2}=(n+1)\binom{n}{2k} n ( 2 k n − 1 ) + ( n − 2 k + 2 ) ( 2 k − 2 n − 1 ) = ( n + 1 ) ( 2 k n ) λ \lambda λ (18) . The last assertion is Zhu’s theorem [70 ] with [47 ] .
As a numerical check, we test (18) 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 ) \mathcal{M}_n(\mu) M n ( μ ) (16) 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 , (19) 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*{(19)} 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 , (19) 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 ) \mathcal{M}_n(\mu) M n ( μ ) M n ( μ ) \mathcal{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 (16) of the elements of M 4 ( μ ) \mathcal{M}_4(\mu) M 4 ( μ ) M 6 ( μ ) \mathcal{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 , τ 6 , Q 6 ) (T,T_4,\tau_6,Q_6) ( T , T 4 , τ 6 , Q 6 ) u u u
Q 5 = 12 T + 2 T 4 − 27 u 2 − 90 u + 105 26 . (20) Q_5=\frac{12T+2T_4-27u^2-90u+105}{26}. \tag*{(20)} Q 5 = 26 12 T + 2 T 4 − 27 u 2 − 90 u + 105 . (20) 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 ∙ , j J E_4\Xi_{\bullet,j}^J E 4 Ξ ∙ , j J ϑ 2 D ( Ξ ∙ , j J ) \vartheta_2D(\Xi_{\bullet,j}^J) ϑ 2 D ( Ξ ∙ , j J ) D J ⋅ D ( Ξ ∙ , j J ) D_J\cdot D(\Xi_{\bullet,j}^J) D J ⋅ D ( Ξ ∙ , j J ) ( D J ) 2 Ξ ∙ , j J (D_J)^2\Xi_{\bullet,j}^J ( D J ) 2 Ξ ∙ , j J e 6 \mathfrak{e}_6 e 6 s l 3 \mathfrak{s}\mathfrak{l}_3 s l 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{s}\mathfrak{l}_6\oplus\mathfrak{s}\mathfrak{l}_2\subset\mathfrak{e}_6 s l 6 ⊕ s l 2 ⊂ e 6 27 = ( 15 , 1 ) ⊕ ( 6 , 2 ) 27=(15,1)\oplus(6,2) 27 = ( 15 , 1 ) ⊕ ( 6 , 2 ) e 6 = ( 35 , 1 ) ⊕ ( 1 , 3 ) ⊕ ( 20 , 2 ) \mathfrak{e}_6=(35,1)\oplus(1,3)\oplus(20,2) 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 4 = ( 1 − u ) 2 2 \operatorname{Tr}_{3}h^4=\frac{(1-u)^2}{2} Tr 3 h 4 = 2 ( 1 − u ) 2 Tr s l 3 ( ad h ) 4 = 9 ( 1 − u ) 2 \operatorname{Tr}_{\mathfrak{sl}_3}(\operatorname{ad}h)^4=9(1-u)^2 Tr sl 3 ( ad h ) 4 = 9 ( 1 − u ) 2 (19) 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 (20) . 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 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\ge1 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\ge0 τ 4 ≥ 0 τ 1 ≥ 0 \tau_1\ge0 τ 1 ≥ 0 T − 9 u + 5 ≤ τ 6 ≤ − 2 T + 18 u + 10 3 T-9u+5\le\tau_6\le\frac{-2T+18u+10}{3} T − 9 u + 5 ≤ τ 6 ≤ 3 − 2 T + 18 u + 10 τ 5 ≥ 0 \tau_5\ge0 τ 5 ≥ 0
9 u − 3 ≤ T ≤ 9 u − 1. 9u-3\le T\le9u-1. 9 u − 3 ≤ T ≤ 9 u − 1. In particular u > 0 u>0 u > 0 i 1 ≥ 1 i_1\ge1 i 1 ≥ 1 2 h ∨ ≥ 6 k − 9 i 2 2h^\vee\ge6k-9i_2 2 h ∨ ≥ 6 k − 9 i 2 k ≤ 2 h ∨ + 9 i 2 6 k\le\frac{2h^\vee+9i_2}{6} k ≤ 6 2 h ∨ + 9 i 2 dim g ≤ 19 \dim\mathfrak{g}\le19 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\le T\le9u-1 9 u − 3 ≤ T ≤ 9 u − 1 k k k 6 i 1 − 3 i 2 ≤ 2 h ∨ ≤ 8 i 1 − i 2 6i_1-3i_2\le2h^\vee\le8i_1-i_2 6 i 1 − 3 i 2 ≤ 2 h ∨ ≤ 8 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\ge\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\ge0 τ 3 ≥ 0 τ 5 ≥ 3 − 9 u \tau_5\ge3-9u τ 5 ≥ 3 − 9 u τ 4 ≥ 0 \tau_4\ge0 τ 4 ≥ 0 τ 1 ≥ 0 \tau_1\ge0 τ 1 ≥ 0 11 u 18 − 7 u 2 + 19 6 ≤ τ 6 ≤ 17 3 − u − 7 9 \frac{11u}{18}-\frac{7u}{2}+\frac{19}{6}\le\tau_6\le\frac{17}{3}-u-\frac{7}{9} 18 11 u − 2 7 u + 6 19 ≤ τ 6 ≤ 3 17 − u − 9 7 τ 6 \tau_6 τ 6 τ 5 ≤ 9 5 ( 1 + u ) \tau_5\le\frac{9}{5}(1+u) τ 5 ≤ 5 9 ( 1 + u ) 3 − 9 u ≤ 9 5 ( 1 + u ) 3-9u\le\frac{9}{5}(1+u) 3 − 9 u ≤ 5 9 ( 1 + u ) u ≥ 1 9 u\ge\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 [Rie26a, Hua26, Jia26]. It uses the classification of holomorphic vertex operator algebras of central charge eight, the theory of extensions [43 , 14 ] , the Witt group [21 ] , the classification of the elements of order three of E 8 E_8 E 8 [52 ] , the results of [26 , 23 , 55 , 29 ] used in Section 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 ) \mathcal{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 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\mathcal{Z}(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) X ( 0 ) X^{(0)} X ( 0 ) s = U 1 ⊂ e 6 ⊕ s l 3 \mathfrak{s} = U_1 \subset\mathfrak{e}_6 \oplus\mathfrak{sl}_3 s = U 1 ⊂ e 6 ⊕ sl 3 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 ) a ≤ k a \le k a ≤ k
Proof : By [25 ] , 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 [20 ] , 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 \le j \le k 0 ≤ j ≤ k [37 ] , 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 \ge1 n ≥ 1 L ( k , j ) L(k,j) L ( k , j ) x ( n ) x(n) x ( n ) n ≥ 1 n \ge1 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 \le 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 0 0 0 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 ) 0 0 0 Proposition 4.4 . The kernel of p 2 ∣ z p_2|_{\mathfrak{z}} p 2 ∣ z R c 1 ≅ 3 R_{c_1} \cong\mathbf{3} 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 (20) 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_a H a
After conjugation, H a H_a H a α i ( H a ) ∈ { 0 , 1 , 2 } \alpha_i(H_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_a ad H a e 6 \mathfrak{e}_6 e 6 H a H_a H a s l 2 \mathfrak{sl}_2 sl 2 ⟨ H a , H a ⟩ \langle H_a,H_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 [16 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 [27 Table 18] ; see also [Gan19, p. 16].
We construct e 6 \mathfrak{e}_6 e 6 [36 ] , 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_a ad H a [ e , f ] = H a [e,f] = H_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 [16 Chapter 8] . Since two subalgebras s l 2 \mathfrak{sl}_2 sl 2 [16 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 \le9i_1 - 5k 4 ≤ 9 i 1 − 5 k i 1 ≥ 1 i_1 \ge1 i 1 ≥ 1 i 1 i_1 i 1 p 1 ( g ) p_1(\mathfrak{g}) p 1 ( g ) 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 τ 6 \tau_6 τ 6 Q 6 Q_6 Q 6 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 27 \mathbf{27} 27 e 6 \mathfrak{e}_6 e 6 i 1 i_1 i 1 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(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) U ⊂ V E 6 ⊕ A 2 U \subset V_{\mathfrak{E}_6\oplus A_2} U ⊂ V E 6 ⊕ A 2 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_{\mathfrak{E}_6\oplus A_2} V E 6 ⊕ A 2
(I) 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 ′′ )
(II) 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 ′′ s l 3 \mathfrak{sl}_3 sl 3 { ( 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 { ( 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 (20) . 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 g 5 ( ad h ) 2 = 8 ( x 2 + y 2 ) , Tr g 5 ( ad h ) 4 = 32 ( x 4 + y 4 ) . \langle h,h\rangle= 2k'x^2 + 2k''y^2,\qquad\operatorname{Tr}_{\mathfrak{g}_5}(\operatorname{ad}h)^2 = 8(x^2+y^2),\qquad\operatorname{Tr}_{\mathfrak{g}_5}(\operatorname{ad}h)^4 = 32(x^4+y^4). ⟨ h , h ⟩ = 2 k ′ x 2 + 2 k ′′ y 2 , Tr g 5 ( ad h ) 2 = 8 ( x 2 + y 2 ) , Tr g 5 ( ad h ) 4 = 32 ( x 4 + y 4 ) . Since p 2 ( h ) = 0 p_2(h)=0 p 2 ( h ) = 0 (20) from normalized elements to
26 Tr R d 5 h ( 0 ) 4 = 12 ⟨ h , h ⟩ Tr g 5 ( ad h ) 2 + 2 Tr g 5 ( ad h ) 4 − 12 ⟨ h , h ⟩ 2 . 26\operatorname{Tr}_{R_{d_5}} h(0)^4 = 12\langle h,h\rangle\operatorname{Tr}_{\mathfrak{g}_5}(\operatorname{ad}h)^2 + 2\operatorname{Tr}_{\mathfrak{g}_5}(\operatorname{ad}h)^4 - 12\langle h,h\rangle^2. 26 Tr R d 5 h ( 0 ) 4 = 12 ⟨ h , h ⟩ Tr g 5 ( ad h ) 2 + 2 Tr g 5 ( 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 . (21) \operatorname{Tr}_{R_{d_5}} H'(0)^2 H''(0)^2 = \frac{8(2(k' + k'') - k'k'')}{13}. \tag*{(21)} Tr R d 5 H ′ ( 0 ) 2 H ′′ ( 0 ) 2 = 13 8 ( 2 ( k ′ + k ′′ ) − k ′ k ′′ ) . (21) 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)^2 H''(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 Lemma 8.3 and the bijection between conjugacy classes and nilpotent orbits [16 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\mathfrak{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 ′ ′ ) p_1(\mathfrak{g}'') p 1 ( g ′′ ) p 2 ( g ′ ′ ) p_2(\mathfrak{g}'') 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}' g ′ g ′ ′ \mathfrak{g}'' 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' + yh'') 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'(0)^p H''(0)^q Tr R d j H ′ ( 0 ) p H ′′ ( 0 ) q p + q ≤ 6 p+q \le6 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'}\bigoplus_{b=0}^{k''}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 \le m_{ab} \le\lfloor\dim R/((a+1)(b+1))\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 (16) by the fourteenth functional w 0 ′ ′ = [ q 5 / 3 ] W 0 w''_0=[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'+yh'') W = W n ( x h ′ + y h ′′ ) w 0 ′ ′ w''_0 w 0 ′′ Tr U 2 H ′ ( 0 ) p H ′ ′ ( 0 ) q \operatorname{Tr}_{U_2}H'(0)^pH''(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\le p+q\le6 2 ≤ p + q ≤ 6 J m J^m J m 0 ≤ m ≤ 2 0\le m\le2 0 ≤ m ≤ 2 53 × 144 53\times144 53 × 144 41 × 72 41\times72 41 × 72 29 × 36 29\times36 29 × 36 10 10 10 10 10 10 7 7 7 4 4 4 4 4 4 7 7 7 w 0 ′ ′ w''_0 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_{E_6}\oplus A_2)_2\cong(27\otimes27,1)\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}\le m_{ab}\le 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 ′
Remark 8.6 Case (I) is the structure announced in [Rie26a, Hua26]: the level one factor comes from a root s l 2 \mathfrak{sl}_2 sl 2 e 6 \mathfrak{e}_6 e 6 39 39 39 A 5 A_5 A 5 A 2 A_2 A 2 Table 2 in case (I) coincide with those listed in the working manuscript [65 ] , and the decomposition of U 2 U_2 U 2 [65 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), which is realized assuming ( H J ) (\mathrm{H}_J) ( H J ) 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 0 , c 0 , c 1 , c 2 } z\in\{0,b,a_0,c_0,c_1,c_2\} z ∈ { 0 , b , a 0 , 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 would be 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 1 ⋅ 39 = 3 ⋅ 13 1 \cdot39 = 3 \cdot13 1 ⋅ 39 = 3 ⋅ 13 U 2 U_2 U 2
The generation problem
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^\prime= 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^\prime_2 V 2 ′ Table 2 , containing ( A ) 2 (A)_2 ( A ) 2 V ′ V^\prime V ′ V 2 ′ V^\prime_2 V 2 ′
Proof: The first assertion is Theorem 8.5 . A U 1 U_1 U 1 V 2 ′ V^\prime_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^\prime_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^\prime_2 \ominus U_2 V 2 ′ ⊖ U 2 U U U M b M_b M b U U U U 1 ⊕ U 2 U_1 \oplus U_2 U 1 ⊕ U 2 U U U U 2 U_2 U 2 U 1 U_1 U 1 U U U 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^\prime,0) \otimes L_{\mathfrak{sl}_2}(k^{\prime\prime},0) A ≅ L sl 2 ( k ′ , 0 ) ⊗ L sl 2 ( k ′′ , 0 ) ( k ′ , k ′ ′ ) = ( 1 , 39 ) (k^\prime,k^{\prime\prime})=(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 18 5 \frac{18}{5} 5 18 ω − ω k ′ − ω k ′ ′ \omega-\omega_{k^\prime}-\omega_{k^{\prime\prime}} ω − ω k ′ − ω k ′′
Proof: The first assertion follows from Theorem 8.5 and [25 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^\prime} ω k ′ ω k ′ ′ \omega_{k^{\prime\prime}} 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^\prime} \oplus\mathbb{C}\omega_{k^{\prime\prime}} 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}^\prime\oplus\mathfrak{g}^{\prime\prime} g ′ ⊕ g ′′ g ′ \mathfrak{g}^\prime g ′ g ′ ′ \mathfrak{g}^{\prime\prime} 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 U 2 U_2 U 2
We keep the hypotheses of Theorem 8.5 . We write V ′ = V E 6 ⊕ A 2 V^{\prime}=V_{E_6\oplus A_2} V ′ = V E 6 ⊕ A 2 V ′ = U ⊕ M b V^{\prime}=U\oplus M_b V ′ = U ⊕ M b Proposition 4.1 , A ⊂ U A\subset U A ⊂ U U 1 = g ′ ⊕ g ′ ′ U_1=\mathfrak{g}^{\prime}\oplus\mathfrak{g}^{\prime\prime} U 1 = g ′ ⊕ g ′′ C ^ = Com V ′ ( A ) \widehat{C}=\operatorname{Com}_{V^{\prime}}(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\coloneqq\omega-\omega_{k^{\prime}}-\omega_{k^{\prime\prime}} 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^{\prime} V ′ C C C U 2 U_2 U 2 [50 ] in Theorem 10.24. We denote the normalized invariant bilinear form on V ′ V^{\prime} 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 [35 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^{\prime} 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 \epsilon(\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 \epsilon(\alpha_i,\alpha_i)=-1 ϵ ( α i , α i ) = − 1 ϵ ( α i , α j ) = 1 \epsilon(\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 ) (22) 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*{(22)} 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 ) (22) 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}\binom{m+j-1}{j}\left((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 ) ) 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 ) = weight L(0)=\text{weight} L ( 0 ) = weight 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 LDL^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^{\prime} V ′ H ′ H^{\prime} H ′ H ′ ′ H^{\prime\prime} 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^{\prime} V ′ R b R_b R b g ′ ⊕ g ′ ′ \mathfrak{g}^{\prime}\oplus\mathfrak{g}^{\prime\prime} 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. We use it for homogeneous inputs of weights at most four and only when the output weight 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^{\prime} 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 = d e f a ( 1 ) b a\mathbin{\cdot}b\mathrel{\overset{\mathrm{def}}{=}}a_{(1)}b a ⋅ b = def a ( 1 ) b B B B e = d e f 1 2 ω C e\mathrel{\overset{\mathrm{def}}{=}}\frac{1}{2}\omega_C e = def 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^{\prime}_2 V 2 ′ g ′ ⊕ g ′ ′ \mathfrak{g}^{\prime}\oplus\mathfrak{g}^{\prime\prime} g ′ ⊕ g ′′ V 2 ′ V^{\prime}_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^{\prime}=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 = d e f λ 2 + 4 μ ≠ 0 \Delta\mathrel{\overset{\mathrm{def}}{=}}\lambda^2+4\mu\ne0 Δ = def 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 ) . (23) c_{1,2}=\frac{c_C}{2}\left(1\mp\frac{\lambda}{\sqrt{\Delta}}\right). \tag*{(23)} c 1 , 2 = 2 c C ( 1 ∓ Δ λ ) . (23) 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 ) v = 2 f v=2f v = 2 f ω = ω C − v \omega=\omega_C-v ω = 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 ) (23) 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 3219 19 \frac{3219}{19} 19 3219 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
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 [11 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 1 x\in U_1 x ∈ U 1 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 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\oplus 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 = f ′ ( 0 ) a f ′ ′ ( 0 ) b \theta_{ab}=f'(0)^a f''(0)^b θ ab = f ′ ( 0 ) a f ′′ ( 0 ) b
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 ) (\omega_1)_{(1)} ( ω 1 ) ( 1 ) ( ω 2 ) ( 1 ) (\omega_2)_{(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')=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'\oplus\mathfrak g'' 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'\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 1 ⊥ B\cap C_1^\perp B ∩ C 1 ⊥ 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 z z z z u T P u − 1 = 0 z u^{T}Pu-1=0 z u T P u − 1 = 0 2 2 2 2 2 2 [ u , X u ] [u,Xu] [ u , X u ] Q \mathbb{Q} Q { 1 } \{1\} { 1 }
Step 2: the admissible spaces. For the subalgebra g \mathfrak{g} g 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 ( 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 g 10 \mathfrak{g}_{10} g 10
Remark 10.6 Theorem 10.5 is a computer-assisted result. Step 1 uses the exact rational equations displayed above. In Step 2, characteristic-polynomial factors and their multiplicities are retained in the eigenspace computation; a multiple nonlinear factor or an inconclusive enclosure cannot be discarded. The stored calculation for subalgebra g 9 \mathfrak{g}_{9} g 9 x C x_{C} x C 409 \sqrt{409} 409
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 S ⊂ V ′ \mathcal{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 \mathfrak{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)\mathbf{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 ′ , f ′ , e ′ ′ , f ′ ′ e',f',e'',f'' e ′ , f ′ , e ′′ , 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)\mathbf{1} x ( − 2 ) 1
Remark 10.8 The modular ranks give lower bounds for the characteristic-zero ranks under the reduction conventions above. The membership tests and the reduced module decomposition are statements over the residue fields. Theorems 10.9 and 10.19 establish the corresponding characteristic-zero closure and dimensions. Proposition 10.7 by itself does not prove that S S S
Case (II): exact weight-two closure, unitarity and symmetries
Let K = Q ( 13 , − 3 ) \mathbb{K} = \mathbb{Q}(\sqrt{13},\sqrt{-3}) K = Q ( 13 , − 3 ) V K ′ V'_K V K ′ K \mathbb{K} K U j , K = U j ∩ V K ′ U_{j,\mathbb{K}} = U_j \cap V'_K U j , K = U j ∩ V K ′ K \mathbb{K} K C \mathbb{C} C K \mathbb{K} K V 2 ′ V'_2 V 2 ′ K \mathbb{K} K c ↦ c ‾ c \mapsto\overline{c} c ↦ c K \mathbb{K} K − 3 ↦ − − 3 \sqrt{-3} \mapsto-\sqrt{-3} − 3 ↦ − − 3 v ↦ v ‾ v \mapsto\overline{v} v ↦ v 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 2 120 120 120 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 2 480 = 4 ⋅ 120 480 = 4 \cdot120 480 = 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 \ge0 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 ) ( 1 ) U 2 ⊂ U 2 ((A)_2)_{(1)}U_2 \subset U_2 (( A ) 2 ) ( 1 ) U 2 ⊂ U 2 ( ( A ) 2 ) ( 2 ) U 2 ⊂ U 1 ((A)_2)_{(2)}U_2 \subset U_1 (( A ) 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 ) g \mathfrak{g} g 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 [22 ] and Carpi, Kawahigashi, Longo and Weiner [13 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\colon 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 ‾ h 1 ( − n 1 ) ⋯ h k ( − n k ) e − λ , c ∈ C . (24) \phi_\chi\left(c h_1(-n_1)\cdots h_k(-n_k)e^\lambda\right) = (-1)^k\chi(\lambda)\overline{c}h_1(-n_1)\cdots h_k(-n_k)e^{-\lambda}, \qquad c \in\mathbb{C}. \tag*{(24)} ϕ χ ( c h 1 ( − n 1 ) ⋯ h k ( − n k ) e λ ) = ( − 1 ) k χ ( λ ) c h 1 ( − n 1 ) ⋯ h k ( − n k ) e − λ , c ∈ C . (24) Theorem 10.10 Let χ \chi χ 2 15 , 1 , 1 , 5 25 , 9 24 , 5 24 \frac{2}{15},1,1,\frac{5}{25},\frac{9}{24},\frac{5}{24} 15 2 , 1 , 1 , 25 5 , 24 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
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 (24) for the trivial character. With a bimultiplicative cocycle we have ε ( − α , − β ) = ε ( α , β ) \varepsilon(-\alpha,-\beta)=\varepsilon(\alpha,\beta) ε ( − α , − β ) = ε ( α , β ) ϕ 1 \phi_1 ϕ 1 [22 Theorem 4.12] , [13 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 ′ \mathfrak{g}' g ′ κ ′ ′ \kappa'' κ ′′ g ′ ′ \mathfrak{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 ′ d α / c α d_\alpha/c_\alpha d α / c α
Exact computation over K \mathbb{K} K v v v U 2 U_2 U 2 ϕ χ ( v ) \phi_\chi(v) ϕ χ ( v ) U 2 U_2 U 2
the two choices attached to the subalgebra 9; since 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 [13 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 [13 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 ′ [34 ] .
Remark 10.11 Theorem 10.10 does not use any information on S S S S S S [65 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 ) [32 ] 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 [18 Lemma 4.2] , [47 ] .
Proposition 10.12. 1. The two subspaces U 2 U_2 U 2 θ χ : v ↦ ϕ χ ( v ‾ ) \theta_{\chi}:v\mapsto\phi_{\chi}(\overline{v}) θ 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 θ χ ( W ) = ϕ χ ( W ‾ ) = ϕ χ ( W ) ‾ = W ‾ \theta_{\chi}(W)=\phi_{\chi}(\overline{W})=\overline{\phi_{\chi}(W)}=\overline{W} θ 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 C x_C C x C g ( x C ) = ϵ C x C g(x_C)=\epsilon_C x_C g ( x C ) = ϵ 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 (23) . 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 ⊥ B B B 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 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 p 3 = 8388571 p_3=8388571 p 3 = 8388571 p 3 p_3 p 3
The space S 3 g e n S^{\mathrm{gen}}_3 S 3 gen u ( 1 ) S 3 g e n ⊂ S 3 g e n u(1)S^{\mathrm{gen}}_3 \subset S^{\mathrm{gen}}_3 u ( 1 ) S 3 gen ⊂ S 3 gen u ∈ U 2 u \in U_2 u ∈ U 2 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^{\mathrm{gen}}_3 S 3 = S 3 gen [65 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 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^{\mathrm{gen}}_3+M_3=V'_3 S 3 gen + M 3 = V 3 ′ M b M_b M b § 4 , and the sums are direct by dimension count.
The space S 4 g e n S^{\mathrm{gen}}_4 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 ) . \begin{aligned}
&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).
\end{aligned} 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^{\mathrm{gen}}_4 \subset S^{\mathrm{gen}}_3 y ( 1 ) S 4 gen ⊂ S 3 gen y ( 2 ) S 4 g e n ⊂ U 2 y(2)S^{\mathrm{gen}}_4 \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 3 g e n u(1)S^{\mathrm{gen}}_4 \subset S^{\mathrm{gen}}_3 u ( 1 ) S 4 gen ⊂ S 3 gen u ( 2 ) S 4 g e n ⊂ S 3 g e n u(2)S^{\mathrm{gen}}_4 \subset S^{\mathrm{gen}}_3 u ( 2 ) S 4 gen ⊂ S 3 gen u ( 3 ) S 4 g e n ⊂ U 2 u(3)S^{\mathrm{gen}}_4 \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 K K p 3 p_3 p 3 K K K V ′ = S ⊕ M b V'=S\oplus M_b V ′ = S ⊕ M b Proposition 4.1 : by unitarity, V ′ = S ⊕ S ⊥ V'=S\oplus S^\perp V ′ = 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 + 14 4 = 4 \frac{3}{4}+\frac{14}{4}=4 4 3 + 4 14 = 4
Case (II): strong generation and C 2 C_2 C 2
In this subsection U 2 U_2 U 2 g \mathfrak{g} g Proposition 10.12 and the Galois symmetry 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 \mathbb{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)\mathbf{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 and the decomposition into integrable modules for A = L s l 2 ( 3 , 0 ) ⊗ L s l 2 ( 13 , 0 ) A = L_{\mathfrak{sl}_2}(3,0) \otimes L_{\mathfrak{sl}_2}(13,0) A = L sl 2 ( 3 , 0 ) ⊗ L sl 2 ( 13 , 0 ) 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 , by ordering the six currents according to their negative modes.
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 χ 2 = ∥ L C ( 1 ) v ∥ H χ 2 + 2 h C ∥ v ∥ H χ 2 . \lVert L_C(-1)v\rVert_{H_{\chi}}^2 = \lVert L_C(1)v\rVert_{H_{\chi}}^2 + 2h_C\lVert v\rVert_{H_{\chi}}^2. ∥ L C ( − 1 ) v ∥ H χ 2 = ∥ L C ( 1 ) v ∥ H χ 2 + 2 h C ∥ v ∥ H χ 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 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 g \mathfrak{g} g 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. An absent record is never treated as a zero vector. The complete verification reports no failure, and the exact coordinate data and verification results are included in the supplementary material. Missing or truncated blocks, duplicate labels, a failed minor or a nonzero residual cause failure. Consequently all three requirements in Definition 10.16 hold.
Finally, coefficient conjugation and σ \sigma σ U 2 U_2 U 2 K \mathbb{K} K Theorem 10.24 .
This applies to W = N 4 ′ W = N'_4 W = N 4 ′ X = ( U 2 ) ( 0 ) S 3 + ( U 2 ) ( − 1 ) U 2 X = (U_2)_{(0)}S_3 + (U_2)_{(-1)}U_2 X = ( U 2 ) ( 0 ) S 3 + ( U 2 ) ( − 1 ) U 2 W W W X X X
Theorem 10.19. The following statements hold over K \mathbb{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 \mathbb{K} K 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 \mathbb{K} \cong\mathbb{Q}^4 K ≅ Q 4 b i b_i b i 13 b i \sqrt{13}b_i 13 b i − 3 b i \sqrt{-3}b_i − 3 b i − 39 b i \sqrt{-39}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 ) e_{(0)} e ( 0 ) f ( 0 ) ′ f'_{(0)} f ( 0 ) ′ e ( 0 ) ′ ′ e''_{(0)} e ( 0 ) ′′ f ( 0 ) ′ ′ f''_{(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 ) 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\mathbb{C}1 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(1)(u_{(0)}v) = 2u_{(1)}v + (L(1)u)_{(0)}v + u_{(0)}L(1)v, 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 . L(2)(u_{(0)}v) = 3u_{(2)}v + 3(L(1)u)_{(1)}v. L ( 2 ) ( u ( 0 ) 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_{(1)}(u_{(0)}v) = (y_{(0)}u)_{(1)}v + (y_{(1)}u)_{(0)}v + u_{(0)}y_{(1)}v, 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 . y_{(2)}(u_{(0)}v) = (y_{(0)}u)_{(2)}v + 2(y_{(1)}u)_{(1)}v. y ( 2 ) ( u ( 0 ) 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 ( − 1 ) u_{(0)},u_{(-1)} u ( 0 ) , 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.
The next lemma is a variant of [65 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 ) 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\geq0 n ≥ 0 w ∈ N ( G ) w\in N(G) w ∈ N ( G ) wt g + wt w ≤ T \operatorname{wt}g+\operatorname{wt}w\leq 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\leq 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)}(z'_{(s+i)}v)-(-1)^p z'_{(s-p-i)}(k_{(i)}v)\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)}(k_{(i)}v)\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 + wt k − i − 1 ≤ T \operatorname{wt}z'+\operatorname{wt}(k_{(i)}v)=\operatorname{wt}z'+\operatorname{wt}v+\operatorname{wt}k-i-1\leq T wt z ′ + wt ( k ( i ) v ) = wt z ′ + wt v + wt k − i − 1 ≤ 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 ) 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
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
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 u , w ∈ U 2 u,w\in U_2 u , w ∈ 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 τ M_\tau M τ 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 a ∈ U 2 a\in U_2 a ∈ U 2 g ∈ U 1 g\in U_1 g ∈ U 1 g ∈ D 3 g\in D_3 g ∈ D 3 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 ) g = y ( n − 1 ) g a_{(n)}g=y_{(n-1)}g a ( n ) g = y ( n − 1 ) g 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\mathbb{C}1 S 4 − n ⊂ C 1 L ( − 1 ) U 1 L(-1)U_1 L ( − 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 [65 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 ) ′ 1 4 v = e'_{(-1)}{}^{4}_{1} v = e ( − 1 ) ′ 1 4 f ′ ( 1 ) v = 0 f'(1)v = 0 f ′ ( 1 ) v = 0 ϕ χ ( e ′ ) = − f ′ ′ \phi_{\chi}(e') = -f'' ϕ χ ( e ′ ) = − f ′′ H χ ( e ( − m ) ′ u , u ′ ) = H χ ( u , f ( − m ) ′ ′ u ′ ) H_{\chi}(e'_{(-m)}u,u') = H_{\chi}(u,f''_{(-m)}u') H χ ( e ( − m ) ′ u , u ′ ) = H χ ( u , f ( − m ) ′′ u ′ ) H χ ( v , v ) = H χ ( e ( − 1 ) ′ 1 3 , f ′ ( 1 ) v ) = 0 H_{\chi}(v,v) = H_{\chi}(e'_{(-1)}{}^{3}_{1},f'(1)v) = 0 H χ ( v , v ) = H χ ( e ( − 1 ) ′ 1 3 , f ′ ( 1 ) v ) = 0 v = 0 v = 0 v = 0 e ( − 1 ) ′ ′ 1 14 = 0 e''_{(-1)}{}^{14}_{1} = 0 e ( − 1 ) ′′ 1 14 = 0 [ e ′ ] 4 = [ e ′ ′ ] 14 = 0 [e']^{4} = [e'']^{14} = 0 [ e ′ ] 4 = [ 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 [50 ] 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_{\mathbb{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 ′ K \mathbb{K} K C \mathbb{C} C K \mathbb{K} K K \mathbb{K} K 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 [65 Proposition 4.2] is needed. In case (I), the space S 4 S_4 S 4 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 \mathbb{K} K Proposition 10.17 . Corollary 10.22 gives an alternative dimension argument when an ambient Haagerup algebra or the stated external existence result is available. 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 [50 ] 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 [ 2 , 6 ] [2,6] [ 2 , 6 ] [ 1 , 7 ] [1,7] [ 1 , 7 ] K \mathbb{K} K [50 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 Z(\mathcal{H}_3) Rep ( V ) ≃ Z ( H 3 ) [50 Theorem 1.1] , and that every simple V V V V V V [50 Corollary 6.8] . His construction is dated October 4, 2026; it precedes the present work and the announcements [Rie26a, Hua26], and it makes no uniqueness assertion [50 Section 1.2] . Below, “assuming (HJ _J J
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 ψ \psi ψ [50 ] 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=\epsilon(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 [50 Section 3.1] .
The isometry. Let φ \varphi φ 1 , … , 8 1,\ldots,8 1 , … , 8 [50 ] (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 , B H P_H,B_H P 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)=\epsilon(\psi_0p,\psi_0q)\epsilon_{\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 \epsilon(\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 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 K K U 2 U_2 U 2 g \mathfrak{g} g K 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,K}=U_2\cap V'_K U 2 , K = U 2 ∩ V K ′ A 2 , K = A 2 ∩ V K ′ A_{2,K}=A_2\cap V'_K A 2 , K = A 2 ∩ V K ′ K K K V ′ V' V ′ W K W_K W K K K K U 2 , K U_{2,K} U 2 , K A 2 , K A_{2,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_K\subset U_{2,K} W K ⊂ U 2 , K ψ ( V ) \psi(V) ψ ( V )
At p 3 = 8388571 p_3=8388571 p 3 = 8388571 13 ↦ 1649360 \sqrt{13}\mapsto1649360 13 ↦ 1649360 − 3 ↦ 2036966 \sqrt{-3}\mapsto2036966 − 3 ↦ 2036966
A nonzero 120 × 120 120\times120 120 × 120 K K K dim K W K ≥ 120 = dim K U 2 , K \dim_K W_K\ge120=\dim_K U_{2,K} dim K W K ≥ 120 = dim K U 2 , K W K = U 2 , K W_K=U_{2,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), the algebras attached to the two subspaces of the subalgebra g \mathfrak{g} g Proposition 10.12 ) are positive strongly rational, with Rep ( S ) ≃ Z ( H 3 ) \operatorname{Rep}(S)\simeq Z(\mathcal{H}_3) Rep ( S ) ≃ Z ( H 3 ) Theorem 8.5 is realized.
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 ′ e' e ′ f ′ f' f ′ H ′ H' H ′ e ′ ′ e'' e ′′ f ′ ′ f'' f ′′ H ′ ′ H'' H ′′ Q \mathbb{Q} Q V Q ′ V'_{\mathbb{Q}} V Q ′ U 2 U_2 U 2 K K K σ ∈ Gal ( K / Q ) \sigma\in\operatorname{Gal}(K/\mathbb{Q}) σ ∈ Gal ( K / Q ) 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'_K=V'_{\mathbb{Q}}\otimes_{\mathbb{Q}}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 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 ′ θ χ \theta_\chi θ χ Proposition 10.12 exchanges 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|\le8 ∣ 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=\mathfrak{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 ′ ′ = 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 . (25) \begin{aligned}
e'&=e^{\alpha_4},\\
e''&=5e^{\alpha_1}+4e^{\alpha_6}+3e^{\alpha_2+\alpha_3+\alpha_4}+e^{\alpha_2+\alpha_4+\alpha_5}\\
&\quad+4e^{\alpha_3+\alpha_4+\alpha_5}+e^{\alpha_7}+e^{\alpha_8}. \tag*{(25)}
\end{aligned} e ′ e ′′ = e α 4 , = 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 . (25) and in case (II),
e ′ = e − α 2 + e α 3 − 5 3 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 . (26) \begin{aligned}
e'&=e^{-\alpha_2}+e^{\alpha_3}-\frac{5}{3}e^{\alpha_5},\\
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}\\
&\quad+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*{(26)}
\end{aligned} e ′ e ′′ = e − α 2 + e α 3 − 3 5 e α 5 , = 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 . (26) 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 (25) 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 } (26) 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|\leq2\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 Proposition 10.12 , and the four others act trivially.
Let g g g E 6 E_6 E 6 ( α 1 ↔ α 6 , α 3 ↔ α 5 ) (\alpha_1\leftrightarrow\alpha_6,\ \alpha_3\leftrightarrow\alpha_5) ( α 1 ↔ α 6 , 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)=(\frac45,1,-\frac53,-1,-\frac35,\frac54,1,1) ( 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 cocycle identity displayed below constructs z z z z ′ z' z ′ (26) and the Cartan elements gives z , z ′ ∈ Z z,z'\in Z z , z ′ ∈ Z h 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_{\mathfrak{e}_6}(1,0)\otimes L_{\mathfrak{sl}_3}(1,0) V ′ ≅ L e 6 ( 1 , 0 ) ⊗ L sl 3 ( 1 , 0 ) [52 ] . 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 ) , ( 1 , 3 ) \tau=(0,4),(1,3) τ = ( 0 , 4 ) , ( 1 , 3 ) ( 2 , 0 ) , ( 1 , 1 ) , ( 1 , 5 ) , ( 2 , 2 ) , ( 2 , 4 ) , ( 3 , 1 ) (2,0),(1,1),(1,5),(2,2),(2,4),(3,1) ( 2 , 0 ) , ( 1 , 1 ) , ( 1 , 5 ) , ( 2 , 2 ) , ( 2 , 4 ) , ( 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 Proposition 10.12 , so it is s s s c c c c ( λ + μ ) = c ( λ ) c ( μ ) ϵ ( λ , μ ) ϵ ( μ , λ ) c(\lambda+\mu)=c(\lambda)c(\mu)\epsilon(\lambda,\mu)\epsilon(\mu,\lambda) c ( λ + μ ) = c ( λ ) c ( μ ) ϵ ( λ , μ ) ϵ ( μ , λ ) U 1 U_1 U 1 z z z U 2 U_2 U 2 U 1 U_1 U 1 K \mathbb{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 1 , … , M r M_0=S,M_1,\ldots,M_r M 0 = S , 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 U S^S,S^U S S , 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_id_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 [47 , 18 ] . 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 ) (\mathrm{H}_{\mathrm{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 1 U_1 U 1 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 ) 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\simeq S^{(9,0)}=\psi(V) S ≃ S ( 9 , 0 ) = ψ ( V ) Theorem 10.24 ), and, assuming ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) 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 ) Corollary 10.25 ), so its vacuum character is also 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\simeq 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 \mathfrak{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 ) 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}(\mathbb{K}/\mathbb{Q}) Gal ( K / Q ) Lemma 10.28 gives a second, analytic proof of the equality 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 g \mathfrak{g} g 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 9 \frac{32}{9} 9 32 b 0 = e b_0 = e b 0 = e b 1 , … , b 4 b_1,\ldots,b_4 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,\qquad(0,4)^2,\qquad(2,2),\qquad\left(\frac{51}{19},\frac{25}{19}\right),\qquad\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 C 2 C_2 C 2 g \mathfrak{g} g 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 C C C U 1 U_1 U 1 C 4 C_4 C 4 g ′ ⊕ g ′ ′ \mathfrak{g}' \oplus\mathfrak{g}'' g ′ ⊕ g ′′ ( 0 , 0 ) (0,0) ( 0 , 0 ) 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\left\{0,2,4,\frac{25}{19},\frac{50}{19}\right\} 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 { 0 , − 22 5 } \{0,-\frac{22}{5}\} { 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 ′
::: 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 C = ω 1 ⊗ 1 + 1 ⊗ ω 2 \omega_C = \omega^1 \otimes1 + 1 \otimes\omega^2 ω C = 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 95 \frac{182}{95} 95 182 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 ) 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 C C C S 5 S_5 S 5 dim S 5 = dim U 5 = 29512 \dim S_5=\dim U_5=29512 dim S 5 = dim U 5 = 29512
Case (I): uniqueness
Proposition 10.34 Assume the hypotheses of Theorem 8.5 in case (I). Then C 2 C_2 C 2 2 2 2 5 5 5 6 6 6 Table 3 . The subalgebras 5 5 5 6 6 6 2400 1271 \frac{2400}{1271} 1271 2400 70 31 \frac{70}{31} 31 70 [65 Section 1] .
Proof. There is a direct exact exclusion that avoids choosing eigenvectors. For candidates 0 0 0 3 3 3 4 4 4 ( 1 , 13 ) (1,13) ( 1 , 13 ) 1 1 1 ( 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 Step 1 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) and 4 by the types (0, 12), (1, 1), (1, 7) and (1, 13), exactly.
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 = 5 2 h_1=\frac{5}{2} h 1 = 2 5 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=0 C ′ z = 0 C ′ ′ z = 84 z C''z=84z 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 \pm\frac{3\sqrt{-19758}}{148} ± 148 3 − 19758 ± 2 − 4403 37 \pm\frac{2\sqrt{-4403}}{37} ± 37 2 − 4403 w ′ = f ( 0 ) ′ w w'=f'_{(0)}w w ′ = f ( 0 ) ′ w w , f ( 0 ) ′ ′ w , f ( 0 ) ′ w , f ( 0 ) ′ f ( 0 ) ′ ′ w w,f''_{(0)}w,f'_{(0)}w,f'_{(0)}f''_{(0)}w w , f ( 0 ) ′′ w , 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 (C''-84)z ( C ′′ − 84 ) z 28 z 28z 28 z ( 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 excludes, with certified nonzero coefficients, the two crossed pairs of lines of types (1, 13) and (1, 1), and the two other pairs, those of the two choices of U 2 U_2 U 2
We now compare with the data of [Rie26a]. There, U R = ⟨ A , e , f , v ⟩ ⊂ V E 6 ⊕ A 2 U^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 e,f e , f e ∈ Com ( A ) 2 e\in\operatorname{Com}(A)_2 e ∈ 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 ) [65 Section 1] ; the coordinates are specified in the certificates accompanying [65 ] , and the same vectors are given in [Rie26a, Section “The explicit candidate”], in the bases fixed in [65 ] .
Proposition 10.36 There is an isomorphism of vertex operator algebras ψ R \psi_R ψ R V ′ V' V ′ e ′ , f ′ , H ′ , e ′ ′ , f ′ ′ , H ′ ′ e',f',H',e'',f'',H'' e ′ , f ′ , H ′ , e ′′ , f ′′ , H ′′ e e e f f f Table 3 , which they span, and v v v ( 1 , 13 ) (1,13) ( 1 , 13 )
Proof: Exact computation, as for Theorem 10.24 . In [65 ] 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 ) K \mathbb{K} K w = s 3 s 4 s 1 s 3 ∈ W ( E 6 ) w=s_{3}s_{4}s_{1}s_{3}\in W(E_{6}) w = s 3 s 4 s 1 s 3 ∈ W ( E 6 ) [65 ] onto H ′ H' H ′ H ′ ′ H'' H ′′ Theorem 10.24 ; and the torus character with values 5 5 5 − 3 2 -\frac{\sqrt{3}}{2} − 2 3 − 2 3 -2\sqrt{3} − 2 3 1 1 1 − 2 3 3 -\frac{2\sqrt{3}}{3} − 3 2 3 4 4 4 1 1 1 1 1 1 ω V ′ \omega_{V'} ω V ′ A A A B B B ψ R ( e ) \psi_R(e) ψ R ( e ) ψ R ( f ) \psi_R(f) ψ R ( f ) Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 ) 5 5 5 ( 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_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 6 6 6 0 0 0 3 3 3 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 , 1 3 , 1 , 3 , 5 4 , 1 , 1 ) (c_1,\ldots,c_8)=(\frac{4}{5},-1,\frac{1}{3},1,3,\frac{5}{4},1,1) ( c 1 , … , c 8 ) = ( 5 4 , − 1 , 3 1 , 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|\leq8 ∣ Z ∣ ≤ 8 Lemma 10.26 . The explicitly given lifts z 1 z_1 z 1 z A z_A z A ρ \rho ρ ρ \rho ρ (25) ; 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 Proposition 10.27 . The multiplicity spaces have dimension 2 2 2 ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 0 , 4 ) (0,4) ( 0 , 4 ) 1 1 1 88 88 88 12 12 12 8 8 8 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}_{\mathbb{R}}) ( H R ) U U U U R U^{\mathbb{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 ′′ 10.35 , C 2 = U 2 ∩ Com ( A ) C_2=U_2\cap\operatorname{Com}(A) C 2 = U 2 ∩ Com ( A ) 5 5 5 6 6 6 U U U z A ( U ) z_A(U) z A ( U ) 5 5 5 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 Proposition 10.36 . By Lemma 10.4 and the proof of Proposition 10.36 , the line 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^{\mathbb{R}}) U ⊃ ψ R ( U R ) Theorem 7.1 , applied to U U U ( H R ) (\mathrm{H}_{\mathbb{R}}) ( H R ) U R U^{\mathbb{R}} U R X 0 V O X_0^{\mathrm{VO}} X 0 VO U = ψ R ( U R ) U=\psi_R(U^{\mathbb{R}}) U = ψ R ( U R )
Once C 2 C_2 C 2 ( 1 , 13 ) (1,13) ( 1 , 13 ) U R U^{\mathbb{R}} U R A A A e e e f f f v v v 1 1 1 2 2 2 4 4 4 ( 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^{\mathbb{R}})\subset V' ψ R ( U R ) ⊂ V ′ 120 120 120 956 956 956 6010 6010 6010
Proof : The coordinates of 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 ′ 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 ′ W_2\subset V' W 2 ⊂ V ′ 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_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) X 0 ( 0 ) X_0^{(0)} X 0 ( 0 ) q 11 / 3 q^{11/3} q 11/3 ch ψ R ( U R ) = X 0 ( 0 ) \operatorname{ch}\psi_{\mathrm{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 ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R ) ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) 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 [48 ] has currents of levels ( 1 , 39 ) (1,39) ( 1 , 39 ) U R U^{\mathrm{R}} U R Theorem 10.38 , assuming ( H R ) (\mathrm{H}_{\mathrm{R}}) ( H R )
The remaining inputs are 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 10.35 ), and the explicit automorphisms and identifications (Theorem 10.24 , Propositions 10.12 , 10.27 , 10.36 and 10.37). The existence statements of [Rie26a, Jia26] identify the two candidates as positive strongly rational Haagerup algebras. They supply their vacuum characters in the first proofs of uniqueness, and also give the equality of global dimensions needed for the alternative proof in Lemma 10.28 . Without them, the same arguments show that a Haagerup algebra of central charge eight contains, after an automorphism of V E 6 ⊕ A 2 V_{E_6\oplus A_2} V E 6 ⊕ A 2 ψ R ( U R ) \psi_{\mathrm{R}}(U^{\mathrm{R}}) ψ R ( U R ) ψ ( V ) \psi(V) ψ ( V ) X 0 ( 0 ) X_0^{(0)} X 0 ( 0 )
:::
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}(\mathbb{K}/\mathbb{Q})=\{1,\sigma,\tau,\sigma\tau\} G = Gal ( K / Q ) = { 1 , σ , τ , σ τ } τ \tau τ g ~ = g ⊗ 1 \widetilde{g}=g\otimes1 g = g ⊗ 1 V K ′ = V Q ′ ⊗ K V'_{\mathbb{K}}=V'_{\mathbb{Q}}\otimes\mathbb{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
defined over Q \mathbb{Q} Q z 2 = z ′ 2 = 1 z^2=z^{\prime2}=1 z 2 = z ′2 = 1 z z ′ = z ′ z zz^{\prime}=z^{\prime}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^{\prime}\widetilde{\tau} ψ τ = z z ′ τ U 2 , K = U 2 ∩ V K ′ U_{2,\mathbb{K}}=U_2\cap V_{\mathbb{K}}^{\prime} 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 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 v ′ = i v v^{\prime}=iv v ′ = i v K \mathbb{K} K ψ τ ( v ′ ) = ψ σ ( v ′ ) = − v ′ \psi_{\tau}(v^{\prime})=\psi_{\sigma}(v^{\prime})=-v^{\prime} 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^{\prime}) ψ σ ( v ′ ) v ′ v^{\prime} v ′
Put U 1 , K = U 1 ∩ V K ′ U_{1,\mathbb{K}}=U_1\cap V_{\mathbb{K}}^{\prime} U 1 , K = U 1 ∩ V K ′ U K ⊂ V K ′ U_{\mathbb{K}}\subset V_{\mathbb{K}}^{\prime} 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,\mathbb{K}}\oplus U_{2,\mathbb{K}} U 1 , K ⊕ U 2 , K e e e f f f v ′ v^{\prime} v ′ U = U K ⊗ K C U=U_{\mathbb{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_{\mathbb{K}} U K U Q = { u ∈ U K : ψ g ( u ) = u , g ∈ G } U_{\mathbb{Q}}=\{u\in U_{\mathbb{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^{\prime}=-\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^{\prime} z ′ z 1 z_1 z 1 z A z_A z A Z Z Z V K ′ V_{\mathbb{K}}^{\prime} V K ′ V K ′ ; V_{\mathbb{K}}^{\prime}; V K ′ ; K \mathbb{K} K U 2 , K = U 2 ( 9 , 0 ) ∩ V K ′ U_{2,\mathbb{K}}=U_2(9,0)\cap V_{\mathbb{K}}^{\prime} 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,\mathbb{K}} U 2 , K K \mathbb{K} K Propositions 10.12 and 10.27 and Remark 10.30 to the chosen K \mathbb{K} K e e e f f f v ′ v^{\prime} v ′ z z z z ′ z^{\prime} 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_{\mathbb{K}}^{\prime} V K ′ U 2 , K U_{2,\mathbb{K}} U 2 , K e e e f f f v ′ v^{\prime} v ′ U K U_{\mathbb{K}} U K H 1 ( G , GL n ( K ) ) = 1 H^1(G,\operatorname{GL}_n(\mathbb{K}))=1 H 1 ( G , GL n ( K )) = 1 [68 Chapter X, Proposition 3] ), each U K , n U_{\mathbb{K},n} U K , n K \mathbb{K} K U K , n = ( U K , n ) G ⊗ Q K U_{\mathbb{K},n}=(U_{\mathbb{K},n})^G\otimes_{\mathbb{Q}}\mathbb{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^{\prime} − 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_{\mathbb{K}} U K K \mathbb{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 K \mathbb{K} K U 2 , K U_{2,\mathbb{K}} U 2 , K Q \mathbb{Q} Q U 2 ( 9 , 0 ) U_2(9,0) U 2 ( 9 , 0 ) V Q ′ V_{\mathbb{Q}}^{\prime} V Q ′ f 1 f_1 f 1 V ′ V^{\prime} V ′ Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 ) K \mathbb{K} K Aut ( U K ) \operatorname{Aut}(U_{\mathbb{K}}) Aut ( U K )
An independent route to rationality: partial results
In this section, S ⊂ V ′ = V E 6 ⊕ A 2 S\subset V^{\prime}=V_{E_6\oplus A_2} S ⊂ V ′ = V E 6 ⊕ A 2 U 1 = g ′ ′ φ ′ ′ U_1=g^{\prime\prime}\varphi^{\prime\prime} U 1 = g ′′ φ ′′ U 2 U_2 U 2 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 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
We compare what remains to be done with the proof of [65 ] for case (I), which we use as a model of proof. The categorical steps not proved here are listed in Remark 11.2. The algebraic steps used below are identified where they enter. Section and page numbers refer to the working manuscript [65 ] and its package of computational certificates. Evans [30 p. 3] cites the three constructions [48 , 50 Rie26a] ; that of Jiao is the case (II) algebra (Theorem 10.24).
The structure of the proof in case (I)
The proof strategy of [65 ] separates the construction, strong generation, C 2 C_2 C 2 C 2 C_2 C 2 S S S Proposition 10.17 . Rationality still requires a characteristic-zero argument for its Zhu algebra, and the categorical identification retains the unresolved steps of Remark 11.2 .
Construction and positivity [65 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
Strong generation [65 Section 4, pp. 20–24] . The finite mode closure in weights 2 and 3 [65 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 [65 Proposition 4.2] ; and a general lemma, triangular strong generation [65 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 [65 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 [65 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 S S S 10.21 show that S S S C 2 C_2 C 2 [56 Main Theorem 3 (Corollary 5.19)] , S S S A ( S ) A(S) A ( S ) [70 ] . Alternatively, since V ′ V' V ′ S S S V ′ = S ⊕ S ′ V' = S \oplus S' V ′ = S ⊕ S ′ [57 ] shows that S S S S S S V ′ V' V ′ C 2 C_2 C 2 [15 ] require that every simple V ′ V' V ′ Rep ( S ) ≅ Z ( H 3 ) \operatorname{Rep}(S) \cong Z(H_3) Rep ( S ) ≅ Z ( H 3 ) S S S L s l 2 ( 3 , 0 ) ⊠ L s l 2 ( 13 , 0 ) L_{\mathfrak{sl}_2}(3,0) \boxtimes 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) \boxtimes 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) \boxtimes L(39,39) L ( 1 , 1 ) ⊠ L ( 39 , 39 ) 1 4 + 39 10 = 10 \frac{1}{4}+\frac{39}{10}=10 4 1 + 10 39 = 10 S S S V ( 3 , 13 ) V(3,13) V ( 3 , 13 ) J J J S S S A ⊕ J A \oplus J A ⊕ J A + A^+ A + [65 Section 7] ; in case (I) the current appears only in weight ten.
Modules and the categorical identification [65 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 (e,a) \in L^*/L \cong\mathbb{Z}_3 ( e , a ) ∈ L ∗ / L ≅ Z 3
exactly six more [65 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 Z(D_{\pm}) Rep ( U ) ≃ Z ( D ± ) D ± = Rep ( U ) A ± D_{\pm} = \operatorname{Rep}(U)_{A_{\pm}} D ± = Rep ( U ) A ± [65 Lemma 12.2] [21 ] . 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 [32 Theorems 2 and 3] is distinct from proving this exhaustion. The dimension comparison used to select the relevant categories, followed by the Morita equivalence of H 2 \mathcal{H}_2 H 2 H 3 \mathcal{H}_3 H 3 [41 Theorem 1.4] , is among the categorical steps to be transferred. For S S S
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
Proposition 4.1 suggests using a holomorphic extension to identify Rep ( S ) \operatorname{Rep}(S) Rep ( S ) V E 8 = S ⊕ M b ⊕ 2 M a V_{E_8} = S \oplus M_b \oplus2M_a V E 8 = S ⊕ M b ⊕ 2 M a
There are two even unimodular overlattices of L L L L ∗ / L ≅ Z 3 2 L^*/L \cong\mathbb{Z}_3^2 L ∗ / L ≅ Z 3 2 q ( e , a ) = 2 e 2 + a 2 3 q(e,a) = \frac{2e^2+a^2}{3} q ( e , a ) = 3 2 e 2 + a 2 A ± A_{\pm} A ± X ± X_{\pm} X ±
The exhaustion statement (C) of Remark 11.2 is an assumption: every fusion category with the specified fusion rules and untwisted pointed part must belong to the listed family. The classification within that family [32 ] does not by itself prove this statement.
The fusion rules of D ± D_{\pm} D ± V E 8 V_{E_8} V E 8 [65 Section 12] uses the full branching of the nine lattice modules and one dimension computed from the modular matrix, hence the twelve-module list.
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 p 3 p_3 p 3 R b R_b R b 80 , 1250 , 10630 80, 1250, 10630 80 , 1250 , 10630 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 [Rie26a, Hua26].
Proof. Proposition 10.17 and Theorem 10.21 give C 2 C_2 C 2 [56 Main Theorem 3 (Corollary 5.19)] applies under (Z) and proves strong rationality. For the last assertion, an isomorphism with an algebra of [Rie26a, Hua26] would force dim S 1 = 6 \dim S_1=6 dim S 1 = 6 S 1 S_1 S 1 U 1 U_1 U 1 U 1 U_1 U 1 U 1 U_1 U 1
Under the hypotheses of Theorem 11.1 and (M), Rep ( S ) \operatorname{Rep}(S) Rep ( S ) [47 ] . The nonvacuum lowest weights are positive, so the quantum dimensions are positive [18 Lemma 4.2] and equal the categorical dimensions [47 ] . 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 ± \mathbf{1}\oplus M_b\oplus2X_\pm 1 ⊕ M b ⊕ 2 X ± A ± A_\pm A ± Rep ( S ) ≃ Z ( D ± ) \operatorname{Rep}(S)\simeq Z(\mathcal{D}_\pm) Rep ( S ) ≃ Z ( D ± ) D ± = Rep ( S ) A ± \mathcal{D}_\pm=\operatorname{Rep}(S)_{A_\pm} D ± = Rep ( S ) A ± [21 ] . To identify this center with Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [65 Sections 11.3 and 12–13] , one must still establish the fusion rules of D ± \mathcal{D}_\pm D ± Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 )
Hypothesis (C) is the exhaustion statement formulated in [65 Proposition 12.5] ; classification within the family of [32 Theorem 2] does not itself prove it. Assuming (HJ _J J
The Zhu algebra of S S S
We now adapt to S S S [65 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 [70 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 ] [70 Lemma 2.1.1] and, by [70 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 ] . (27) [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*{(27)} [ h ( − 1 ) x ] = − [ x ] ∗ [ h ] , [ h ( 0 ) x ] = [ h ] ∗ [ x ] − [ x ] ∗ [ h ] , [ a ( 0 ) x ] = [ a ] ∗ [ x ] − [ x ] ∗ [ a ] − [ a ( 1 ) x ] . (27) 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 ′′ ( 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 (27) , 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 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 ′ L(-1)e' L ( − 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 ′′ (27) .
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 ′ V' V ′ 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
∥ L − 1 C x ∥ χ 2 = ∥ L 1 C x ∥ χ 2 + 2 H χ ( L 0 C x , x ) . (28) \lVert L_{-1}^C x\rVert_\chi^2=\lVert L_1^C x\rVert_\chi^2+2H_\chi(L_0^C x,x). \tag*{(28)} ∥ L − 1 C x ∥ χ 2 = ∥ L 1 C x ∥ χ 2 + 2 H χ ( L 0 C x , x ) . (28) 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 (28) 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\ge0 λ ≥ 0 L 0 C L_0^C L 0 C A A A S S S
For x ∈ Ker L 0 C x\in\operatorname{Ker}L_0^C x ∈ Ker L 0 C L 1 C x = 0 L_1^C x=0 L 1 C x = 0 (28) gives L − 1 C x = 0 L_{-1}^C x=0 L − 1 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\ge0 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 . (29) m\ge\frac{a^2}{12}+\frac{b^2}{52}. \tag*{(29)} m ≥ 12 a 2 + 52 b 2 . (29) 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 4 ′ V_4' V 4 ′
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 (30) , 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 . (30) (u_{(1,7)})_{(-1)}u_{(2,6)}=\left(\frac{507}{5}\sqrt{13}+\frac{3549}{5}\sqrt{-3}\right)e^w. \tag*{(30)} ( u ( 1 , 7 ) ) ( − 1 ) u ( 2 , 6 ) = ( 5 507 13 + 5 3549 − 3 ) e w . (30) 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 4 4 4 ⟨ 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 (22) 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 . (31) \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*{(31)}
\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 . (31) This proves (30) 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 (29) . 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 4 4 4 ( 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 3 3 3 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_0e^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 J 0 = C 1 J_0=\mathbb{C}1 J 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 [19 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 \leq i \leq3 0 ≤ i ≤ 3 0 ≤ j ≤ 13 0 \leq j \leq13 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 ) 0 0 0 1 1 1 2 2 2 4 4 4 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 + [19 , 43 ] 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 = 6 5 ) , T 13 = V ( 0 , 8 ) ⊕ V ( 3 , 5 ) ( h = 4 3 ) . (32) \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=\tfrac{6}{5}),\qquad T_{13}=V(0,8)\oplus V(3,5) \quad(h=\tfrac{4}{3}). \tag*{(32)}
\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 = 5 6 ) , T 13 = V ( 0 , 8 ) ⊕ V ( 3 , 5 ) ( h = 3 4 ) . (32) 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_{ij}^{(t)} E ij ( t ) E 11 ( t ) E_{11}^{(t)} E 11 ( t ) 1 = ∑ t , i E i 1 ( t ) E 11 ( t ) E 1 i ( t ) 1=\sum_{t,i}E_{i1}^{(t)}E_{11}^{(t)}E_{1i}^{(t)} 1 = ∑ t , i E i 1 ( t ) E 11 ( t ) E 1 i ( t ) 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 [65 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 0 0 c 1 c_1 c 1 d 1 d_1 d 1 d 4 d_4 d 4 d 5 d_5 d 5 d 6 d_6 d 6 c 2 c_2 c 2 d 2 d_2 d 2 b b b a a a c 3 c_3 c 3 d 3 d_3 d 3 dim D = 762 \dim D=762 dim D = 762 T 12 , T 13 T_{12},T_{13} T 12 , T 13 546 546 546 [65 Proposition 9.5] .
To obtain an upper bound for D D D [65 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 [65 Propositions 8.3 and 8.6] . The relations are the projections of the Zhu images of 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 (27) 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]\ast[b]+[b]\ast[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]\ast[b]-[b]\ast[a]-[c_-(a,b)] [ 2 1 ( a ( 0 ) b − b ( 0 ) a ) ] = [ a ] ∗ [ b ] − [ b ] ∗ [ a ] − [ c − ( a , b )] 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_{\mathbb{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 \mathfrak p p F p \mathbb{F}_p F p [7 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. The verification record in the supplementary material specifies which stages were independently checked.
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
Remark 11.7 Theorem 11.6 does not prove (Z). A kernel computed modulo p p p D p D_p D p D D D
To obtain such a presentation one must compute the three families of relations over K, check the integrality of the eliminations and rewriting rules at a chosen prime, and prove a spanning bound over the resulting local ring. With these data, a descent argument of the form [65 Lemma 9.6] would give semisimplicity of the upper presentation and hence of its quotient D D D [65 Sections 9.8–9.10] .
The block sizes and vertex ranks computed here agree with the lowest weight spaces of Theorem 8.5 and Table 2 . An exact version would bound the number of simple S S S A + A^+ A + D D D
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 [22 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 )) 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
What remains
The following unresolved steps would produce a proof independent of the existence results granted above.
The exact version of Theorem 11.6 : the three families of relations over K (by pivots modulo p p p Theorem 10.19 ), the eliminations and quadratic rules over K and their integrality at a prime of K above p p p [65 Lemma 9.6] . This would prove (Z), hence the rationality of S S S
Determine the restrictions in (M). The zero-mode action on a lattice module’s lowest weight space identifies the simple submodules generated by that space, by Lemma 11.8 . It does not exclude additional simple summands beginning in higher degrees. In particular, the lowest weight space of W 0 , 0 W_{0,0} W 0 , 0 S S S S S S L L L L ∗ L^* L ∗
Establish that the six additional blocks, with lowest weights 5 13 \frac{5}{13} 13 5 6 13 \frac{6}{13} 13 6 7 13 \frac{7}{13} 13 7 8 13 \frac{8}{13} 13 8 11 13 \frac{11}{13} 13 11 15 13 \frac{15}{13} 13 15 D D D [65 Section 11] ; its hypotheses and conclusion must be verified for S S S
After these steps, Theorem 11.1 gives strong rationality. The categorical transfer listed in Remark 11.2 remains a separate task, including the exhaustion statement (C).
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 [55 , 46 , 60 ] , and we use the notions of genus of Höhn [45 ] (the bulk genus, given by the representation category and the central charge) and of Moriwaki [59 ] and Möller–Rayhaun [60 ] (the hyperbolic genus). We call a Haagerup algebra U U U Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U) \cong\mathcal{Z}(\mathcal{H}_3) Rep ( U ) ≅ Z ( H 3 ) Proposition 4.1 and Theorem 8.5 , after an automorphism of V ′ V^{\prime} V ′ U ⊂ V ′ U \subset V^{\prime} U ⊂ V ′ U 1 = g ′ ⊕ g ′ ′ U_1 = \mathfrak{g}^{\prime} \oplus\mathfrak{g}^{\prime\prime} U 1 = g ′ ⊕ g ′′ ( k ′ , k ′ ′ ) = ( 1 , 39 ) (k^{\prime}, k^{\prime\prime}) = (1,39) ( k ′ , k ′′ ) = ( 1 , 39 ) ( 3 , 13 ) (3,13) ( 3 , 13 ) h = C h ′ ⊕ C h ′ ′ \mathfrak{h} = \mathbb{C}h^{\prime} \oplus\mathbb{C}h^{\prime\prime} h = C h ′ ⊕ C h ′′ U 1 U_1 U 1 V 1 ′ V^{\prime}_1 V 1 ′ h ′ ∈ g ′ h^{\prime} \in\mathfrak{g}^{\prime} h ′ ∈ g ′ h ′ ′ ∈ g ′ ′ h^{\prime\prime} \in\mathfrak{g}^{\prime\prime} h ′′ ∈ g ′′ ⟨ h ′ , h ′ ⟩ = 2 k ′ \langle h^{\prime},h^{\prime}\rangle= 2k^{\prime} ⟨ h ′ , h ′ ⟩ = 2 k ′ ⟨ h ′ ′ , h ′ ′ ⟩ = 2 k ′ ′ \langle h^{\prime\prime},h^{\prime\prime}\rangle= 2k^{\prime\prime} ⟨ h ′′ , h ′′ ⟩ = 2 k ′′ ⟨ h ′ , h ′ ′ ⟩ = 0 \langle h^{\prime},h^{\prime\prime}\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^{\prime},m^{\prime\prime}) = (\langle h^{\prime},\lambda\rangle,\langle h^{\prime\prime},\lambda\rangle) ( m ′ , m ′′ ) = (⟨ h ′ , λ ⟩ , ⟨ h ′′ , λ ⟩) λ \lambda λ h \mathfrak{h} h v v v h ′ ( 0 ) v = m ′ v h^{\prime}(0)v = m^{\prime}v h ′ ( 0 ) v = m ′ v h ′ ′ ( 0 ) v = m ′ ′ v h^{\prime\prime}(0)v = m^{\prime\prime}v h ′′ ( 0 ) v = m ′′ v
∣ λ ∣ 2 = m ′ 2 2 k ′ + m ′ ′ 2 2 k ′ ′ . |\lambda|^2 = \frac{m^{\prime2}}{2k^{\prime}} + \frac{m^{\prime\prime2}}{2k^{\prime\prime}}. ∣ 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 ≔ d e f Com U ( M ( h ) ) \Omega_U \mathrel{\mathop{\coloneqq}^{\mathrm{def}}} \operatorname{Com}_U(M(\mathfrak{h})) Ω U : = def Com U ( M ( h )) the Heisenberg commutant of U U U L L L Com U ( Ω U ) ≅ V L \operatorname{Com}_U(\Omega_U) \cong V_L Com U ( Ω U ) ≅ V L [60 Definition 2.12] . Throughout this section L L L E 6 ⊕ A 2 E_6 \oplus A_2 E 6 ⊕ A 2 [60 Remark 2.13] . For an even lattice L L L C ( L ′ / L ) \mathcal{C}(L^{\prime}/L) C ( L ′ / L ) ( L ′ / L , q L ) (L^{\prime}/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^{\prime}/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^{\prime},m^{\prime\prime}) \in\mathbb{Z}^2 : m^{\prime} \equiv m^{\prime\prime} \bmod2\} Γ 0 = {( m ′ , m ′′ ) ∈ Z 2 : m ′ ≡ m ′′ mod 2 } ( k ′ , k ′ ′ ) ∈ { ( 1 , 39 ) , ( 3 , 13 ) } (k^{\prime},k^{\prime\prime}) \in\{(1,39),(3,13)\} ( k ′ , k ′′ ) ∈ {( 1 , 39 ) , ( 3 , 13 )}
L k ′ , k ′ ′ ≔ d e f Γ 0 ∗ = { ( k ′ a , k ′ ′ c ) : a , c ∈ Z , a ≡ c m o d 2 } , L_{k^{\prime},k^{\prime\prime}} \mathrel{\mathop{\coloneqq}^{\mathrm{def}}} \Gamma_0^* = \{(k^{\prime}a,k^{\prime\prime}c) : a,c \in\mathbb{Z},\ a \equiv c \bmod2\}, L k ′ , k ′′ : = def 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^{\prime},0) ( 2 k ′ , 0 ) ( k ′ , k ′ ′ ) (k^{\prime},k^{\prime\prime}) ( 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 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{bmatrix}2 & 1 \\ 1 & 20\end{bmatrix} [ 2 1 1 20 ] [ 6 3 3 8 ] \begin{bmatrix}6 & 3 \\ 3 & 8\end{bmatrix} [ 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^{\prime} = \Gamma_0 L ′ = Γ 0 L ′ / L ≅ Z / 39 L^{\prime}/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}}^{\prime}/L_{\mathrm{II}} \to L_{\mathrm{I}}^{\prime}/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^{\prime}/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}}^{\prime}/L_{\mathrm{II}}) O ( L II ′ / L II ) w ′ = diag ( − 1 , 1 ) w^{\prime}=\operatorname{diag}(-1,1) w ′ = diag ( − 1 , 1 ) w ′ ′ = diag ( 1 , − 1 ) w^{\prime\prime}=\operatorname{diag}(1,-1) w ′′ = diag ( 1 , − 1 ) h h h g ′ \mathfrak{g}^{\prime} g ′ g ′ ′ \mathfrak{g}^{\prime\prime} 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}}^{\prime}/L_{\mathrm{I}}) O ( L I ′ / L I ) { ± 1 } \{\pm1\} { ± 1 } w ′ w^{\prime} w ′ w ′ ′ w^{\prime\prime} 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 [63 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 ) [17 ] .
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 \ge0\}. 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 [54 Section 4.7] .
Proposition 12.2 Let U U U
The Heisenberg commutant Ω U \Omega_U Ω U ( Ω U ) 1 = 0 (\Omega_U)_1 = 0 ( Ω U ) 1 = 0 τ \tau τ C ( L ′ / L ) \mathcal{C}(L'/L) C ( L ′ / L ) Rep ( Ω U ) \operatorname{Rep}(\Omega_U) Rep ( Ω U ) τ \tau τ
U = ⨁ α ∈ L ′ / L Ω U τ ( α ) ⊗ V α + L U = \bigoplus_{\alpha\in L'/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 Rep ( Ω U ) ≅ Z ( H 3 ) ⊠ C ( L ′ / L ) \operatorname{Rep}(\Omega_U) \cong\mathcal{Z}(\mathcal{H}_3) \boxtimes\mathcal{C}(L'/L) Rep ( Ω U ) ≅ Z ( H 3 ) ⊠ C ( L ′ / L )
Every irreducible Ω U \Omega_U Ω U α ≠ 0 \alpha\ne0 α = 0 Ω U τ ( α ) \Omega_U^{\tau(\alpha)} Ω U τ ( α ) U U U h \mathfrak{h} h λ \lambda λ ∣ λ ∣ 2 2 \frac{|\lambda|^2}{2} 2 ∣ λ ∣ 2
Γ ( U ) = Γ 0 \Gamma(U) = \Gamma_0 Γ ( U ) = Γ 0 L = L k ′ , k ′ ′ L = L_{k',k''} L = L k ′ , k ′′ L I L_{\mathrm{I}} L I L I I L_{\mathrm{II}} L II
For λ = ( k ′ , k ′ ′ ) \lambda= (k', k'') λ = ( k ′ , k ′′ ) e λ e^\lambda e λ V L ⊂ U V_L \subset U V L ⊂ U ( 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 ) k ′ + k ′ ′ 4 \frac{k'+k''}{4} 4 k ′ + k ′′ A A A U U U 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 ′′ )
Proof. 1. By [60 Definition 2.12, Proposition 2.14] , which rest on [55 ] , Ω U \Omega_U Ω U V L V_L V L U U U Ω U \Omega_U Ω U [12 ] , 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'/L A U ⊂ L ′ / L L ′ / L L'/L L ′ / L Rep ( U ) ≅ Z ( H 3 ) \operatorname{Rep}(U) \cong\mathcal{Z}(\mathcal{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 [18 ] , and they are the categorical dimensions of Rep ( U ) \operatorname{Rep}(U) Rep ( U ) [47 ] ; alternatively, Z ( H 3 ) \mathcal{Z}(\mathcal{H}_3) Z ( H 3 ) [60 Corollary 4.10] , applied with V = W = U V = W = U V = W = U A U = L ′ / L A_U = L'/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\mathcal{C}(L'/L) Rep ( Ω U ) ≅ Rep ( U ) ⊠ C ( L ′ / L ) Ω U \Omega_U Ω U L = 6 L = 6 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'/L L ′ / L L ′ / L L'/L L ′ / L β ∈ L ′ / L \beta\in L'/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 [43 , 14 ] , 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)^{-\frac{1}{2}}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 12 y \eta(iy)^{-2}=y\eta(i/y)^{-2}\sim ye^{\frac{2\pi}{12y}} η ( i y ) − 2 = y η ( i / y ) − 2 ∼ y e 12 y 2 π ( det L ) − 1 2 e 2 π ⋅ 2 24 y (\det L)^{-\frac{1}{2}}e^{\frac{2\pi\cdot2}{24y}} ( det L ) − 2 1 e 24 y 2 π ⋅ 2 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 [70 ] 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 ν ) e 2 π y ( 1 24 − h ν ) \operatorname{ch}_{\nu}(i/y)\geq\dim(\nu_{h_\nu})e^{\frac{2\pi}{y}(\frac{1}{24}-h_\nu)} ch ν ( i / y ) ≥ dim ( ν h ν ) e y 2 π ( 24 1 − h ν ) h ν ≥ 0 h_\nu\geq0 h ν ≥ 0 α ≠ 0 \alpha\neq0 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}\geq\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\mathbb{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\geq0 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\geq1 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)\geq\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)},e''_{(0)} e ( 0 ) ′ , e ( 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 [65 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^\flat L ♭ { L , L ♭ } = { L I , L I I } \{L,L^\flat\}=\{L_I,L_{II}\} { L , L ♭ } = { L I , L I I } ψ : L ′ / L → ( L ♭ ) ′ / L ♭ \psi:L'/L\to(L^\flat)'/L^\flat 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^\flat})\cong\operatorname{Rep}(\Omega_U)\boxtimes\mathcal C((L^\flat)'/L^\flat) Rep ( Ω U ⊗ V L ♭ ) ≅ Rep ( Ω U ) ⊠ C (( L ♭ ) ′ / L ♭ ) [34 , 14 ] . The isometry ψ \psi ψ Φ ψ : C ( L ′ / L ) → C ( ( L ♭ ) ′ / L ♭ ) \Phi_\psi:\mathcal C(L'/L)\to\mathcal C((L^\flat)'/L^\flat) Φ ψ : C ( L ′ / L ) → C (( L ♭ ) ′ / L ♭ ) [28 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 [43 ] , U U U A U = ⨁ α ∈ L ′ / L τ ( α ) ⊠ α A_U=\bigoplus_{\alpha\in L'/L}\tau(\alpha)\boxtimes\alpha A U = ⨁ α ∈ L ′ / L 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 ♮ Ω U ⊗ V L ♮ \Omega_U\otimes V_{L^\natural} Ω U ⊗ V L ♮ Φ ( A U ) \Phi(A_U) Φ ( A U ) [43 , 14 ] . 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 ) 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^\natural U ♮ ψ \psi ψ ( U ♮ ) ♮ ≅ U (U^\natural)^\natural\cong U ( U ♮ ) ♮ ≅ U U ↦ U ♮ U\mapsto U^\natural U ↦ U ♮ Ω U \Omega_U Ω U U U U U ♮ U^\natural U ♮
U U U U ♮ U^\natural U ♮ U U U U ♮ U^\natural U ♮
Proof. 1. Extension and representation category. Put B = Ω U ⊗ V L ♮ B=\Omega_U\otimes V_{L^\natural} B = Ω U ⊗ V L ♮ A = Φ ( A U ) A=\Phi(A_U) A = Φ ( A U ) B B B [43 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 ) [43 , 14 ] . Their category is ribbon equivalent, through Φ \Phi Φ A U 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^\natural} 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(\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^\natural}\frac{|\gamma|^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^\natural} 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^\natural})_1\oplus\bigoplus_{\alpha\ne0}(\Omega_U^{\tau(\alpha)}\otimes V_{\psi(\alpha)+L^\natural})_1 W 1 = 1 ⊗ ( V L ♮ ) 1 ⊕ ⨁ α = 0 ( Ω U τ ( α ) ⊗ V ψ ( α ) + 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^\natural})_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 [53 ] .
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^\natural} Ω U ⊗ V L ♮ C 2 C_2 C 2 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^\natural} Ω U ⊗ V L ♮ B = Ω U ⊗ V L ♮ B=\Omega_U \otimes V_{L^\natural} B = Ω U ⊗ V L ♮ M ′ M' M ′ W W W B B B N N N M M M 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(\frac{K}{W\ N}\right) Hom B ( M ′ , K ) ⟶ I B ( W N K ) 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 ) [43 , 14 ] , it is a local module for the connected étale algebra Φ ( A U ) \Phi(A_U) Φ ( A U ) [21 ] . Hence this submodule is a finite direct sum of irreducible W W W M M M W W W [60 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_U^\nu\otimes V_{\gamma+L^\natural} Ω U ν ⊗ V γ + L ♮ h ν ≥ 0 h_\nu\geq0 h ν ≥ 0 h X ≥ 0 h_X\geq0 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_U^\nu\otimes V_{L^\natural} Ω U ν ⊗ V L ♮ h ν = 0 h_\nu=0 h ν = 0 A U A_U A U X ♭ = Φ − 1 ( X ) X^\flat=\Phi^{-1}(X) X ♭ = Φ − 1 ( X ) U U U ( ν ⊠ 0 ) = ν ⊠ 0 (\nu\boxtimes0)=\nu\boxtimes0 ( ν ⊠ 0 ) = ν ⊠ 0 Ω U ν ⊗ V L \Omega_U^\nu\otimes V_L Ω U ν ⊗ V L h X ♭ ≤ h ν = 0 h_{X^\flat}\leq 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(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 [23 ] 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 0 \mathbb{C}e^0 C e 0 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 ∈ L ♮ Vac Ω U ( Ω U τ ( α ) ) ⊗ V ψ ( α ) + L ♮ = V L ♮ \operatorname{Com}_W(\Omega_U)=\bigoplus_{\alpha\in L^\natural}\operatorname{Vac}_{\Omega_U}(\Omega_U^{\tau(\alpha)})\otimes V_{\psi(\alpha)+L^\natural}=V_{L^\natural} Com W ( Ω U ) = ⨁ α ∈ L ♮ Vac Ω U ( Ω U τ ( α ) ) ⊗ V ψ ( α ) + L ♮ = V L ♮ W W W Ω U \Omega_U Ω U L ♮ L^\natural L ♮ W W W L I L_I L I L I I L_{II} L I I L ♮ L^\natural L ♮ L L L W W W
The map τ : L ′ / L ⟶ P \tau:L'/L\longrightarrow 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'_{II}/L_{II}\longrightarrow 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_U^{\tau_{II}(\alpha)}\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 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\longrightarrow 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 [23 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 ♮ ) ♮ ≅ U (U^\natural)^\natural\cong U ( U ♮ ) ♮ ≅ U [23 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 [60 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 [60 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 [60 Corollary 4.10] , V = ⨁ α Ω σ ( α ) ⊗ V α + N V = \bigoplus_\alpha\Omega^\sigma(\alpha) \otimes V_{\alpha+N} V = ⨁ α Ω 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 1. 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 R ) \psi_R(U_R^R) ψ R ( U R 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 [Rie26a], 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 [50 ] , 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.
Thus the existence of a Haagerup algebra in one case implies its existence in the other case, and its uniqueness in one case implies its uniqueness in the other case. Each of the announcements [Rie26a] and [50 ] separately gives exactly two Haagerup algebras; the identification of both of them with the algebras of Riesen and of Jiao (part 3, and Theorem 10.40 ) still uses both. In particular, assuming ( H J ) (\mathrm{H}_{\mathrm{J}}) ( H J ) ψ R ( U R ) \psi_R(U^R) ψ R ( U R ) S ♮ S^\natural S ♮ V ′ V' V ′ ⟨ A , e , f , v ⟩ \langle A,e,f,v\rangle ⟨ A , e , f , v ⟩ X 0 ( 0 ) X_0^{(0)} X 0 ( 0 )
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^{-|\mu|^2/2} q − ∣ μ ∣ 2 /2 f α f_\alpha f α Ω U τ ( α ) \Omega_U^\tau(\alpha) Ω U τ ( α ) α = μ + L \alpha=\mu+L α = μ + L U 2 U_2 U 2 U 3 U_3 U 3 μ \mu μ ψ \psi ψ f α f_\alpha f α f ψ ( α ) f_{\psi(\alpha)} f ψ ( α ) Theorem 12.4 , part 4: it is a test, in low weights, of the isomorphism of the Heisenberg commutants of any two Haagerup algebras of the two cases, which Theorem 12.3 proves for U U U U ♯ U^\sharp U ♯
Remark 12.6 Let V V V Rep ( V ) ≅ Z ( H 3 ) \operatorname{Rep}(V)\cong Z(\mathcal{H}_3) Rep ( V ) ≅ Z ( H 3 ) Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [40 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 ) [29 ] ; since c ( V ) ≡ 0 ≡ c ( Z ( H 3 ) ) ( m o d 8 ) c(V)\equiv0\equiv c(Z(\mathcal{H}_3))\pmod{8} c ( V ) ≡ 0 ≡ c ( Z ( H 3 )) ( mod 8 ) S S S V V V S S S s s s § 2 [60 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^j_{h_{\min}}\right)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 [60 Proposition 3.10] with a weaker sign condition: Z ( H 3 ) Z(\mathcal{H}_3) Z ( H 3 ) [60 ] (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 1 1 13 \sqrt{13} 13 − 3 \sqrt{-3} − 3 − 39 \sqrt{-39} − 39 Q \mathbb{Q} Q Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
For a matrix integral at a prime p \mathfrak{p} p K \mathbb{K} K K \mathbb{K} K K \mathbb{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.
Table 8.
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.
Table 5. Correspondence between numbered statements and computational scripts.
Bibliography 