Introduction
Wilson’s lattice gauge theory [34 ] assigns an element Q e Q_e Q e G ⊂ U ( N ) G \subset\mathrm{U}(N) G ⊂ U ( N ) e e e Z 4 \mathbb{Z}^{4} Z 4 exp ( N β ∑ p Re Tr Q p ) \exp\left(N\beta\sum_p \operatorname{Re}\operatorname{Tr} Q_p\right) exp ( N β ∑ p Re Tr Q p ) Q p Q_p Q p p p p 1 ∣ β ∣ |\beta| ∣ β ∣ [29 , 30 ] . The question is how far into the coupling range such statements can be proved with explicit constants, since the problem of the continuum limit [22 ] begins where strong coupling ends and Bałaban’s weak-coupling analysis [3 , 4 , 5 ] takes over.
Two methods give explicit ranges. The Bakry–Émery method [1 , 2 ] of Shen, Zhu and Zhu [31 ] bounds the Hessian of the action; in [9 ] we showed that the sharp bound is 4 d = 16 4d = 16 4 d = 16 ∣ β ∣ < 1 / 32 |\beta| < 1/32 ∣ β ∣ < 1/32 [14 , 15 ] bounds the influence of each link on the conditional law of another; for SU(2) the single-link conditional law is a von Mises–Fisher law on the three-sphere, the sharp Kantorovich modulus of that family is one quarter [9 Thm. 7] , and every link has eighteen partners, which gives β W < 2 / 9 \beta_W < 2/9 β W < 2/9
Both limits come from the same count. A link of Z 4 \mathbb{Z}^4 Z 4 [13 ] .2 Lemma 2.2 ), and both methods improve.
Theorem 1.1 (SU(2)). Let G = S U ( 2 ) G = \mathrm{SU}(2) G = SU ( 2 ) 0 ≤ β W < 1 3 0 \le\beta_W < \frac{1}{3} 0 ≤ β W < 3 1 O O O W Λ ( O ) W_\Lambda(O) W Λ ( O ) Z 4 \mathbb{Z}^4 Z 4 ω ( O ) \omega(O) ω ( O ) ω \omega ω − log ( 3 β W ) -\log(3\beta_W) − log ( 3 β W )
m ≥ − log ( 3 β W ) > 0. m \ge-\log(3\beta_W) > 0. m ≥ − log ( 3 β W ) > 0. For − 1 3 < β W < 0 -\frac{1}{3} < \beta_W < 0 − 3 1 < β W < 0
The step behind Theorem 1.1 is short once the line field is in place.3 1 3 \frac{1}{3} 3 1
Theorem 1.2 (Ceiling). Let F ′ F' F ′ Z 4 \mathbb{Z}^4 Z 4 F ′ F' F ′ ∣ β ∣ |\beta| ∣ β ∣ β W ≥ 1 3 \beta_W \ge\frac{1}{3} β W ≥ 3 1 β W = 1 3 \beta_W = \frac{1}{3} β W = 3 1
The ceiling concerns coefficients between single links. Take instead as a block the three free links of one plaquette. The free links of the line field split into such triples with period two (Lemma 5.1 ), the coupling inside the block plaquette is no longer paid for, and each link keeps ten outside partners. The price is the sensitivity of a law on three spheres instead of one, which we bound analytically by integrating out the middle link of the block.
Theorem 1.3 (SU(2) with plaquette blocks). Let G = SU ( 2 ) G = \operatorname{SU}(2) G = SU ( 2 ) and 0 ≤ β W ≤ 0.345 0 \le\beta_{\mathrm W} \le0.345 0 ≤ β W ≤ 0.345 The number α ( β W ) \alpha(\beta_{\mathrm W}) α ( β W ) of (4) is increasing, α ( 1 3 ) ≤ 0.9561 \alpha(\frac{1}{3}) \le0.9561 α ( 3 1 ) ≤ 0.9561 and α ( 0.345 ) ≤ 0.9973 \alpha(0.345) \le0.9973 α ( 0.345 ) ≤ 0.9973 The conclusions of Theorem 1.1 hold with decay at every rate below − 1 2 log α ( β W ) -\frac{1}{2}\log\alpha(\beta_{\mathrm W}) − 2 1 log α ( β W ) and a transfer-matrix gap
m ≥ − 1 2 log α ( β W ) > 0 ; m \ge-\frac{1}{2}\log\alpha(\beta_{\mathrm W}) > 0; m ≥ − 2 1 log α ( β W ) > 0 ; at β W = 1 3 \beta_{\mathrm W} = \frac{1}{3} β W = 3 1 this gives m ≥ 0.0224 m \ge0.0224 m ≥ 0.0224 For − 0.345 ≤ β W < 0 -0.345 \le\beta_{\mathrm W} < 0 − 0.345 ≤ β W < 0 all statements except reflection positivity and the gap hold.
At small coupling the rate of Theorem 1.1 is the better one, so both bounds stand side by side; Theorem 1.3 matters past β W = 1 3 \beta_{\mathrm W} = \frac{1}{3} β W = 3 1 Theorem 1.1 has closed.
The curvature method gains as well, and for every N N N Hess S 0 \operatorname{Hess} S_0 Hess S 0 ∑ p Re Tr Q p \sum_p \operatorname{Re}\operatorname{Tr} Q_p ∑ p Re Tr Q p G E G^E G E F F F
Theorem 1.4 (Curvature in the gauge). For every N N N every field Q Q Q and every tangent vector v = ( X e Q e ) v=(X_eQ_e) v = ( X e Q e ) with X e = 0 X_e=0 X e = 0 for e ∈ F e\in F e ∈ F
∣ Hess S 0 ( v , v ) ∣ ≤ 29 2 ∣ v ∣ 2 , \left|\operatorname{Hess} S_0(v,v)\right| \le\frac{29}{2}|v|^2, ∣ Hess S 0 ( v , v ) ∣ ≤ 2 29 ∣ v ∣ 2 , against the sharp constant 16 without gauge fixing. Consequently, for G = SU ( N ) G=\operatorname{SU}(N) G = SU ( N ) and ∣ β ∣ < 1 29 |\beta|<\frac{1}{29} ∣ β ∣ < 29 1 or G = SO ( N ) G=\operatorname{SO}(N) G = SO ( N ) N ≥ 3 N\ge3 N ≥ 3 and ∣ β ∣ < N − 2 58 N |\beta|<\frac{N-2}{58N} ∣ β ∣ < 58 N N − 2 the conclusions of Theorem 1.1 hold with some rate c > 0 c>0 c > 0 in place of − log ( 3 β W ) -\log(3\beta_{\mathrm W}) − log ( 3 β W ) For SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) this is β W < 9 29 = 0.310 \beta_{\mathrm W}<\frac{9}{29}=0.310 β W < 29 9 = 0.310
No Bakry–Émery argument for the ungauged measure can pass ∣ β ∣ = 1 32 |\beta|=\frac{1}{32} ∣ β ∣ = 32 1 [9 Thm. C] ; the gauge fixing changes the measure, not the metric, which is why it can. For SU ( 2 ) \operatorname{SU}(2) SU ( 2 ) β W < 4 29 \beta_{\mathrm W}<\frac{4}{29} β W < 29 4 Theorem 1.1 . For SU ( 3 ) \operatorname{SU}(3) SU ( 3 )
Theorem 1.5 (SU(3)). Let G = SU ( 3 ) G=\operatorname{SU}(3) G = SU ( 3 ) and 0 ≤ β W ≤ 3 4 0\le\beta_{\mathrm W}\le\frac{3}{4} 0 ≤ β W ≤ 4 3 Then the conclusions of Theorem 1.1 hold, with decay at every rate below − log ( 1.33319 β W ) -\log(1.33319\beta_{\mathrm W}) − log ( 1.33319 β W ) and a transfer-matrix gap
m ≥ − log ( 1.33319 β W ) > 0. m \ge-\log(1.33319\beta_{\mathrm W}) > 0. m ≥ − log ( 1.33319 β W ) > 0. The proof of Theorem 1.5 bounds the Kantorovich modulus of the one-link measure ν b ( d U ) ∝ e Re Tr ( U b ) d U \nu_b(dU)\propto e^{\operatorname{Re}\operatorname{Tr}(Ub)}dU ν b ( d U ) ∝ e Re Tr ( U b ) d U SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) b b b ν b \nu_b ν b Lemma 7.3 ), a corrector chosen box by box among polynomials of degree two in the matrix entries (Lemma 7.4 ), the reduction of b b b ν b \nu_b ν b Lemma 7.6 ) and an interpolation inequality on boxes of staple sums (Lemma 7.7 ). The remaining step evaluates the resulting bound on 2341 boxes with rigorous error bounds; the theorem rests on that computation, which is described in full in § 8.7 and Appendix § 16 .
The plaquette blocks of Theorem 1.3 work for SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) Λ ( 2 β W ) \Lambda(2\beta_{\mathrm W}) Λ ( 2 β W ) 2 r Λ ( r ) < 1 2r\Lambda(r)<1 2 r Λ ( r ) < 1 Theorem 1.5 they give a much larger gap, and they go past it.
Theorem 1.6 (SU(3) with plaquette blocks). Let G = SU ( 3 ) G=\operatorname{SU}(3) G = SU ( 3 ) and 0 < β W ≤ 19 25 = 0.76 0<\beta_{\mathrm W}\le\frac{19}{25}=0.76 0 < β W ≤ 25 19 = 0.76 Then α 3 ( β W ) < 1 \alpha_3(\beta_{\mathrm W})<1 α 3 ( β W ) < 1 for the number of (15) , with α 3 ( 0.72 ) ≤ 0.9182 \alpha_3(0.72)\le0.9182 α 3 ( 0.72 ) ≤ 0.9182 α 3 ( 3 4 ) ≤ 0.9653 \alpha_3(\frac{3}{4})\le0.9653 α 3 ( 4 3 ) ≤ 0.9653 and α 3 ( 0.76 ) ≤ 0.99950 \alpha_3(0.76)\le0.99950 α 3 ( 0.76 ) ≤ 0.99950 The conclusions of Theorem 1.1 hold with decay at every rate below − 1 2 log α 3 ( β W ) -\frac{1}{2}\log\alpha_3(\beta_{\mathrm W}) − 2 1 log α 3 ( β W ) and a transfer-matrix gap
m ≥ − 1 2 log α 3 ( β W ) > 0 ; m \ge-\frac{1}{2}\log\alpha_3(\beta_{\mathrm W}) > 0; m ≥ − 2 1 log α 3 ( β W ) > 0 ; Table 1. Explicit ranges of uniqueness, decay and a transfer-matrix gap for four-dimensional lattice Yang–Mills theory. The first three rows hold for the ungauged measure; the last three are the results of this paper.
Figure 1. Ranges of Table 1. The height of each bar is the largest inverse coupling β W \beta_W β W Theorem 1.2 and the orange bar, plaquette blocks, passes it (Theorem 1.3 ); for SU(3) the red bar is Theorem 1.5 and the orange bar, plaquette blocks, gives a larger gap there and passes it (Theorem 1.6 ).
at β W = 3 4 \beta_W = \frac{3}{4} β W = 4 3 m ≥ 0.0177 m \ge0.0177 m ≥ 0.0177 1.1 ⋅ 10 − 4 1.1 \cdot10^{-4} 1.1 ⋅ 1 0 − 4 Theorem 1.5 , and at β W = 0.76 \beta_W = 0.76 β W = 0.76 Theorem 1.5 no longer applies, m ≥ 2.5 ⋅ 10 − 4 m \ge2.5 \cdot10^{-4} m ≥ 2.5 ⋅ 1 0 − 4
All six theorems are statements at fixed lattice spacing. They do not touch the continuum limit, the crossover to Bałaban’s regime, or a gap in physical units, which together are the Clay problem; § 10 says where each method stops and what remains of the four open problems of [9 ] . The obstruction proved in [10 ] , that a lattice gap uniform in lattice units forces trivial continuum two-point functions, applies to all of them.
§ 2 constructs the line field. § 3 shows that a Dobrushin bound for the gauge-fixed specification gives the full conclusion of Theorem 1.1 . § 4 , § 5 , § 7 , § 8 and § 9 prove Theorems 1.1 –1.6 , and § 10 measures what is left. The appendices contain the residue table of the line field, the comparison lemma and its version for blocks, the blocks of the local Hessian inequality and the details of the certified computation.
The line field
Write e 0 , … , e 3 e_0,\ldots,e_3 e 0 , … , e 3 Z 4 \mathbb{Z}^4 Z 4 ( x , μ ) (x,\mu) ( x , μ ) x x x x + e μ x+e_\mu x + e μ direction field is a map σ : Z 4 → { 0 , 1 , 2 , 3 } \sigma:\mathbb{Z}^4\to\{0,1,2,3\} σ : Z 4 → { 0 , 1 , 2 , 3 } F = { ( x , σ ( x ) ) : x ∈ Z 4 } F=\{(x,\sigma(x)):x\in\mathbb{Z}^4\} F = {( x , σ ( x )) : x ∈ Z 4 } π x ( μ ) = σ ( x + e μ ) \pi_x(\mu)=\sigma(x+e_\mu) π x ( μ ) = σ ( x + e μ ) x x x { μ , ν } \{\mu,\nu\} { μ , ν } ( x , μ ) (x,\mu) ( x , μ ) ( x + e μ , ν ) (x+e_\mu,\nu) ( x + e μ , ν ) ( x + e ν , μ ) (x+e_\nu,\mu) ( x + e ν , μ ) ( x , ν ) (x,\nu) ( x , ν )
[ σ ( x ) ∈ { μ , ν } ] + [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] (1) [\sigma(x)\in\{\mu,\nu\}]+[\pi_x(\mu)=\nu]+[\pi_x(\nu)=\mu] \tag*{(1)} [ σ ( x ) ∈ { μ , ν }] + [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] (1) links of F F F
Lemma 2.1. Every plaquette of Z 4 \mathbb{Z}^4 Z 4 F F F x x x π x \pi_x π x σ ( x ) \sigma(x) σ ( x )
Proof. Fix x x x s = σ ( x ) s=\sigma(x) s = σ ( x ) R R R { s , ν } \{s,\nu\} { s , ν } (1) is 1 + [ π x ( s ) = ν ] + [ π x ( ν ) = s ] 1+[\pi_x(s)=\nu]+[\pi_x(\nu)=s] 1 + [ π x ( s ) = ν ] + [ π x ( ν ) = s ] ν ∈ R \nu\in R ν ∈ R π x ( s ) = s \pi_x(s)=s π x ( s ) = s π x ( ν ) ≠ s \pi_x(\nu)\ne s π x ( ν ) = s ν ∈ R \nu\in R ν ∈ R π x \pi_x π x s s s R R R R R R { μ , ν } ⊂ R \{\mu,\nu\}\subset R { μ , ν } ⊂ R [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] [\pi_x(\mu)=\nu]+[\pi_x(\nu)=\mu] [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] μ → π x ( μ ) \mu\to\pi_x(\mu) μ → π x ( μ ) μ ∈ R \mu\in R μ ∈ R π x ( μ ) ∈ R ∖ { μ } \pi_x(\mu)\in R\setminus\{\mu\} π x ( μ ) ∈ R ∖ { μ } R R R R R R
From now on σ \sigma σ Table 2 of Appendix A, read from x x x π x \pi_x π x π x \pi_x π x σ ( x ) \sigma(x) σ ( x ) F F F 4
Table 2. The line field: x x x x 0 x 1 x 2 x 3 x_0x_1x_2x_3 x 0 x 1 x 2 x 3 σ ( x ) \sigma(x) σ ( x ) σ ( x + e μ ) \sigma(x+e_\mu) σ ( x + e μ ) μ = 0 , 1 , 2 , 3 \mu=0,1,2,3 μ = 0 , 1 , 2 , 3
Lemma 2.2. (i) F F F F F F σ ( x ) \sigma(x) σ ( x )
(ii) For every box Λ \Lambda Λ F ∩ E ( Λ ) F\cap E(\Lambda) F ∩ E ( Λ )
(iii) A link e = ( x , μ ) ∉ F e=(x,\mu)\notin F e = ( x , μ ) ∈ / F F F F e e e { μ , σ ( x ) } \{\mu,\sigma(x)\} { μ , σ ( x )} { μ , π x ( μ ) } \{\mu,\pi_x(\mu)\} { μ , π x ( μ )} { μ , ρ } \{\mu,\rho\} { μ , ρ } F F F e e e
(iv) Every link outside F F F F F F
Proof. (i) Since π x \pi_x π x σ ( x ) \sigma(x) σ ( x ) σ ( x + e σ ( x ) ) = σ ( x ) \sigma(x+e_{\sigma(x)})=\sigma(x) σ ( x + e σ ( x ) ) = σ ( x ) σ \sigma σ x − e μ ≡ x + e μ x-e_\mu\equiv x+e_\mu x − e μ ≡ x + e μ σ ( x − e σ ( x ) ) = σ ( x ) \sigma(x-e_{\sigma(x)})=\sigma(x) σ ( x − e σ ( x ) ) = σ ( x ) ( x − e σ ( x ) , σ ( x ) ) (x-e_{\sigma(x)},\sigma(x)) ( x − e σ ( x ) , σ ( x )) F F F ( x − e ν , ν ) ∈ F (x-e_\nu,\nu)\in F ( x − e ν , ν ) ∈ F σ ( x − e ν ) = ν \sigma(x-e_\nu)=\nu σ ( x − e ν ) = ν x − e ν x-e_\nu x − e ν σ ( x ) = ν \sigma(x)=\nu σ ( x ) = ν
(ii) A line meets a box in a segment, and distinct lines are disjoint.
(iii) The plaquettes of e e e ν ≠ μ \nu\ne\mu ν = μ x x x x − e ν x-e_\nu x − e ν { μ , ν } \{\mu,\nu\} { μ , ν } F F F e e e F F F x x x x + e μ x+e_\mu x + e μ x x x s = σ ( x ) ≠ μ s=\sigma(x)\ne\mu s = σ ( x ) = μ x x x x − e s x-e_s x − e s { μ , s } \{\mu,s\} { μ , s } x + e μ x+e_\mu x + e μ t = π x ( μ ) t=\pi_x(\mu) t = π x ( μ ) R ∖ { s } R\setminus\{s\} R ∖ { s } π x \pi_x π x R R R x x x x − e t x-e_t x − e t { μ , t } \{\mu,t\} { μ , t } s ≠ t s\ne t s = t F F F { μ , ρ } \{\mu,\rho\} { μ , ρ } F F F μ \mu μ ρ \rho ρ e e e ρ \rho ρ e e e
Figure 2. (a) The line field in the slice x 3 = 0 x_3=0 x 3 = 0 F F F 0 , 1 , 2 0,1,2 0 , 1 , 2 3 3 3 Lemma 2.2 (i)), and the segments that leave the box are its boundary stubs. (b) One plaquette with its fixed link e 1 e_1 e 1 Lemma 6.2 for the three free links: 21 10 \frac{21}{10} 10 21 e 1 e_1 e 1 5 4 \frac{5}{4} 4 5 (7) once Y 1 = 0 Y_1=0 Y 1 = 0
(iv) Each of the six plaquettes contains e e e F F F
Gauge fixing and the free-boundary state
Let G ⊂ U ( N ) G\subset U(N) G ⊂ U ( N ) d U dU d U Λ \Lambda Λ
W Λ ( d Q ) = 1 Z Λ exp ( N β ∑ p ∈ P ( Λ ) Re Tr Q p ) ∏ e ∈ E ( Λ ) d Q e . W_\Lambda(dQ)=\frac{1}{Z_\Lambda}\exp\left(N\beta\sum_{p\in P(\Lambda)}\operatorname{Re}\operatorname{Tr}Q_p\right)\prod_{e\in E(\Lambda)}dQ_e. W Λ ( d Q ) = Z Λ 1 exp N β p ∈ P ( Λ ) ∑ Re Tr Q p e ∈ E ( Λ ) ∏ d Q e . The measure is invariant under the gauge transformations Q ( x , y ) ↦ g x Q ( x , y ) g y − 1 Q_{(x,y)}\mapsto g_xQ_{(x,y)}g_y^{-1} Q ( x , y ) ↦ g x Q ( x , y ) g y − 1 g ∈ G Λ g\in G^\Lambda g ∈ G Λ F F F A A A O O O O F O^F O F O O O F F F
Lemma 3.1 (Gauge fixing). Let μ A F \mu_A^F μ A F G A ∩ E ( Λ ) G^{A\cap E(\Lambda)} G A ∩ E ( Λ ) exp ( N β ∑ p ∈ P ( Λ ) Re Tr Q p F ) \exp\left(N\beta\sum_{p\in P(\Lambda)}\operatorname{Re}\operatorname{Tr}Q_p^F\right) exp ( N β ∑ p ∈ P ( Λ ) Re Tr Q p F ) O O O G E ( Λ ) G^{E(\Lambda)} G E ( Λ )
W Λ ( O ) = μ A F ( O F ) . W_\Lambda(O)=\mu_A^F(O^F). W Λ ( O ) = μ A F ( O F ) . Proof. Each segment of F ∩ E ( Λ ) F\cap E(\Lambda) F ∩ E ( Λ ) x 0 , x 1 , … , x k x_0,x_1,\ldots,x_k x 0 , x 1 , … , x k F F F g x 0 = 1 g_{x_0}=1 g x 0 = 1 g x i + 1 = g x i Q ( x i , x i + 1 ) g_{x_{i+1}}=g_{x_i}Q_{(x_i,x_{i+1})} g x i + 1 = g x i Q ( x i , x i + 1 ) Q e g = 1 Q_e^g=1 Q e g = 1 F ∩ E ( Λ ) F\cap E(\Lambda) F ∩ E ( Λ ) F F F Q A ↦ Q A g Q_A\mapsto Q_A^g Q A ↦ Q A g O O O
No leftover holonomy appears because F ∩ E ( Λ ) F \cap E(\Lambda) F ∩ E ( Λ ) μ Λ F \mu^{F}_{\Lambda} μ Λ F γ F \gamma^{F} γ F G A G^{A} G A e ∈ A e \in A e ∈ A γ e F \gamma^{F}_{e} γ e F Q e Q_e Q e exp ( N β ∑ p ∋ e Re Tr Q p F ) \exp(N\beta\sum_{p\ni e} \operatorname{Re} \operatorname{Tr} Q^{F}_{p}) exp ( N β ∑ p ∋ e Re Tr Q p F ) γ F \gamma^{F} γ F δ \delta δ G G G D D D SU ( 2 ) \operatorname{SU}(2) SU ( 2 ) S 3 S^{3} S 3 D = 2 D=2 D = 2 SU ( N ) \operatorname{SU}(N) SU ( N ) e ′ e' e ′ e e e
C e e ′ = sup W 1 ( γ e F ( ⋅ ∣ Q ) , γ e F ( ⋅ ∣ Q ′ ) ) δ ( Q e ′ , Q e ′ ′ ) . C_{ee'}=\sup\frac{W_{1}(\gamma^{F}_{e}(\cdot\mid Q),\gamma^{F}_{e}(\cdot\mid Q'))}{\delta(Q_{e'},Q'_{e'})}. C e e ′ = sup δ ( Q e ′ , Q e ′ ′ ) W 1 ( γ e F ( ⋅ ∣ Q ) , γ e F ( ⋅ ∣ Q ′ )) . the supremum over pairs of configurations that differ only at e ′ e' e ′ e e e e ′ e' e ′
Proposition 3.2. Suppose that every row and every column of ( C e e ′ ) (C_{ee'}) ( C e e ′ ) α < 1 \alpha<1 α < 1
(i) γ F \gamma^{F} γ F μ F \mu_{F} μ F
(ii) for every gauge-invariant local O O O ω ( O ) = lim W Λ ( O ) \omega(O)=\lim W_{\Lambda}(O) ω ( O ) = lim W Λ ( O ) Z 4 \mathbb{Z}^{4} Z 4 μ F ( O F ) \mu_{F}(O^{F}) μ F ( O F )
(iii) ω \omega ω
(iv) if β ≥ 0 \beta\geq0 β ≥ 0 ω \omega ω
(v) for gauge-invariant local O , O ′ O,O' O , O ′ O O O ∣ Cov ω ( O , O ′ ) ∣ ≤ C O , O ′ e − λ dist ( O , O ′ ) |\operatorname{Cov}_{\omega}(O,O')|\leq C_{O,O'}e^{-\lambda\operatorname{dist}(O,O')} ∣ Cov ω ( O , O ′ ) ∣ ≤ C O , O ′ e − λ dist ( O , O ′ ) λ < − log α \lambda<-\log\alpha λ < − log α
(vi) if β ≥ 0 \beta\geq0 β ≥ 0 ω \omega ω − log α -\log\alpha − log α
Proof. (i) Gibbs measures exist by compactness, and the comparison lemma of Appendix B with S = ∅ S=\varnothing S = ∅
(ii) Extend μ Λ F \mu^{F}_{\Lambda} μ Λ F Λ \Lambda Λ S S S P ( Λ ) P(\Lambda) P ( Λ ) E ( Λ ) E(\Lambda) E ( Λ ) e ∉ S e\notin S e ∈ / S e e e P ( Λ ) P(\Lambda) P ( Λ ) Q e Q_e Q e γ e F \gamma^{F}_{e} γ e F T T T O F O^{F} O F λ < − log α \lambda<-\log\alpha λ < − log α e − λ dist ( ⋅ , T ) e^{-\lambda\operatorname{dist}(\cdot,T)} e − λ dist ( ⋅ , T )
∣ μ Λ F ( O F ) − μ F ( O F ) ∣ ≤ L θ D 1 − θ ∑ z ∈ S e − λ dist ( z , T ) , \left|\mu^{F}_{\Lambda}(O^{F})-\mu_{F}(O^{F})\right|
\leq L\frac{\theta D}{1-\theta}\sum_{z\in S}e^{-\lambda\operatorname{dist}(z,T)}, μ Λ F ( O F ) − μ F ( O F ) ≤ L 1 − θ θ D z ∈ S ∑ e − λ dist ( z , T ) , with θ = α e λ < 1 \theta=\alpha e^{\lambda}<1 θ = α e λ < 1 L L L O F O^{F} O F C T R 3 e − λ R C_{T}R^{3}e^{-\lambda R} C T R 3 e − λ R R R R T T T Λ \Lambda Λ O O O Lemma 3.1 gives W Λ ( O ) = μ Λ F ( O F ) W_{\Lambda}(O)=\mu^{F}_{\Lambda}(O^{F}) W Λ ( O ) = μ Λ F ( O F )
(iii) W Λ ( τ v O ) = W Λ − v ( O ) W_{\Lambda}(\tau_{v}O)=W_{\Lambda-v}(O) W Λ ( τ v O ) = W Λ − v ( O ) Λ − v \Lambda-v Λ − v Z 4 \mathbb{Z}^{4} Z 4 W Λ W_{\Lambda} W Λ
(iv) For β ≥ 0 \beta\geq0 β ≥ 0 W Λ W_{\Lambda} W Λ [29 ] , and reflection positivity passes to the limit along such boxes, which is ω \omega ω
(v) Cov ω ( O , O ′ ) = Cov μ F ( O F , O ′ F ) \operatorname{Cov}_{\omega}(O,O')=\operatorname{Cov}_{\mu_{F}}(O^{F},O'^{F}) Cov ω ( O , O ′ ) = Cov μ F ( O F , O ′ F ) F F F
(vi) Let ϑ \vartheta ϑ τ t \tau_t τ t O O O ∣ ω ( ϑ ( O ‾ ) τ t O ) − ∣ ω ( O ) ∣ 2 ∣ ≤ C O e − λ t |\omega(\vartheta(\overline{O})\tau_tO)-|\omega(O)|^{2}|\leq C_Oe^{-\lambda t} ∣ ω ( ϑ ( O ) τ t O ) − ∣ ω ( O ) ∣ 2 ∣ ≤ C O e − λ t O O O L 2 ( ω ) L^{2}(\omega) L 2 ( ω ) G E G^{E} G E L 2 L^{2} L 2 [9 Prop. 23] the spectrum of the transfer matrix on the complement of the vacuum lies in [ 0 , e − λ ] [0,e^{-\lambda}] [ 0 , e − λ ] λ < − log α \lambda<-\log\alpha λ < − log α
The proposition replaces the torus limits and the uniqueness of the ungauged Gibbs measure used in [9 Thm. F] by the free-boundary limit, which has all the properties the reconstruction of the transfer matrix needs.5
SU(2) up to one third
Proof of Theorem 1.1 . Identify S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) e e e p ∋ e p \ni e p ∋ e s p s_p s p Re ( Q e s p ) = 1 2 Tr Q p \operatorname{Re}(Q_e s_p) = \frac{1}{2}\operatorname{Tr} Q_p Re ( Q e s p ) = 2 1 Tr Q p F F F N β Re Tr Q p = 4 β ⟨ Q e , s ˉ p ⟩ N\beta\operatorname{Re}\operatorname{Tr} Q_p = 4\beta\langle Q_e,\bar{s}_p\rangle N β Re Tr Q p = 4 β ⟨ Q e , s ˉ p ⟩ R 4 \mathbb{R}^4 R 4 Q e Q_e Q e ν b ( d x ) ∝ e ⟨ x , b ⟩ d x \nu_b(dx) \propto e^{\langle x,b\rangle}dx ν b ( d x ) ∝ e ⟨ x , b ⟩ d x S 3 S^3 S 3 b = 4 β ∑ p ∋ e s ˉ p b = 4\beta\sum_{p\ni e}\bar{s}_p b = 4 β ∑ p ∋ e s ˉ p e ′ e' e ′ δ \delta δ b b b 4 ∣ β ∣ δ 4|\beta|\delta 4∣ β ∣ δ [9 Thm. 7] , W 1 ( ν b , ν b ′ ) ≤ 1 4 ∣ b − b ′ ∣ W_1(\nu_b,\nu_{b'}) \le\frac{1}{4}|b-b'| W 1 ( ν b , ν b ′ ) ≤ 4 1 ∣ b − b ′ ∣ C e e ′ ≤ ∣ β ∣ C_{ee'} \le|\beta| C e e ′ ≤ ∣ β ∣ Lemma 2.2 (iv) every row and column has twelve nonzero entries, so α ≤ 12 ∣ β ∣ = 3 ∣ β W ∣ \alpha\le12|\beta| = 3|\beta_W| α ≤ 12∣ β ∣ = 3∣ β W ∣ ∣ β W ∣ < 1 3 |\beta_W| < \frac{1}{3} ∣ β W ∣ < 3 1
Proof of Theorem 1.2 . For the line field, every staple of an active link contains two active links, so the six staples can be chosen with sum zero while the staple through a given partner e ′ e' e ′ b = 0 b = 0 b = 0 ν b \nu_b ν b h h h 1 4 ∣ h ∣ \frac{1}{4}|h| 4 1 ∣ h ∣ [9 Lemma 6(i)] , so C e e ′ = ∣ β ∣ C_{ee'} = |\beta| C e e ′ = ∣ β ∣ 12 ∣ β ∣ = 3 ∣ β W ∣ 12|\beta| = 3|\beta_W| 12∣ β ∣ = 3∣ β W ∣
Now let F ′ F' F ′ C ′ C' C ′ w w w C ′ w ≤ α w C'w \le\alpha w C ′ w ≤ α w Λ n \Lambda_n Λ n n n n C n C_n C n E ( Λ n ) E(\Lambda_n) E ( Λ n ) C n w ≤ α w C_nw \le\alpha w C n w ≤ α w ρ ( C n ) ≤ α \rho(C_n) \le\alpha ρ ( C n ) ≤ α C n C_n C n ρ ( C n ) \rho(C_n) ρ ( C n ) 4 n 4 − O ( n 3 ) 4n^4 - O(n^3) 4 n 4 − O ( n 3 ) n 4 n^4 n 4 a a a 3 n 4 − O ( n 3 ) 3n^4 - O(n^3) 3 n 4 − O ( n 3 ) a p a_p a p p p p P = 6 n 4 − O ( n 3 ) P = 6n^4 - O(n^3) P = 6 n 4 − O ( n 3 ) ∑ p a p = 6 a − O ( n 3 ) \sum_p a_p = 6a - O(n^3) ∑ p a p = 6 a − O ( n 3 ) ∣ β ∣ ∑ p a p ( a p − 1 ) |\beta|\sum_p a_p(a_p - 1) ∣ β ∣ ∑ p a p ( a p − 1 )
∣ β ∣ a ∑ p a p ( a p − 1 ) ≥ ∣ β ∣ P a a ˉ ( a ˉ − 1 ) = 6 ∣ β ∣ ( a ˉ − 1 ) − o ( 1 ) , \frac{|\beta|}{a}\sum_p a_p(a_p-1) \ge\frac{|\beta|P}{a}\bar{a}(\bar{a}-1) = 6|\beta|(\bar{a}-1)-o(1), a ∣ β ∣ p ∑ a p ( a p − 1 ) ≥ a ∣ β ∣ P a ˉ ( a ˉ − 1 ) = 6∣ β ∣ ( a ˉ − 1 ) − o ( 1 ) , with a ˉ = ∑ p a p / P = 6 a / P − o ( 1 ) ≥ 3 − o ( 1 ) \bar{a} = \sum_p a_p/P = 6a/P-o(1) \ge3-o(1) a ˉ = ∑ p a p / P = 6 a / P − o ( 1 ) ≥ 3 − o ( 1 ) α ≥ 12 ∣ β ∣ − o ( 1 ) \alpha\ge12|\beta|-o(1) α ≥ 12∣ β ∣ − o ( 1 ) n n n α ≥ 12 ∣ β ∣ = 3 ∣ β W ∣ \alpha\ge12|\beta| = 3|\beta_W| α ≥ 12∣ β ∣ = 3∣ β W ∣
Without gauge fixing the same count gives eighteen partners and the ceiling β W = 2 9 \beta_W = \frac{2}{9} β W = 9 2 [9 Prop. 13] ; the temporal gauge, a family of lines in one direction, leaves fourteen. The line field is optimal among forests: twelve is the mean number of partners forced by the number of free links, and it is attained at every link.
SU(2) past one third: plaquette blocks
Throughout this section G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) ∣ x − y ∣ |x-y| ∣ x − y ∣ x ˉ \bar{x} x ˉ ⟨ x , y ⟩ = Re ( x y ˉ ) \langle x,y\rangle=\operatorname{Re}(x\bar{y}) ⟨ x , y ⟩ = Re ( x y ˉ ) R 4 \mathbb{R}^4 R 4 d x dx d x S 3 S^3 S 3 Re ( x y z ) = Re ( y z x ) = ⟨ y , z ˉ x ˉ ⟩ \operatorname{Re}(xyz)=\operatorname{Re}(yzx)=\langle y,\bar{z}\bar{x}\rangle Re ( x y z ) = Re ( y z x ) = ⟨ y , z ˉ x ˉ ⟩
Z 0 ( c ) = ∫ S 3 e ⟨ c , x ⟩ d x = ∑ j ≥ 0 ( ∣ c ∣ 2 / 4 ) j j ! ( j + 1 ) ! , c ∈ R 4 , Z_0(c)=\int_{S^3}e^{\langle c,x\rangle}\,dx=\sum_{j\ge0}\frac{(|c|^2/4)^j}{j!(j+1)!},\qquad c\in\mathbb{R}^4, Z 0 ( c ) = ∫ S 3 e ⟨ c , x ⟩ d x = j ≥ 0 ∑ j ! ( j + 1 )! ( ∣ c ∣ 2 /4 ) j , c ∈ R 4 , let ν c ( d x ) ∝ e ⟨ c , x ⟩ d x \nu_c(dx)\propto e^{\langle c,x\rangle}dx ν c ( d x ) ∝ e ⟨ c , x ⟩ d x g = log Z 0 g=\log Z_0 g = log Z 0 ∇ g ( c ) \nabla g(c) ∇ g ( c ) Hess g ( c ) \operatorname{Hess}g(c) Hess g ( c ) ν c \nu_c ν c
The blocks
A block plaquette is a plaquette of the set B \mathcal{B} B Table 3 of Appendix C, and its block is the set of its three active links.
Table 3. The sixteen block plaquettes modulo two: the base point x x x x 0 x 1 x 2 x 3 x_{0}x_{1}x_{2}x_{3} x 0 x 1 x 2 x 3
Lemma 5.1. The set B \mathcal{B} B 2 Z 4 2\mathbb{Z}^{4} 2 Z 4
(i) the blocks partition the active links;
(ii) every plaquette p ∉ B p \notin\mathcal{B} p ∈ / B
(iii) every active link lies in exactly five plaquettes outside B \mathcal{B} B B \mathcal{B} B
(iv) if an active link j j j B B B B \mathcal{B} B B B B 2 2 2 j j j
Proof. All four statements are invariant under 2 Z 4 2\mathbb{Z}^{4} 2 Z 4 Z 4 \mathbb{Z}^{4} Z 4 ( Z / P Z ) 4 (\mathbb{Z}/P\mathbb{Z})^{4} ( Z / P Z ) 4 P ≥ 6 P \ge6 P ≥ 6
Fix a block plaquette p p p e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 1 2 Tr Q p = Re ( y 1 y 2 y 3 ) \frac{1}{2}\operatorname{Tr} Q_p = \operatorname{Re}(y_1y_2y_3) 2 1 Tr Q p = Re ( y 1 y 2 y 3 ) y i y_i y i Q e i Q_{e_i} Q e i Q e i − 1 Q_{e_i}^{-1} Q e i − 1 e i e_i e i e i e_i e i Lemma 5.1 (ii) and contributes 4 β ⟨ y i , s ˉ ⟩ 4\beta\langle y_i,\bar{s}\rangle 4 β ⟨ y i , s ˉ ⟩ s s s y i y_i y i
μ b ( d y ) ∝ exp ( ∑ i = 1 3 ⟨ y i , b i ⟩ + 4 β Re ( y 1 y 2 y 3 ) ) d y 1 d y 2 d y 3 . (2) \mu_b(dy) \propto\exp\left(\sum_{i=1}^{3}\langle y_i,b_i\rangle+ 4\beta\operatorname{Re}(y_1y_2y_3)\right)dy_1\,dy_2\,dy_3. \tag*{(2)} μ b ( d y ) ∝ exp ( i = 1 ∑ 3 ⟨ y i , b i ⟩ + 4 β Re ( y 1 y 2 y 3 ) ) d y 1 d y 2 d y 3 . (2) with b i b_i b i 4 β s ˉ 4\beta\bar{s} 4 β s ˉ ∣ b i ∣ ≤ 20 ∣ β ∣ |b_i| \le20|\beta| ∣ b i ∣ ≤ 20∣ β ∣ S 3 S^3 S 3 y y y Q Q Q ( S 3 ) 3 (S^3)^3 ( S 3 ) 3 d ( y , y ′ ) = ∑ i ∣ y i − y i ′ ∣ d(y,y')=\sum_i|y_i-y_i'| d ( y , y ′ ) = ∑ i ∣ y i − y i ′ ∣
The sensitivity of one block
Let λ ∗ \lambda_* λ ∗ τ \tau τ g ( c ) g(c) g ( c ) ∣ c ∣ ≤ 24 ∣ β ∣ |c| \le24|\beta| ∣ c ∣ ≤ 24∣ β ∣ M M M D 3 g ( c ) D^3g(c) D 3 g ( c )
w = 8 β 2 ( τ − λ ∗ ) , s = 1 4 + ( e w − 1 ) e w τ , w=8\beta^2(\tau-\lambda_*),\qquad s=\frac{1}{4}+(e^w-1)e^w\sqrt{\tau}, w = 8 β 2 ( τ − λ ∗ ) , s = 4 1 + ( e w − 1 ) e w τ , C = 20 ∣ β ∣ ( 4 ∣ β ∣ τ s + 16 β 2 M ) , K = ( 1 + ∣ β ∣ ) s 1 − C , (3) C=20|\beta|(4|\beta|\tau s+16\beta^2M),\qquad K=\frac{(1+|\beta|)s}{1-C}, \tag*{(3)} C = 20∣ β ∣ ( 4∣ β ∣ τ s + 16 β 2 M ) , K = 1 − C ( 1 + ∣ β ∣ ) s , (3) assuming C < 1 C<1 C < 1
Lemma 5.2. Let ν \nu ν λ = e R ν / ν ( e R ) \lambda=e^R\nu/\nu(e^R) λ = e R ν / ν ( e R ) osc R ≤ ω \operatorname{osc} R\le\omega osc R ≤ ω f , h f,h f , h
Cov λ ( f , h ) ≤ Cov ν ( f , h ) + ( e ω − 1 ) e ω sd ν ( f ) sd ν ( h ) . \operatorname{Cov}_{\lambda}(f,h)\le\operatorname{Cov}_{\nu}(f,h)+(e^\omega-1)e^\omega\operatorname{sd}_{\nu}(f)\operatorname{sd}_{\nu}(h). Cov λ ( f , h ) ≤ Cov ν ( f , h ) + ( e ω − 1 ) e ω sd ν ( f ) sd ν ( h ) . Proof. We may assume 0 ≤ R ≤ ω 0\le R\le\omega 0 ≤ R ≤ ω ρ = e R / ν ( e R ) \rho=e^R/\nu(e^R) ρ = e R / ν ( e R ) [ e − ω , e ω ] [e^{-\omega},e^\omega] [ e − ω , e ω ] δ = ∥ ρ − 1 ∥ ∞ ≤ e ω − 1 \delta=\|\rho-1\|_\infty\le e^\omega-1 δ = ∥ ρ − 1 ∥ ∞ ≤ e ω − 1 f ~ = f − ν f \tilde{f}=f-\nu f f ~ = f − ν f h ~ = h − ν h \tilde{h}=h-\nu h h ~ = h − ν h
Cov λ ( f , h ) − Cov ν ( f , h ) = ν ( ( ρ − 1 ) f ~ h ~ ) \operatorname{Cov}_{\lambda}(f,h)-\operatorname{Cov}_{\nu}(f,h)=\nu((\rho-1)\tilde{f}\tilde{h}) Cov λ ( f , h ) − Cov ν ( f , h ) = ν (( ρ − 1 ) f ~ h ~ ) − ν ( ( ρ − 1 ) f ~ ) ν ( ( ρ − 1 ) h ~ ) , \qquad-\nu((\rho-1)\tilde{f})\nu((\rho-1)\tilde{h}), − ν (( ρ − 1 ) f ~ ) ν (( ρ − 1 ) h ~ ) , and the right side is at most ( δ + δ 2 ) sd ν ( f ) sd ν ( h ) (\delta+\delta^2)\operatorname{sd}_{\nu}(f)\operatorname{sd}_{\nu}(h) ( δ + δ 2 ) sd ν ( f ) sd ν ( h )
Lemma 5.3. Let β ≥ 0 \beta\ge0 β ≥ 0 a , c 0 ∈ R 4 a,c_0\in\mathbb{R}^{4} a , c 0 ∈ R 4 ∣ a ∣ , ∣ c 0 ∣ ≤ 20 β |a|,|c_0|\le20\beta ∣ a ∣ , ∣ c 0 ∣ ≤ 20 β
κ ( d x ) ∝ exp ( ⟨ x , a ⟩ + g ( c 0 + 4 β x ˉ ) ) d x on S 3 . \kappa(dx)\propto\exp\left(\langle x,a\rangle+g(c_0+4\beta\bar{x})\right)dx\quad\text{on }S^3. κ ( d x ) ∝ exp ( ⟨ x , a ⟩ + g ( c 0 + 4 β x ˉ ) ) d x on S 3 . For every 1 1 1 f f f h ∈ R 4 h\in\mathbb{R}^{4} h ∈ R 4
(i) Cov κ ( f , ⟨ x , h ⟩ ) ≤ s ∣ h ∣ \operatorname{Cov}_{\kappa}(f,\langle x,h\rangle)\le s|h| Cov κ ( f , ⟨ x , h ⟩) ≤ s ∣ h ∣
(ii) if c 0 c_0 c 0 k k k ∣ d d t κ ( f ) ∣ ≤ ( 4 β τ s + 16 β 2 M ) ∣ k ∣ \left|\frac{d}{dt}\kappa(f)\right|\le(4\beta\tau s+16\beta^2M)|k| d t d κ ( f ) ≤ ( 4 β τ s + 16 β 2 M ) ∣ k ∣
Proof. (i) Taylor’s formula at c 0 c_0 c 0 g ( c 0 + 4 β u ) = g ( c 0 ) + 4 β ⟨ ∇ g ( c 0 ) , u ⟩ + R ( u ) g(c_0+4\beta u)=g(c_0)+4\beta\langle\nabla g(c_0),u\rangle+R(u) g ( c 0 + 4 β u ) = g ( c 0 ) + 4 β ⟨ ∇ g ( c 0 ) , u ⟩ + R ( u ) R ( u ) = 8 β 2 Hess g ( ξ ) ( u , u ) R(u)=8\beta^2\operatorname{Hess}g(\xi)(u,u) R ( u ) = 8 β 2 Hess g ( ξ ) ( u , u ) ξ \xi ξ ∣ ξ ∣ ≤ 24 β |\xi|\le24\beta ∣ ξ ∣ ≤ 24 β u = x ˉ u=\bar{x} u = x ˉ R ( x ˉ ) ∈ [ 8 β 2 λ ∗ , 8 β 2 τ ] R(\bar{x})\in[8\beta^2\lambda_*,8\beta^2\tau] R ( x ˉ ) ∈ [ 8 β 2 λ ∗ , 8 β 2 τ ] osc R ≤ w \operatorname{osc}R\le w osc R ≤ w ⟨ ∇ g ( c 0 ) , x ˉ ⟩ = ⟨ x , ∇ g ( c 0 ) ⟩ \langle\nabla g(c_0),\bar{x}\rangle=\langle x,\nabla g(c_0)\rangle ⟨ ∇ g ( c 0 ) , x ˉ ⟩ = ⟨ x , ∇ g ( c 0 )⟩ κ \kappa κ e R ν / ν ( e R ) e^{R\nu}/\nu(e^R) e R ν / ν ( e R ) ν = ν c \nu=\nu_c ν = ν c c = a + 4 β ∇ g ( c 0 ) c=a+4\beta\nabla g(c_0) c = a + 4 β ∇ g ( c 0 ) [9 Thm. 7] gives
Cov ν ( f , ⟨ x , h ⟩ ) = d d t ν c + t h ( f ) ≤ 1 4 ∣ h ∣ . \operatorname{Cov}_{\nu}(f,\langle x,h\rangle)=\frac{d}{dt}\nu_{c+th}(f)\le\frac{1}{4}|h|. Cov ν ( f , ⟨ x , h ⟩) = d t d ν c + t h ( f ) ≤ 4 1 ∣ h ∣. A 1-Lipschitz function on a set of diameter two has sd ≤ 1 \operatorname{sd}\le1 sd ≤ 1 sd ν ⟨ x , h ⟩ ≤ τ ∣ h ∣ \operatorname{sd}_{\nu}\langle x,h\rangle\le\sqrt{\tau}|h| sd ν ⟨ x , h ⟩ ≤ τ ∣ h ∣ ∣ c ∣ ≤ ∣ a ∣ + 4 β ≤ 24 β |c|\le|a|+4\beta\le24\beta ∣ c ∣ ≤ ∣ a ∣ + 4 β ≤ 24 β
(ii) The derivative is Cov κ ( f , ⟨ ∇ g ( c 0 + 4 β x ˉ ) , k ⟩ ) \operatorname{Cov}_{\kappa}(f,\langle\nabla g(c_0+4\beta\bar{x}),k\rangle) Cov κ ( f , ⟨ ∇ g ( c 0 + 4 β x ˉ ) , k ⟩) ∇ g ( c 0 + 4 β u ) = ∇ g ( c 0 ) + 4 β Hess g ( c 0 ) u + N ( u ) \nabla g(c_0+4\beta u)=\nabla g(c_0)+4\beta\operatorname{Hess}g(c_0)u+N(u) ∇ g ( c 0 + 4 β u ) = ∇ g ( c 0 ) + 4 β Hess g ( c 0 ) u + N ( u ) ⟨ x , h ⟩ \langle x,h\rangle ⟨ x , h ⟩ h = 4 β Hess g ( c 0 ) k h=4\beta\operatorname{Hess}g(c_0)k h = 4 β Hess g ( c 0 ) k ∣ h ∣ ≤ 4 β τ ∣ k ∣ |h|\le4\beta\tau|k| ∣ h ∣ ≤ 4 β τ ∣ k ∣ 4 β τ s ∣ k ∣ 4\beta\tau s|k| 4 β τ s ∣ k ∣ ∣ u ∣ , ∣ u ′ ∣ ≤ 1 |u|,|u'|\le1 ∣ u ∣ , ∣ u ′ ∣ ≤ 1
∥ Hess g ( c 0 + 4 β v ) − Hess g ( c 0 ) ∥ ≤ 4 β M ∣ v ∣ ≤ 4 β M \left\|\operatorname{Hess}g(c_0+4\beta v)-\operatorname{Hess}g(c_0)\right\|\le4\beta M|v|\le4\beta M ∥ Hess g ( c 0 + 4 β v ) − Hess g ( c 0 ) ∥ ≤ 4 β M ∣ v ∣ ≤ 4 β M on the segment between them, so ∣ N ( u ) − N ( u ′ ) ∣ ≤ 16 β 2 M ∣ u − u ′ ∣ |N(u)-N(u')|\le16\beta^2M|u-u'| ∣ N ( u ) − N ( u ′ ) ∣ ≤ 16 β 2 M ∣ u − u ′ ∣ ⟨ N ( x ˉ ) , k ⟩ \langle N(\bar{x}),k\rangle ⟨ N ( x ˉ ) , k ⟩ 16 β 2 M ∣ k ∣ 16\beta^2M|k| 16 β 2 M ∣ k ∣ f f f 16 β 2 M ∣ k ∣ 16\beta^2M|k| 16 β 2 M ∣ k ∣
Lemma 5.4. Let C < 1 C<1 C < 1 b b b b ′ b' b ′ ∣ b i ∣ , ∣ b i ′ ∣ ≤ 20 ∣ β ∣ |b_i|,|b_i'|\le20|\beta| ∣ b i ∣ , ∣ b i ′ ∣ ≤ 20∣ β ∣ b i b_i b i
W 1 ( μ b , μ b ′ ) ≤ K ∣ b i − b i ′ ∣ W_1(\mu_b,\mu_{b'})\le K|b_i-b_i'| W 1 ( μ b , μ b ′ ) ≤ K ∣ b i − b i ′ ∣ in the metric d d d
Proof. The map y 1 ↦ − y 1 y_1\mapsto-y_1 y 1 ↦ − y 1 μ b \mu_b μ b β \beta β − β -\beta − β b 1 b_1 b 1 − b 1 -b_1 − b 1 β ≥ 0 \beta\ge0 β ≥ 0 Re ( y 1 y 2 y 3 ) \operatorname{Re}(y_1y_2y_3) Re ( y 1 y 2 y 3 ) i = 1 i=1 i = 1 f f f d d d ε = ∣ b 1 − b 1 ′ ∣ \varepsilon=|b_1-b_1'| ε = ∣ b 1 − b 1 ′ ∣
The middle link. Given y 1 , y 3 y_1,y_3 y 1 , y 3 y 2 y_2 y 2 ν c \nu_c ν c c = b 2 + 4 β y 3 y 1 ‾ c=b_2+4\beta\overline{y_3y_1} c = b 2 + 4 β y 3 y 1 b 1 b_1 b 1 y 1 y_1 y 1 η \eta η c c c 4 β η 4\beta\eta 4 β η ν c \nu_c ν c β η \beta\eta β η W 1 W_1 W 1 F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] F(y_1,y_3)=\mathbb{E}[f\mid y_1,y_3] F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] ( 1 + β ) (1+\beta) ( 1 + β ) μ b ( f ) = μ b 13 ( F ) \mu_b(f)=\mu_b^{13}(F) μ b ( f ) = μ b 13 ( F ) μ b 13 \mu_b^{13} μ b 13 ( y 1 , y 3 ) (y_1,y_3) ( y 1 , y 3 )
μ b 13 ∝ e ⟨ y 1 , b 1 ⟩ + ⟨ y 3 , b 3 ⟩ Z 0 ( b 2 + 4 β y 3 y 1 ‾ ) . \mu_b^{13}\propto e^{\langle y_1,b_1\rangle+\langle y_3,b_3\rangle}Z_0(b_2+4\beta\overline{y_3y_1}). μ b 13 ∝ e ⟨ y 1 , b 1 ⟩ + ⟨ y 3 , b 3 ⟩ Z 0 ( b 2 + 4 β y 3 y 1 ) . Since ∣ b 2 + 4 β y 3 y 1 ‾ ∣ = ∣ b 2 y 3 + 4 β y 1 ‾ ∣ = ∣ y 1 b 2 + 4 β y 3 ‾ ∣ |b_2+4\beta\overline{y_3y_1}|=|b_2y_3+4\beta\overline{y_1}|=|y_1b_2+4\beta\overline{y_3}| ∣ b 2 + 4 β y 3 y 1 ∣ = ∣ b 2 y 3 + 4 β y 1 ∣ = ∣ y 1 b 2 + 4 β y 3 ∣
κ 1 ( d y 1 ∣ y 3 ) ∝ e ⟨ y 1 , b 1 ⟩ + g ( b 2 y 3 + 4 β y 1 ‾ ) d y 1 , \kappa_1(dy_1\mid y_3)\propto e^{\langle y_1,b_1\rangle+g(b_2y_3+4\beta\overline{y_1})}\,dy_1, κ 1 ( d y 1 ∣ y 3 ) ∝ e ⟨ y 1 , b 1 ⟩ + g ( b 2 y 3 + 4 β y 1 ) d y 1 , κ 3 ( d y 3 ∣ y 1 ) ∝ e ⟨ y 3 , b 3 ⟩ + g ( y 1 b 2 + 4 β y 3 ‾ ) d y 3 , \kappa_3(dy_3\mid y_1)\propto e^{\langle y_3,b_3\rangle+g(y_1b_2+4\beta\overline{y_3})}\,dy_3, κ 3 ( d y 3 ∣ y 1 ) ∝ e ⟨ y 3 , b 3 ⟩ + g ( y 1 b 2 + 4 β y 3 ) d y 3 , both of the form of Lemma 5.3, with ∣ c 0 ∣ = ∣ b 2 ∣ |c_0|=|b_2| ∣ c 0 ∣ = ∣ b 2 ∣
Two sites. Integrating Lemma 5.3(i) along the segment from b 1 b_1 b 1 b 1 ′ b_1' b 1 ′ W 1 ( κ 1 , κ 1 ′ ) ≤ s ε W_1(\kappa_1,\kappa_1')\le s\varepsilon W 1 ( κ 1 , κ 1 ′ ) ≤ s ε y 3 y_3 y 3 y 3 y_3 y 3 y 3 ′ y_3' y 3 ′ c 0 = b 2 y 3 c_0=b_2y_3 c 0 = b 2 y 3 ∣ b 2 ∣ ∣ y 3 − y 3 ′ ∣ ≤ 20 β ∣ y 3 − y 3 ′ ∣ |b_2||y_3-y_3'|\le20\beta|y_3-y_3'| ∣ b 2 ∣∣ y 3 − y 3 ′ ∣ ≤ 20 β ∣ y 3 − y 3 ′ ∣ ∣ c 0 ∣ ≤ 20 β |c_0|\le20\beta ∣ c 0 ∣ ≤ 20 β W 1 ( κ 1 ( ⋅ ∣ y 3 ) , κ 1 ′ ( ⋅ ∣ y 3 ′ ) ) ≤ C ∣ y 3 − y 3 ′ ∣ W_1(\kappa_1(\cdot\mid y_3),\kappa_1'(\cdot\mid y_3'))\le C|y_3-y_3'| W 1 ( κ 1 ( ⋅ ∣ y 3 ) , κ 1 ′ ( ⋅ ∣ y 3 ′ )) ≤ C ∣ y 3 − y 3 ′ ∣ κ 3 \kappa_3 κ 3 b 1 b_1 b 1 π \pi π μ b 13 \mu_b^{13} μ b 13 μ b ′ 13 \mu_{b'}^{13} μ b ′ 13 d 1 + d 3 d_1+d_3 d 1 + d 3 d k = E π ∣ y k − y k ′ ∣ d_k=\mathbb{E}_{\pi}|y_k-y_k'| d k = E π ∣ y k − y k ′ ∣ ( y 1 , y 1 ′ ) (y_1,y_1') ( y 1 , y 1 ′ ) κ 1 ( ⋅ ∣ y 3 ) \kappa_1(\cdot\mid y_3) κ 1 ( ⋅ ∣ y 3 ) κ 1 ′ ( ⋅ ∣ y 3 ′ ) \kappa_1'(\cdot\mid y_3') κ 1 ′ ( ⋅ ∣ y 3 ′ ) d 1 ≤ s ε + C d 3 d_1\le s\varepsilon+Cd_3 d 1 ≤ s ε + C d 3 d 3 ≤ C d 1 d_3\le Cd_1 d 3 ≤ C d 1 d 1 ≤ s ε / ( 1 − C 2 ) d_1\le s\varepsilon/(1-C^2) d 1 ≤ s ε / ( 1 − C 2 )
∣ μ b ( f ) − μ b ′ ( f ) ∣ ≤ ( 1 + β ) ( d 1 + d 3 ) ≤ ( 1 + β ) s 1 − C ε = K ε . |\mu_b(f)-\mu_{b'}(f)|\le(1+\beta)(d_1+d_3)\le\frac{(1+\beta)s}{1-C}\varepsilon=K\varepsilon. ∣ μ b ( f ) − μ b ′ ( f ) ∣ ≤ ( 1 + β ) ( d 1 + d 3 ) ≤ 1 − C ( 1 + β ) s ε = K ε . The constants.
Lemma 5.5. Write g ( c ) = φ ( ∣ c ∣ 2 ) g(c)=\varphi(|c|^2) g ( c ) = φ ( ∣ c ∣ 2 ) r = ∣ c ∣ r=|c| r = ∣ c ∣ c ≠ 0 c\ne0 c = 0 Hess g ( c ) \operatorname{Hess}g(c) Hess g ( c ) 2 φ ′ ( r 2 ) 2\varphi'(r^2) 2 φ ′ ( r 2 ) 2 φ ′ ( r 2 ) + 4 r 2 φ ′ ′ ( r 2 ) 2\varphi'(r^2)+4r^2\varphi''(r^2) 2 φ ′ ( r 2 ) + 4 r 2 φ ′′ ( r 2 ) D 3 g ( c ) D^3g(c) D 3 g ( c )
max ( ∣ A + Γ ∣ , 2 3 ∣ A ∣ ) , A = 12 r φ ′ ′ ( r 2 ) , Γ = 8 r 3 φ ′ ′ ′ ( r 2 ) . \max\left(|A+\Gamma|,\frac{2}{3}|A|\right),\qquad A=12r\varphi''(r^2),\qquad\Gamma=8r^3\varphi'''(r^2). max ( ∣ A + Γ∣ , 3 2 ∣ A ∣ ) , A = 12 r φ ′′ ( r 2 ) , Γ = 8 r 3 φ ′′′ ( r 2 ) . On the ball r ≤ 2.07 r\le2.07 r ≤ 2.07 Hess g \operatorname{Hess}g Hess g 0.158491 0.158491 0.158491 0.25000001 0.25000001 0.25000001 D 3 g D^3g D 3 g 0.0615586 0.0615586 0.0615586
Proof. Differentiating φ ( ∣ c ∣ 2 ) \varphi(|c|^2) φ ( ∣ c ∣ 2 ) D 2 g ( v , v ) = 2 φ ′ ∣ v ∣ 2 + 4 φ ′ ′ ⟨ c , v ⟩ 2 D^2g(v,v)=2\varphi'|v|^2+4\varphi''\langle c,v\rangle^2 D 2 g ( v , v ) = 2 φ ′ ∣ v ∣ 2 + 4 φ ′′ ⟨ c , v ⟩ 2 D 3 g ( v , v , v ) = 12 φ ′ ′ ⟨ c , v ⟩ ∣ v ∣ 2 + 8 φ ′ ′ ′ ⟨ c , v ⟩ 3 D^3g(v,v,v)=12\varphi''\langle c,v\rangle|v|^2+8\varphi'''\langle c,v\rangle^3 D 3 g ( v , v , v ) = 12 φ ′′ ⟨ c , v ⟩ ∣ v ∣ 2 + 8 φ ′′′ ⟨ c , v ⟩ 3 ∣ v ∣ = 1 |v|=1 ∣ v ∣ = 1 t = ⟨ c / r , v ⟩ ∈ [ − 1 , 1 ] t=\langle c/r,v\rangle\in[-1,1] t = ⟨ c / r , v ⟩ ∈ [ − 1 , 1 ] A t + Γ t 3 At+\Gamma t^3 A t + Γ t 3 t = ± 1 t=\pm1 t = ± 1 2 3 ∣ A ∣ ∣ t ∣ ≤ 2 3 ∣ A ∣ \frac{2}{3}|A||t|\le\frac{2}{3}|A| 3 2 ∣ A ∣∣ t ∣ ≤ 3 2 ∣ A ∣ [6 ] the supremum of a symmetric trilinear form on the diagonal of the unit sphere is its norm. The numerical bounds are obtained in ball arithmetic [23 ] from the series of Z 0 Z_0 Z 0 2 ⋅ 10 4 2\cdot10^4 2 ⋅ 1 0 4 [ 0 , 2.07 ] [0,2.07] [ 0 , 2.07 ]
The proof of Theorem 1.3.
Let γ B \gamma^B γ B B B B γ B B \gamma^B_B γ B B B B B (2) . It comes from the same interaction as γ F \gamma^F γ F
C B j = sup W 1 ( γ B B ( ⋅ ∣ Q ) , γ B B ( ⋅ ∣ Q ′ ) ) ∣ Q j − Q j ′ ∣ C_{Bj}=\sup\frac{W_1(\gamma^B_B(\mathord\cdot\mid Q),\gamma^B_B(\mathord\cdot\mid Q'))}{|Q_j-Q'_j|} C B j = sup ∣ Q j − Q j ′ ∣ W 1 ( γ B B ( ⋅ ∣ Q ) , γ B B ( ⋅ ∣ Q ′ )) for an active link j ∉ B j\notin B j ∈ / B j j j W 1 W_1 W 1 d d d
Proposition 5.6. Suppose that ∑ B C B j ≤ α < 1 \sum_B C_{Bj}\le\alpha<1 ∑ B C B j ≤ α < 1 j j j − 1 2 log α -\frac{1}{2}\log\alpha − 2 1 log α − log α -\log\alpha − log α
Proof. The proof of Proposition 3.2 goes through with Lemma B.3 of Appendix B in place of Lemma B.1, and with the set S S S H H H
Proof of Theorem 1.3. Let ∣ β ∣ ≤ 0.345 / 4 |\beta|\le0.345/4 ∣ β ∣ ≤ 0.345/4 r ≤ 2.07 r\le2.07 r ≤ 2.07 ∣ c ∣ ≤ 24 ∣ β ∣ |c|\le24|\beta| ∣ c ∣ ≤ 24∣ β ∣ (3) is increasing in ∣ β ∣ |\beta| ∣ β ∣
α ( β W ) = 40 ∣ β ∣ K . (4) \alpha(\beta_W)=40|\beta|K. \tag*{(4)} α ( β W ) = 40∣ β ∣ K . (4) Changing an active link j j j η \eta η b i b_i b i i i i 4 ∣ β ∣ η 4|\beta|\eta 4∣ β ∣ η ∑ B C B j ≤ 40 ∣ β ∣ K \sum_B C_{Bj}\le40|\beta|K ∑ B C B j ≤ 40∣ β ∣ K (3) in ball arithmetic at β W = 0.345 \beta_W=0.345 β W = 0.345 w ≤ 0.0054459 w\le0.0054459 w ≤ 0.0054459 s ≤ 0.252746 s\le0.252746 s ≤ 0.252746 C ≤ 0.050243 C\le0.050243 C ≤ 0.050243 K ≤ 0.289069 K\le0.289069 K ≤ 0.289069 α ≤ 0.99729 \alpha\le0.99729 α ≤ 0.99729 β W = 1 3 \beta_W=\frac{1}{3} β W = 3 1 α ≤ 0.95602 \alpha\le0.95602 α ≤ 0.95602
Remark 5.7. Most of the range is kept. A second bound for the same sensitivity, which keeps the law of the middle link instead of linearising it, evaluated in floating point on a quadrature grid and maximised over the fields b b b 0.2783 0.2783 0.2783 β W = 0.34 \beta_W=0.34 β W = 0.34 0.2881 0.2881 0.2881 40 ∣ β ∣ 40|\beta| 40∣ β ∣ β W = 0.357 \beta_W=0.357 β W = 0.357
Curvature in the gauge
In this section G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) S O ( N ) \mathrm{SO}(N) SO ( N ) U ( N ) \mathrm{U}(N) U ( N ) ⟨ X , Y ⟩ = − Tr ( X Y ) \langle X,Y\rangle=-\operatorname{Tr}(XY) ⟨ X , Y ⟩ = − Tr ( X Y ) Ric S U ( N ) = N 2 \operatorname{Ric}_{\mathrm{SU}(N)}=\frac{N}{2} Ric SU ( N ) = 2 N Ric S O ( N ) = N − 2 4 \operatorname{Ric}_{\mathrm{SO}(N)}=\frac{N-2}{4} Ric SO ( N ) = 4 N − 2 Q Q Q v = ( X e Q e ) v=(X_eQ_e) v = ( X e Q e ) ∣ v ∣ 2 = ∑ e ∣ X e ∣ 2 |v|^2=\sum_e|X_e|^2 ∣ v ∣ 2 = ∑ e ∣ X e ∣ 2 x x x μ < ν \mu<\nu μ < ν e 1 = ( x , μ ) e_1=(x,\mu) e 1 = ( x , μ ) e 2 = ( x + e μ , ν ) e_2=(x+e_\mu,\nu) e 2 = ( x + e μ , ν ) e 3 = ( x + e ν , μ ) e_3=(x+e_\nu,\mu) e 3 = ( x + e ν , μ ) e 4 = ( x , ν ) e_4=(x,\nu) e 4 = ( x , ν ) x x x
Y 1 = X e 1 , Y 2 = Q e 1 X e 2 Q e 1 ∗ , Y 3 = Q e 4 X e 3 Q e 4 ∗ , Y 4 = X e 4 , (5) Y_1=X_{e_1},\quad Y_2=Q_{e_1}X_{e_2}Q_{e_1}^*,\quad Y_3=Q_{e_4}X_{e_3}Q_{e_4}^*,\quad Y_4=X_{e_4}, \tag*{(5)} Y 1 = X e 1 , Y 2 = Q e 1 X e 2 Q e 1 ∗ , Y 3 = Q e 4 X e 3 Q e 4 ∗ , Y 4 = X e 4 , (5) and write W = Q p W=Q_p W = Q p Re Tr Q p \operatorname{Re}\operatorname{Tr} Q_p Re Tr Q p Q e ( t ) = e t X e Q e Q_e(t)=e^{tX_e}Q_e Q e ( t ) = e t X e Q e [9 Eq. (5)]
f p = ∑ i = 1 4 Re Tr ( Y i 2 W ) + 2 Re Tr ( Y 1 Y 2 W ) + 2 Re Tr ( Y 3 Y 4 W ) − 2 ∑ i ∈ { 1 , 2 } , j ∈ { 3 , 4 } Re Tr ( Y i W Y j ) , (6) f_p=\sum_{i=1}^{4}\operatorname{Re}\operatorname{Tr}(Y_i^2W)+2\operatorname{Re}\operatorname{Tr}(Y_1Y_2W)+2\operatorname{Re}\operatorname{Tr}(Y_3Y_4W)-2\sum_{\substack{i\in\{1,2\},\ j\in\{3,4\}}}\operatorname{Re}\operatorname{Tr}(Y_iWY_j), \tag*{(6)} f p = i = 1 ∑ 4 Re Tr ( Y i 2 W ) + 2 Re Tr ( Y 1 Y 2 W ) + 2 Re Tr ( Y 3 Y 4 W ) − 2 i ∈ { 1 , 2 } , j ∈ { 3 , 4 } ∑ Re Tr ( Y i W Y j ) , (6) so that Hess S 0 ( v , v ) = ∑ p f p \operatorname{Hess} S_0(v,v)=\sum_p f_p Hess S 0 ( v , v ) = ∑ p f p p p p
c p = 2 ⟨ Y 1 , Y 4 ⟩ − 2 ⟨ Y 1 , Y 2 ⟩ + 2 ⟨ Y 2 , W Y 3 W ∗ ⟩ − 2 ⟨ Y 3 , Y 4 ⟩ . (7) c_p=2\langle Y_1,Y_4\rangle-2\langle Y_1,Y_2\rangle+2\langle Y_2,WY_3W^*\rangle-2\langle Y_3,Y_4\rangle. \tag*{(7)} c p = 2 ⟨ Y 1 , Y 4 ⟩ − 2 ⟨ Y 1 , Y 2 ⟩ + 2 ⟨ Y 2 , W Y 3 W ∗ ⟩ − 2 ⟨ Y 3 , Y 4 ⟩ . (7) It collects the inner products of the transported vectors of two links that meet at a right angle at a corner of p p p
Lemma 6.1. On Z 4 \mathbb{Z}^4 Z 4 free boundary , for every field Q Q Q X X X
∑ y [ ∣ D y X ∣ 2 + ∑ μ ∣ X y , μ + Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ ∣ 2 ] = 4 ∣ X ∣ 2 + ∑ p c p , \sum_y\left[|D_yX|^2+\sum_\mu\left|X_{y,\mu}+Q^*_{y-e_\mu,\mu}X_{y-e_\mu,\mu}Q_{y-e_\mu,\mu}\right|^2\right]=4|X|^2+\sum_p c_p, y ∑ [ ∣ D y X ∣ 2 + μ ∑ X y , μ + Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ 2 ] = 4∣ X ∣ 2 + p ∑ c p , where D y X = ∑ μ [ X y , μ − Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ ] D_yX=\sum_\mu[X_{y,\mu}-Q^*_{y-e_\mu,\mu}X_{y-e_\mu,\mu}Q_{y-e_\mu,\mu}] D y X = ∑ μ [ X y , μ − Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ ]
Proof. At a site y y y o μ o_\mu o μ μ \mu μ i μ i_\mu i μ D y X = ∑ μ ( o μ − i μ ) D_yX=\sum_\mu(o_\mu-i_\mu) D y X = ∑ μ ( o μ − i μ ) ∣ o μ ∣ 2 + ∣ i μ ∣ 2 |o_\mu|^2+|i_\mu|^2 ∣ o μ ∣ 2 + ∣ i μ ∣ 2 ∣ D y X ∣ 2 |D_yX|^2 ∣ D y X ∣ 2 ∣ o μ + i μ ∣ 2 |o_\mu+i_\mu|^2 ∣ o μ + i μ ∣ 2 4 ∣ X ∣ 2 4|X|^2 4∣ X ∣ 2 − 2 ⟨ o μ , i μ ⟩ -2\langle o_\mu,i_\mu\rangle − 2 ⟨ o μ , i μ ⟩ ∣ D y X ∣ 2 |D_yX|^2 ∣ D y X ∣ 2 + 2 ⟨ o μ , i μ ⟩ +2\langle o_\mu,i_\mu\rangle + 2 ⟨ o μ , i μ ⟩ ∣ o μ + i μ ∣ 2 |o_\mu+i_\mu|^2 ∣ o μ + i μ ∣ 2 ∣ D y X ∣ 2 |D_yX|^2 ∣ D y X ∣ 2 y y y (7) (this is the proof of [9 Lemma 1] ). In a box two perpendicular ends at y y y □ \square □
Lemma 6.2. Let k ∈ { 1 , 2 , 3 , 4 } k\in\{1,2,3,4\} k ∈ { 1 , 2 , 3 , 4 } k ± 1 ( m o d 4 ) k\pm1\pmod4 k ± 1 ( mod 4 ) k k k k + 2 k+2 k + 2 N N N W ∈ U ( N ) W\in\mathrm{U}(N) W ∈ U ( N ) Y 1 , … , Y 4 ∈ u ( N ) Y_1,\ldots,Y_4\in\mathfrak{u}(N) Y 1 , … , Y 4 ∈ u ( N ) Y k = 0 Y_k=0 Y k = 0
± f p + 9 10 c p ≤ 21 10 ( ∣ Y k − 1 ∣ 2 + ∣ Y k + 1 ∣ 2 ) + 5 4 ∣ Y k + 2 ∣ 2 . \pm f_p+\frac{9}{10}c_p\leq\frac{21}{10}\left(|Y_{k-1}|^2+|Y_{k+1}|^2\right)+\frac{5}{4}|Y_{k+2}|^2. ± f p + 10 9 c p ≤ 10 21 ( ∣ Y k − 1 ∣ 2 + ∣ Y k + 1 ∣ 2 ) + 4 5 ∣ Y k + 2 ∣ 2 . Proof. Replacing W W W − W -W − W f p f_p f p c p c_p c p + + + W = diag ( w a ) W=\operatorname{diag}(w_a) W = diag ( w a ) F = R H S − f p − 9 10 c p \mathcal{F}=\mathrm{RHS}-f_p-\frac{9}{10}c_p F = RHS − f p − 10 9 c p a a a y i = − i ( Y i ) a a y_i=-i(Y_i)_{aa} y i = − i ( Y i ) aa a ≠ b a\ne b a = b z i = ( Y i ) a b z_i=(Y_i)_{ab} z i = ( Y i ) ab ( Y i ) b a = − z i ‾ (Y_i)_{ba}=-\overline{z_i} ( Y i ) ba = − z i □ \square □
Proof of the bound in Theorem 1.4 . Every plaquette has exactly one link of F F F X X X Lemma 2.2 (iii) each active link is adjacent to the fixed link in four of its plaquettes and opposite to it in two. Summing Lemma 6.2 over the plaquettes,
∑ p ( ± f p + 9 10 c p ) ≤ ( 4 ⋅ 21 10 + 2 ⋅ 5 4 ) ∣ X ∣ 2 = 109 10 ∣ X ∣ 2 . \sum_p\left(\pm f_p+\frac{9}{10}c_p\right)\leq\left(4\cdot\frac{21}{10}+2\cdot\frac{5}{4}\right)|X|^2=\frac{109}{10}|X|^2. p ∑ ( ± f p + 10 9 c p ) ≤ ( 4 ⋅ 10 21 + 2 ⋅ 4 5 ) ∣ X ∣ 2 = 10 109 ∣ X ∣ 2 . By Lemma 6.1 , 9 10 ∑ p c p = 9 10 Σ − 36 10 ∣ X ∣ 2 \frac{9}{10}\sum_p c_p=\frac{9}{10}\Sigma-\frac{36}{10}|X|^2 10 9 ∑ p c p = 10 9 Σ − 10 36 ∣ X ∣ 2 Σ ≥ 0 \Sigma\geq0 Σ ≥ 0
± Hess S 0 ( v , v ) + 9 10 Σ ≤ ( 109 10 + 36 10 ) ∣ X ∣ 2 = 29 2 ∣ X ∣ 2 . \pm\operatorname{Hess} S_0(v,v)+\frac{9}{10}\Sigma\leq\left(\frac{109}{10}+\frac{36}{10}\right)|X|^2=\frac{29}{2}|X|^2. ± Hess S 0 ( v , v ) + 10 9 Σ ≤ ( 10 109 + 10 36 ) ∣ X ∣ 2 = 2 29 ∣ X ∣ 2 . In a box, links near the boundary lie in fewer plaquettes and receive less. □ \square □
The factor 9 10 \frac{9}{10} 10 9 36 10 \frac{36}{10} 10 36 Lemma 6.1 is the divergence identity of [9 Lemma 1] with the collinear term written as a square, so nothing is lost by Cauchy–Schwarz.
Corollary 6.3. Let G = SU ( N ) G = \operatorname{SU}(N) G = SU ( N ) ∣ β ∣ < 1 29 |\beta| < \frac{1}{29} ∣ β ∣ < 29 1 G = SO ( N ) G = \operatorname{SO}(N) G = SO ( N ) N ≥ 3 N \ge3 N ≥ 3 ∣ β ∣ < N − 2 58 N |\beta| < \frac{N-2}{58N} ∣ β ∣ < 58 N N − 2 μ Λ F \mu_{\Lambda}^{F} μ Λ F K = Ric G − 29 2 N ∣ β ∣ > 0 K = \operatorname{Ric}_{G} - \frac{29}{2}N|\beta| > 0 K = Ric G − 2 29 N ∣ β ∣ > 0 2 / K 2/K 2/ K 1 / K 1/K 1/ K
Proof. The set { Q F = 1 } \{Q_{F}=1\} { Q F = 1 } Ric G \operatorname{Ric}_{G} Ric G N β Hess S 0 N\beta\operatorname{Hess} S_{0} N β Hess S 0 X F = 0 X_{F}=0 X F = 0 [2 Cor. 5.7.2] . □
Proof of the rest of Theorem 1.4. Decay in boxes. μ Λ F \mu_{\Lambda}^{F} μ Λ F G G G [31 Section 4] holds with the same constants, because fixing links only removes terms, and the Poincaré constant is 1 / K 1/K 1/ K [31 Cor. 4.11 and Rem. 4.12] gives ∣ Cov μ Λ F ( f , g ) ∣ ≤ c 1 e − c dist ( f , g ) ( ∥ f ∥ 1 ∥ g ∥ 1 + ∣ f ∣ 2 ∣ g ∣ 2 ) |\operatorname{Cov}_{\mu_{\Lambda}^{F}}(f,g)| \le c_{1}e^{-c\operatorname{dist}(f,g)}(\|f\|_{1}\|g\|_{1}+|f|_{2}|g|_{2}) ∣ Cov μ Λ F ( f , g ) ∣ ≤ c 1 e − c dist ( f , g ) ( ∥ f ∥ 1 ∥ g ∥ 1 + ∣ f ∣ 2 ∣ g ∣ 2 ) f , g f,g f , g
Limit. For boxes Λ ⊂ Λ ′ \Lambda\subset\Lambda' Λ ⊂ Λ ′ S s = S Λ + s ( S Λ ′ − S Λ ) S_{s}=S_{\Lambda}+s(S_{\Lambda'}-S_{\Lambda}) S s = S Λ + s ( S Λ ′ − S Λ ) Λ ′ \Lambda' Λ ′ s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] μ s \mu_{s} μ s 29 2 N ∣ β ∣ \frac{29}{2}N|\beta| 2 29 N ∣ β ∣ μ s \mu_{s} μ s K K K
d d s μ s ( f ) = ∑ p ∈ P ( Λ ′ ) ∖ P ( Λ ) Cov μ s ( f , N β Re Tr Q Λ ′ F ) , \frac{d}{ds}\mu_{s}(f)=\sum_{p\in\mathcal{P}(\Lambda')\setminus\mathcal{P}(\Lambda)}\operatorname{Cov}_{\mu_{s}}\left(f,N\beta\operatorname{Re}\operatorname{Tr}Q_{\Lambda'}^{F}\right), d s d μ s ( f ) = p ∈ P ( Λ ′ ) ∖ P ( Λ ) ∑ Cov μ s ( f , N β Re Tr Q Λ ′ F ) , ∣ μ Λ ′ F ( f ) − μ Λ F ( f ) ∣ ≤ C f R 3 e − c R , \left|\mu_{\Lambda'}^{F}(f)-\mu_{\Lambda}^{F}(f)\right|\le C_{f}R^{3}e^{-cR}, μ Λ ′ F ( f ) − μ Λ F ( f ) ≤ C f R 3 e − c R , R R R f f f Λ \Lambda Λ
Properties. Invariance, reflection positivity and decay follow as in Proposition 3.2(iii)–(v), and Proposition 23 of [9 ] , applied to smooth gauge-invariant cylinder functions, gives a gap at least c c c
Remark 6.4. Let h F h_{F} h F ± Hess S 0 \pm\operatorname{Hess}S_{0} ± Hess S 0 ∣ β ∣ < 1 / ( 2 h F ) |\beta|<1/(2h_{F}) ∣ β ∣ < 1/ ( 2 h F ) − Hess S 0 -\operatorname{Hess}S_{0} − Hess S 0 X F = 0 X_{F}=0 X F = 0 9 + 17 = 13.123 9+\sqrt{17}=13.123 9 + 17 = 13.123 9 + 17 ≤ h F ≤ 29 2 9+\sqrt{17}\le h_{F}\le\frac{29}{2} 9 + 17 ≤ h F ≤ 2 29 SU ( 2 ) \operatorname{SU}(2) SU ( 2 ) h F = 13.97401 h_{F}=13.97401 h F = 13.97401 14.384 14.384 14.384 ∣ β ∣ = 1 29 |\beta|=\frac{1}{29} ∣ β ∣ = 29 1 1 26.25 \frac{1}{26.25} 26.25 1
SU(3) up to 3/4
In this section G = SU ( 3 ) G=\operatorname{SU}(3) G = SU ( 3 ) β ≥ 0 \beta\ge0 β ≥ 0 β W = 9 β \beta_{W}=9\beta β W = 9 β SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) d ( U , V ) = ∥ U − V ∥ H S d(U,V)=\|U-V\|_{\mathrm{HS}} d ( U , V ) = ∥ U − V ∥ HS ⟨ X , Y ⟩ = − Tr ( X Y ) \langle X,Y\rangle=-\operatorname{Tr}(XY) ⟨ X , Y ⟩ = − Tr ( X Y ) K = conv SU ( 3 ) K=\operatorname{conv}\operatorname{SU}(3) K = conv SU ( 3 ) A ( M ) = 1 2 ( M − M ∗ ) A(M)=\frac{1}{2}(M-M^{*}) A ( M ) = 2 1 ( M − M ∗ ) P P P Ric = 9 2 \operatorname{Ric}=\frac{9}{2} Ric = 2 9 λ F = 8 3 \lambda_{F}=\frac{8}{3} λ F = 3 8 6 6 6
Reduction to one link
The conditional law of an active link is
ν b ( d U ) ∝ e Re Tr ( U b ) d U , b = N β ∑ p ∋ e S p , \nu_{b}(dU)\propto e^{\operatorname{Re}\operatorname{Tr}(Ub)}\,dU,\qquad b=N\beta\sum_{p\ni e}S_{p}, ν b ( d U ) ∝ e Re Tr ( U b ) d U , b = N β p ∋ e ∑ S p , where S p S_{p} S p e e e p p p SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) b ∈ r K b\in rK b ∈ r K r = 6 N β = 2 β W r=6N\beta=2\beta_{W} r = 6 N β = 2 β W h h h φ h ( U ) = Re Tr ( U h ) \varphi_{h}(U)=\operatorname{Re}\operatorname{Tr}(Uh) φ h ( U ) = Re Tr ( U h )
L h ( b ) = sup { Cov ν b ( g , φ h ) : g 1 -Lipschitz for d } , L_{h}(b)=\sup\{\operatorname{Cov}_{\nu_{b}}(g,\varphi_{h}):g\ 1\text{-Lipschitz for }d\}, L h ( b ) = sup { Cov ν b ( g , φ h ) : g 1 -Lipschitz for d } , Λ ( r ) = sup b ∈ r K sup ∥ h ∥ H S = 1 L h ( b ) . \Lambda(r)=\sup_{b\in rK}\sup_{\lVert h\rVert_{\mathrm{HS}}=1}L_h(b). Λ ( r ) = b ∈ r K sup ∥ h ∥ HS = 1 sup L h ( b ) . Changing a partner e ′ e' e ′ d ( Q e ′ , Q e ′ ′ ) d(Q_{e'},Q'_{e'}) d ( Q e ′ , Q e ′ ′ ) b t b_t b t b b b b ′ b' b ′ r K rK r K d d t ν b t ( g ) = Cov ν b t ( g , Re Tr ( U ( b ′ − b ) ) ) \frac{d}{dt}\nu_{b_t}(g)=\operatorname{Cov}_{\nu_{b_t}}(g,\operatorname{Re}\operatorname{Tr}(U(b'-b))) d t d ν b t ( g ) = Cov ν b t ( g , Re Tr ( U ( b ′ − b ))) W 1 ( ν b , ν b ′ ) ≤ N β Λ ( r ) d ( Q e ′ , Q e ′ ′ ) W_1(\nu_b,\nu_{b'})\le N\beta\Lambda(r)d(Q_{e'},Q'_{e'}) W 1 ( ν b , ν b ′ ) ≤ N β Λ ( r ) d ( Q e ′ , Q e ′ ′ )
α ≤ 12 N β Λ ( r ) = 2 r Λ ( r ) = 4 β W Λ ( 2 β W ) . (8) \alpha\le12N\beta\Lambda(r)=2r\Lambda(r)=4\beta_W\Lambda(2\beta_W). \tag*{(8)} α ≤ 12 N β Λ ( r ) = 2 r Λ ( r ) = 4 β W Λ ( 2 β W ) . (8) Since K K K 0 0 0 I , ω I , ω 2 I I,\omega I,\omega^2I I , ω I , ω 2 I ω 3 = 1 \omega^3=1 ω 3 = 1 r ′ K ⊂ r K r'K\subset rK r ′ K ⊂ r K r ′ ≤ r r'\le r r ′ ≤ r Λ \Lambda Λ
Gradients and one-forms
Lemma 7.1. In the left trivialisation, ∇ φ c ( U ) = − P A ( c U ) \nabla\varphi_c(U)=-P A(cU) ∇ φ c ( U ) = − P A ( c U ) V = φ b V=\varphi_b V = φ b Hess V ( X , X ) = Re Tr ( U X 2 b ) \operatorname{Hess}V(X,X)=\operatorname{Re}\operatorname{Tr}(UX^2b) Hess V ( X , X ) = Re Tr ( U X 2 b ) Δ φ c = − λ F φ c \Delta\varphi_c=-\lambda_F\varphi_c Δ φ c = − λ F φ c
Proof. d d t Re Tr ( U e t X c ) ∣ t = 0 = Re Tr ( c U X ) = − ⟨ P A ( c U ) , X ⟩ \left.\frac{d}{dt}\operatorname{Re}\operatorname{Tr}(Ue^{tX}c)\right|_{t=0}=\operatorname{Re}\operatorname{Tr}(cUX)=-\langle P A(cU),X\rangle d t d Re Tr ( U e tX c ) t = 0 = Re Tr ( c U X ) = − ⟨ P A ( c U ) , X ⟩ X ∈ s u ( 3 ) X\in\mathfrak{su}(3) X ∈ su ( 3 ) ∇ X X = 0 \nabla_XX=0 ∇ X X = 0 X X X e t X e^{tX} e tX φ c \varphi_c φ c λ F \lambda_F λ F □ \square □
Lemma 7.2. The bottom of the spectrum of the rough Laplacian ∇ ∗ ∇ \nabla^*\nabla ∇ ∗ ∇ L 2 L^2 L 2 S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) μ 0 = 7 6 \mu_0=\frac{7}{6} μ 0 = 6 7
Proof. Trivialise one-forms on the left, so that a one-form is a function f : S U ( 3 ) → s u ( 3 ) f:\mathrm{SU}(3)\to\mathfrak{su}(3) f : SU ( 3 ) → su ( 3 ) X a X_a X a ∇ X a Y = 1 2 [ X a , Y ] \nabla_{X_a}Y=\frac{1}{2}[X_a,Y] ∇ X a Y = 2 1 [ X a , Y ] ∇ X a X a = 0 \nabla_{X_a}X_a=0 ∇ X a X a = 0 ∇ ∗ ∇ = − ∑ a ( ρ ( X a ) + 1 2 ad X a ) 2 \nabla^*\nabla=-\sum_a(\rho(X_a)+\frac{1}{2}\operatorname{ad}X_a)^2 ∇ ∗ ∇ = − ∑ a ( ρ ( X a ) + 2 1 ad X a ) 2 V π ⊗ V π ∗ ⊗ s u ( 3 ) V_\pi\otimes V_\pi^*\otimes\mathfrak{su}(3) V π ⊗ V π ∗ ⊗ su ( 3 ) L 2 ⊗ s u ( 3 ) L^2\otimes\mathfrak{su}(3) L 2 ⊗ su ( 3 ) A a = ρ π ( X a ) A_a=\rho_\pi(X_a) A a = ρ π ( X a ) B a = ad X a B_a=\operatorname{ad}X_a B a = ad X a T a = A a + B a T_a=A_a+B_a T a = A a + B a A a A_a A a B a B_a B a A a + 1 2 B a = 1 2 ( A a + T a ) A_a+\frac{1}{2}B_a=\frac{1}{2}(A_a+T_a) A a + 2 1 B a = 2 1 ( A a + T a ) A a T a + T a A a = A a 2 + T a 2 − B a 2 A_aT_a+T_aA_a=A_a^2+T_a^2-B_a^2 A a T a + T a A a = A a 2 + T a 2 − B a 2 σ ⊂ π ⊗ ad \sigma\subset\pi\otimes\operatorname{ad} σ ⊂ π ⊗ ad
∇ ∗ ∇ = 1 2 ( C π + C σ ) − 1 4 C a d = 1 2 ( C π + C σ ) − 3 2 , \nabla^*\nabla=\frac{1}{2}(C_\pi+C_\sigma)-\frac{1}{4}C_{\mathrm{ad}}=\frac{1}{2}(C_\pi+C_\sigma)-\frac{3}{2}, ∇ ∗ ∇ = 2 1 ( C π + C σ ) − 4 1 C ad = 2 1 ( C π + C σ ) − 2 3 , with C ( p , q ) = 2 3 ( p 2 + q 2 + p q + 3 p + 3 q ) C(p,q)=\frac{2}{3}(p^2+q^2+pq+3p+3q) C ( p , q ) = 3 2 ( p 2 + q 2 + pq + 3 p + 3 q ) ( p , q ) (p,q) ( p , q ) [17 ] . The components of ( p , q ) ⊗ ( 1 , 1 ) (p,q)\otimes(1,1) ( p , q ) ⊗ ( 1 , 1 ) ( p + 1 , q + 1 ) (p+1,q+1) ( p + 1 , q + 1 ) ( p + 2 , q − 1 ) (p+2,q-1) ( p + 2 , q − 1 ) ( p − 1 , q + 2 ) (p-1,q+2) ( p − 1 , q + 2 ) ( p + 1 , q − 2 ) (p+1,q-2) ( p + 1 , q − 2 ) ( p − 2 , q + 1 ) (p-2,q+1) ( p − 2 , q + 1 ) ( p − 1 , q − 1 ) (p-1,q-1) ( p − 1 , q − 1 ) ( p , q ) (p,q) ( p , q ) p , q p,q p , q π = ( 0 , 0 ) \pi=(0,0) π = ( 0 , 0 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) 3 2 \frac{3}{2} 2 3 π = ( 1 , 0 ) \pi=(1,0) π = ( 1 , 0 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 1 , 0 ) (1,0) ( 1 , 0 ) 31 6 , 19 6 , 7 6 \frac{31}{6},\frac{19}{6},\frac{7}{6} 6 31 , 6 19 , 6 7 ( 0 , 1 ) (0,1) ( 0 , 1 ) p + q ≥ 2 p+q\ge2 p + q ≥ 2 ( p − 1 , q − 1 ) (p-1,q-1) ( p − 1 , q − 1 ) ( p , q ) (p,q) ( p , q ) ( p − 2 , q + 1 ) (p-2,q+1) ( p − 2 , q + 1 ) ( p + 1 , q − 2 ) (p+1,q-2) ( p + 1 , q − 2 ) 1 3 ( 2 ( p 2 + q 2 + p q ) + 3 ( p + q ) − 3 ) − 3 2 ≥ 3 2 \frac{1}{3}(2(p^2+q^2+pq)+3(p+q)-3)-\frac{3}{2}\ge\frac{3}{2} 3 1 ( 2 ( p 2 + q 2 + pq ) + 3 ( p + q ) − 3 ) − 2 3 ≥ 2 3 □ \square □
The gap of the one-link measure
Lemma 7.3. For b ∈ r K b\in rK b ∈ r K L = Δ + ⟨ ∇ V , ∇ ⋅ ⟩ L=\Delta+\langle\nabla V,\nabla\cdot\rangle L = Δ + ⟨ ∇ V , ∇ ⋅ ⟩ V = φ b V=\varphi_b V = φ b L 2 ( ν b ) L^2(\nu_b) L 2 ( ν b )
γ ( r ) = 39 64 μ 0 + 3 2 − r 2 = 283 128 − r 2 . \gamma(r)=\frac{39}{64}\mu_0+\frac{3}{2}-\frac{r}{2}=\frac{283}{128}-\frac{r}{2}. γ ( r ) = 64 39 μ 0 + 2 3 − 2 r = 128 283 − 2 r . Proof. Let − L f = λ f -Lf=\lambda f − L f = λ f λ > 0 \lambda>0 λ > 0 ω = d f \omega=df ω = df η = e V / 2 ω \eta=e^{V/2}\omega η = e V /2 ω L 2 ( d U ) L^2(dU) L 2 ( d U ) ν b \nu_b ν b λ ν b ∣ ω ∣ 2 = ν b ∣ ∇ ω ∣ 2 + ν b ( ( Ric − Hess V ) ( ω , ω ) ) \lambda\nu_b|\omega|^2=\nu_b|\nabla\omega|^2+\nu_b((\operatorname{Ric}-\operatorname{Hess}V)(\omega,\omega)) λ ν b ∣ ω ∣ 2 = ν b ∣∇ ω ∣ 2 + ν b (( Ric − Hess V ) ( ω , ω )) e V / 2 ∇ ω = ∇ η − 1 2 d V ⊗ η e^{V/2}\nabla\omega=\nabla\eta-\frac{1}{2}dV\otimes\eta e V /2 ∇ ω = ∇ η − 2 1 d V ⊗ η
λ ∥ η ∥ 2 = ∥ ∇ η − 1 2 d V ⊗ η ∥ 2 + ∫ ( Ric − Hess V ) ( η , η ) . (9) \lambda\|\eta\|^2=\left\|\nabla\eta-\frac{1}{2}dV\otimes\eta\right\|^2+\int(\operatorname{Ric}-\operatorname{Hess}V)(\eta,\eta). \tag*{(9)} λ ∥ η ∥ 2 = ∇ η − 2 1 d V ⊗ η 2 + ∫ ( Ric − Hess V ) ( η , η ) . (9) Put A = ∇ η \mathcal{A}=\nabla\eta A = ∇ η B = 1 2 d V ⊗ η \mathcal{B}=\frac{1}{2}dV\otimes\eta B = 2 1 d V ⊗ η
⟨ A , B ⟩ = 1 4 ∫ ⟨ ∇ V , ∣ η ∣ 2 ⟩ = λ F 4 ∫ V ∣ η ∣ 2 , \langle\mathcal{A},\mathcal{B}\rangle=\frac{1}{4}\int\langle\nabla V,|\eta|^2\rangle=\frac{\lambda_F}{4}\int V|\eta|^2, ⟨ A , B ⟩ = 4 1 ∫ ⟨ ∇ V , ∣ η ∣ 2 ⟩ = 4 λ F ∫ V ∣ η ∣ 2 , because − Δ V = λ F V -\Delta V=\lambda_F V − Δ V = λ F V ∥ A − B ∥ 2 = ∥ 5 8 A − B ∥ 2 + 39 64 ∥ A ∥ 2 − 3 4 ⟨ A , B ⟩ \lVert A-B\rVert^2=\lVert\frac{5}{8}A-B\rVert^2+\frac{39}{64}\lVert A\rVert^2-\frac{3}{4}\langle A,B\rangle ∥ A − B ∥ 2 = ∥ 8 5 A − B ∥ 2 + 64 39 ∥ A ∥ 2 − 4 3 ⟨ A , B ⟩
∥ A − B ∥ 2 ≥ 39 64 μ 0 ∥ η ∥ 2 − 1 2 ∫ V ∣ η ∣ 2 \lVert A-B\rVert^2\ge\frac{39}{64}\mu_0\lVert\eta\rVert^2-\frac{1}{2}\int V\lvert\eta\rvert^2 ∥ A − B ∥ 2 ≥ 64 39 μ 0 ∥ η ∥ 2 − 2 1 ∫ V ∣ η ∣ 2 by Lemma 7.2. With Ric = 3 2 \operatorname{Ric}=\frac{3}{2} Ric = 2 3 (9) gives λ ∥ η ∥ 2 ≥ ( 39 64 μ 0 + 3 2 ) ∥ η ∥ 2 − ∫ ( Hess V ( η , η ) + 1 2 V ∣ η ∣ 2 ) \lambda\lVert\eta\rVert^2\ge\left(\frac{39}{64}\mu_0+\frac{3}{2}\right)\lVert\eta\rVert^2-\int\left(\operatorname{Hess}V(\eta,\eta)+\frac{1}{2}V\lvert\eta\rvert^2\right) λ ∥ η ∥ 2 ≥ ( 64 39 μ 0 + 2 3 ) ∥ η ∥ 2 − ∫ ( Hess V ( η , η ) + 2 1 V ∣ η ∣ 2 )
Re Tr ( M X 2 ) + 1 2 Re Tr M ≤ r 2 ∣ X ∣ 2 , M = b U ∈ r K (10) \operatorname{Re}\operatorname{Tr}(MX^2)+\frac{1}{2}\operatorname{Re}\operatorname{Tr}M\le\frac{r}{2}\lvert X\rvert^2,\qquad M=bU\in rK \tag*{(10)} Re Tr ( M X 2 ) + 2 1 Re Tr M ≤ 2 r ∣ X ∣ 2 , M = b U ∈ r K (10) since Hess V ( X , X ) = Re Tr ( M X 2 ) \operatorname{Hess}V(X,X)=\operatorname{Re}\operatorname{Tr}(MX^2) Hess V ( X , X ) = Re Tr ( M X 2 ) V = Re Tr M V=\operatorname{Re}\operatorname{Tr}M V = Re Tr M ∣ X ∣ = 1 \lvert X\rvert=1 ∣ X ∣ = 1 M M M M = r W M=rW M = r W W ∈ SU ( 3 ) W\in\operatorname{SU}(3) W ∈ SU ( 3 ) W = diag ( e i θ k ) W=\operatorname{diag}(e^{i\theta_k}) W = diag ( e i θ k ) ∑ k θ k = 0 \sum_k\theta_k=0 ∑ k θ k = 0 c k = cos θ k c_k=\cos\theta_k c k = cos θ k X = i H X=iH X = i H H H H Tr H 2 = 1 \operatorname{Tr}H^2=1 Tr H 2 = 1 d k d_k d k H j k H_{jk} H j k
Re Tr ( W X 2 ) = − ∑ k c k d k 2 − ∑ j < k ( c j + c k ) ∣ H j k ∣ 2 , \operatorname{Re}\operatorname{Tr}(WX^2)=-\sum_k c_kd_k^2-\sum_{j<k}(c_j+c_k)\lvert H_{jk}\rvert^2, Re Tr ( W X 2 ) = − k ∑ c k d k 2 − j < k ∑ ( c j + c k ) ∣ H j k ∣ 2 , with ∑ k d k 2 + 2 ∑ j < k ∣ H j k ∣ 2 = 1 \sum_kd_k^2+2\sum_{j<k}\lvert H_{jk}\rvert^2=1 ∑ k d k 2 + 2 ∑ j < k ∣ H j k ∣ 2 = 1 q o f f = max j < k − 1 2 ( c j + c k ) q_{\mathrm{off}}=\max_{j<k}-\frac{1}{2}(c_j+c_k) q off = max j < k − 2 1 ( c j + c k ) q d i a g = max { − ∑ c k d k 2 : ∑ d k = 0 , ∑ d k 2 = 1 } q_{\mathrm{diag}}=\max\{-\sum c_kd_k^2:\sum d_k=0,\sum d_k^2=1\} q diag = max { − ∑ c k d k 2 : ∑ d k = 0 , ∑ d k 2 = 1 } l l l q o f f + 1 2 ∑ k c k = 1 2 c l ≤ 1 2 q_{\mathrm{off}}+\frac{1}{2}\sum_kc_k=\frac{1}{2}c_l\le\frac{1}{2} q off + 2 1 ∑ k c k = 2 1 c l ≤ 2 1 q d i a g q_{\mathrm{diag}} q diag diag ( c ) \operatorname{diag}(c) diag ( c ) ∑ d k = 0 \sum d_k=0 ∑ d k = 0 ∑ k ( c k − μ ) − 1 = 0 \sum_k(c_k-\mu)^{-1}=0 ∑ k ( c k − μ ) − 1 = 0 3 μ 2 − 2 e 1 μ + e 2 = 0 3\mu^2-2e_1\mu+e_2=0 3 μ 2 − 2 e 1 μ + e 2 = 0 e 1 , e 2 e_1,e_2 e 1 , e 2 c k c_k c k q d i a g = 1 3 ( e 1 2 − 3 e 2 − e 1 ) q_{\mathrm{diag}}=\frac{1}{3}\left(\sqrt{e_1^2-3e_2}-e_1\right) q diag = 3 1 ( e 1 2 − 3 e 2 − e 1 ) q d i a g + 1 2 e 1 ≤ 1 2 q_{\mathrm{diag}}+\frac{1}{2}e_1\le\frac{1}{2} q diag + 2 1 e 1 ≤ 2 1 e 1 2 − 3 e 2 ≤ 1 2 ( 3 − e 1 ) \sqrt{e_1^2-3e_2}\le\frac{1}{2}(3-e_1) e 1 2 − 3 e 2 ≤ 2 1 ( 3 − e 1 ) e 1 2 − 4 e 2 + 2 e 1 − 3 ≤ 0 e_1^2-4e_2+2e_1-3\le0 e 1 2 − 4 e 2 + 2 e 1 − 3 ≤ 0 ∑ θ k = 0 \sum\theta_k=0 ∑ θ k = 0 ∑ k c k 2 = 1 + 2 c 1 c 2 c 3 \sum_kc_k^2=1+2c_1c_2c_3 ∑ k c k 2 = 1 + 2 c 1 c 2 c 3 − 2 ( 1 − c 1 ) ( 1 − c 2 ) ( 1 − c 3 ) ≤ 0 -2(1-c_1)(1-c_2)(1-c_3)\le0 − 2 ( 1 − c 1 ) ( 1 − c 2 ) ( 1 − c 3 ) ≤ 0
The weight 5 8 \frac{5}{8} 8 5 1 2 \frac{1}{2} 2 1 V V V (10) is an equality at W = I W=I W = I r = 3 2 r=\frac{3}{2} r = 2 3 γ = 187 128 \gamma=\frac{187}{128} γ = 128 187 6
A corrector of degree two.
Lemma 7.4. For b ∈ r K b\in rK b ∈ r K ∥ h ∥ = 1 \lVert h\rVert=1 ∥ h ∥ = 1 u u u SU ( 3 ) \operatorname{SU}(3) SU ( 3 )
L h ( b ) ≤ ν b ∣ ∇ u ∣ 2 + Var ν b ( ϕ h + L u ) / γ ( r ) . L_h(b)\le\sqrt{\nu_b\lvert\nabla u\rvert^2}+\sqrt{\operatorname{Var}_{\nu_b}(\phi_h+Lu)/\gamma(r)}. L h ( b ) ≤ ν b ∣ ∇ u ∣ 2 + Var ν b ( ϕ h + Lu ) / γ ( r ) . Proof. Let − L ψ = ϕ h − ν b ϕ h -L\psi=\phi_h-\nu_b\phi_h − L ψ = ϕ h − ν b ϕ h g g g ∣ ∇ g ∣ ≤ 1 \lvert\nabla g\rvert\le1 ∣ ∇ g ∣ ≤ 1 d d d Cov ( g , ϕ h ) = ν b ⟨ ∇ g , ∇ ψ ⟩ ≤ ν b ∣ ∇ ψ ∣ \operatorname{Cov}(g,\phi_h)=\nu_b\langle\nabla g,\nabla\psi\rangle\le\nu_b\lvert\nabla\psi\rvert Cov ( g , ϕ h ) = ν b ⟨ ∇ g , ∇ ψ ⟩ ≤ ν b ∣ ∇ ψ ∣ ψ = u + χ \psi=u+\chi ψ = u + χ ν b ( L u ) = 0 \nu_b(Lu)=0 ν b ( Lu ) = 0 χ \chi χ − L χ = X − ν b X -L\chi=X-\nu_bX − L χ = X − ν b X X = ϕ h + L u X=\phi_h+Lu X = ϕ h + Lu
ν b ∣ ∇ χ ∣ 2 = Cov ( χ , X ) ≤ Var χ Var X ≤ ν b ∣ ∇ χ ∣ 2 Var X / γ \nu_b\lvert\nabla\chi\rvert^2=\operatorname{Cov}(\chi,X)\le\sqrt{\operatorname{Var}\chi\operatorname{Var}X}\le\sqrt{\nu_b\lvert\nabla\chi\rvert^2\operatorname{Var}X/\gamma} ν b ∣ ∇ χ ∣ 2 = Cov ( χ , X ) ≤ Var χ Var X ≤ ν b ∣ ∇ χ ∣ 2 Var X / γ by Lemma 7.3. Hence ν b ∣ ∇ χ ∣ 2 ≤ Var X / γ \nu_b\lvert\nabla\chi\rvert^2\le\operatorname{Var}X/\gamma ν b ∣ ∇ χ ∣ 2 ≤ Var X / γ ν b ∣ ∇ ψ ∣ ≤ ν b ∣ ∇ u ∣ 2 + ν b ∣ ∇ χ ∣ 2 \nu_b\lvert\nabla\psi\rvert\le\sqrt{\nu_b\lvert\nabla u\rvert^2}+\sqrt{\nu_b\lvert\nabla\chi\rvert^2} ν b ∣ ∇ ψ ∣ ≤ ν b ∣ ∇ u ∣ 2 + ν b ∣ ∇ χ ∣ 2
The best u u u ψ \psi ψ u = s ϕ h u=s\phi_h u = s ϕ h ϕ h + L u = ( 1 − s λ F ) ϕ h + s Γ \phi_h+Lu=(1-s\lambda_F)\phi_h+s\Gamma ϕ h + Lu = ( 1 − s λ F ) ϕ h + s Γ Γ = ⟨ ∇ V , ∇ ϕ h ⟩ \Gamma=\langle\nabla V,\nabla\phi_h\rangle Γ = ⟨ ∇ V , ∇ ϕ h ⟩ s = 1 / λ F s=1/\lambda_F s = 1/ λ F ϕ h \phi_h ϕ h β W = 45 64 \beta_W=\frac{45}{64} β W = 64 45 7
We take u u u P 2 \mathcal{P}_2 P 2 u i j u_{ij} u ij U U U u ‾ i j \overline{u}_{ij} u ij U ‾ \overline{U} U ( B k ) (B_k) ( B k ) 1 2 ( m + m ‾ ) \frac{1}{2}(m+\overline{m}) 2 1 ( m + m ) 1 2 i ( m − m ‾ ) \frac{1}{2i}(m-\overline{m}) 2 i 1 ( m − m ) { m , m ‾ } \{m,\overline{m}\} { m , m } m ≠ m ‾ m\ne\overline{m} m = m m m m m = m ‾ m=\overline{m} m = m h ∈ C 3 × 3 h \in\mathbb{C}^{3 \times3} h ∈ C 3 × 3 R 18 \mathbb{R}^{18} R 18 189 × 18 189 \times18 189 × 18 C C C
q h = ∑ k ∑ m = 1 18 C k m h m B k ∈ P 2 , q_h = \sum_k \sum_{m=1}^{18} C_{km}h_mB_k \in\mathcal{P}_2, q h = k ∑ m = 1 ∑ 18 C k m h m B k ∈ P 2 , linear in h h h b = diag ( a 1 , a 2 , x 3 + i y 3 ) b = \operatorname{diag}(a_1,a_2,x_3+iy_3) b = diag ( a 1 , a 2 , x 3 + i y 3 ) V = ∑ k b k φ F k V = \sum_k b_k\varphi_{F_k} V = ∑ k b k φ F k ( b k ) = ( a 1 , a 2 , x 3 , y 3 ) (b_k)=(a_1,a_2,x_3,y_3) ( b k ) = ( a 1 , a 2 , x 3 , y 3 ) F = ( e 11 , e 22 , e 33 , i e 33 ) F=(e_{11},e_{22},e_{33},ie_{33}) F = ( e 11 , e 22 , e 33 , i e 33 )
X h = φ h + L q h = φ h + Δ q h + ∑ k b k ⟨ ∇ φ F k , ∇ q h ⟩ (11) X_h=\varphi_h+Lq_h=\varphi_h+\Delta q_h+\sum_k b_k\langle\nabla\varphi_{F_k},\nabla q_h\rangle \tag*{(11)} X h = φ h + L q h = φ h + Δ q h + k ∑ b k ⟨ ∇ φ F k , ∇ q h ⟩ (11) is a polynomial of degree at most three, affine in b b b ∇ q h \nabla q_h ∇ q h b b b h ↦ ν b ∣ ∇ q h ∣ 2 h\mapsto\nu_b\lvert\nabla q_h\rvert^2 h ↦ ν b ∣ ∇ q h ∣ 2 h ↦ ν b [ ( X h − κ ) 2 ] h\mapsto\nu_b[(X_h-\kappa)^2] h ↦ ν b [( X h − κ ) 2 ] ν b \nu_b ν b
To compute Δ \Delta Δ ⟨ ∇ f , ∇ g ⟩ \langle\nabla f,\nabla g\rangle ⟨ ∇ f , ∇ g ⟩ s u ( 3 ) su(3) s u ( 3 )
e j k − e k j , i ( e j k + e k j ) ( j < k ) , i ( e 11 − e 22 ) , i ( e 11 + e 22 − 2 e 33 ) , e_{jk}-e_{kj},\quad i(e_{jk}+e_{kj})\ (j<k),\quad i(e_{11}-e_{22}),\quad i(e_{11}+e_{22}-2e_{33}), e j k − e k j , i ( e j k + e k j ) ( j < k ) , i ( e 11 − e 22 ) , i ( e 11 + e 22 − 2 e 33 ) , acting as left-invariant fields X a f ( U ) = d d t f ( U e t X a ) ∣ t = 0 X_af(U)=\left.\frac{d}{dt}f(Ue^{tX_a})\right|_{t=0} X a f ( U ) = d t d f ( U e t X a ) t = 0 w a = 1 2 w_a=\frac{1}{2} w a = 2 1 1 6 \frac{1}{6} 6 1 ∑ a w a X a ⊗ X a \sum_a w_aX_a\otimes X_a ∑ a w a X a ⊗ X a ⟨ X , Y ⟩ = − Tr ( X Y ) \langle X,Y\rangle=-\operatorname{Tr}(XY) ⟨ X , Y ⟩ = − Tr ( X Y )
⟨ ∇ f , ∇ g ⟩ = ∑ a w a X a f X a g , Δ = ∑ a w a X a 2 . \langle\nabla f,\nabla g\rangle=\sum_a w_aX_afX_ag,\qquad\Delta=\sum_a w_aX_a^2. ⟨ ∇ f , ∇ g ⟩ = a ∑ w a X a f X a g , Δ = a ∑ w a X a 2 . The fields map monomials to combinations of monomials of the same degree with coefficients in Z [ i ] \mathbb{Z}[i] Z [ i ] Δ φ h = − 3 2 φ h \Delta\varphi_h=-\frac{3}{2}\varphi_h Δ φ h = − 2 3 φ h Lemma 7.1 .
Orbits of staple sums
Lemma 7.5. Let V , W ∈ S U ( 3 ) V,W\in\mathrm{SU}(3) V , W ∈ SU ( 3 )
(i) L h ( V b W ) = L V − 1 h W − 1 ( b ) L_h(VbW)=L_{V^{-1}hW^{-1}}(b) L h ( V bW ) = L V − 1 h W − 1 ( b ) L h ( b ‾ ) = L h ‾ ( b ) L_h(\overline{b})=L_{\overline{h}}(b) L h ( b ) = L h ( b )
(ii) Every b ∈ C 3 × 3 b\in\mathbb{C}^{3\times3} b ∈ C 3 × 3 diag ( a 1 , a 2 , z 3 ) \operatorname{diag}(a_1,a_2,z_3) diag ( a 1 , a 2 , z 3 ) a 1 ≥ a 2 ≥ ∣ z 3 ∣ a_1\ge a_2\ge\lvert z_3\rvert a 1 ≥ a 2 ≥ ∣ z 3 ∣ Im z 3 ≥ 0 \operatorname{Im}z_3\ge0 Im z 3 ≥ 0
(iii) A diagonal matrix lies in K K K T T T S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
Proof. (i) If U U U ν V b W \nu_{VbW} ν V bW W U V WUV W U V ν b \nu_b ν b d d d φ h ( U ) = φ V − 1 h W − 1 ( W U V ) \varphi_h(U)=\varphi_{V^{-1}hW^{-1}}(WUV) φ h ( U ) = φ V − 1 h W − 1 ( W U V ) b ‾ \overline{b} b U ↦ U ‾ U\mapsto\overline{U} U ↦ U b = V 1 Σ W 1 b=V_1\Sigma W_1 b = V 1 Σ W 1 V 1 , W 1 V_1,W_1 V 1 , W 1 Im z 3 ≥ 0 \operatorname{Im}z_3\ge0 Im z 3 ≥ 0 c c c U U U Re Tr ( c U ) \operatorname{Re}\operatorname{Tr}(cU) Re Tr ( c U ) S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) s u ( 3 ) su(3) s u ( 3 ) c U cU c U i λ I i\lambda I iλ I c U = H + i λ I cU=H+i\lambda I c U = H + iλ I H H H H 2 + λ 2 I = c c ∗ H^2+\lambda^2I=cc^* H 2 + λ 2 I = c c ∗ H H H U ∈ T U\in T U ∈ T max S U ( 3 ) Re Tr ( c U ) = max T Re Tr ( c t ) \max_{\mathrm{SU}(3)}\operatorname{Re}\operatorname{Tr}(cU)=\max_T\operatorname{Re}\operatorname{Tr}(ct) max SU ( 3 ) Re Tr ( c U ) = max T Re Tr ( c t ) c c c c c c conv T \operatorname{conv}T conv T K K K
Since every b ∈ r K b\in rK b ∈ r K r r r
Λ ( r ) = sup { sup ∥ h ∥ = 1 L h ( b ) : b ∈ R r } , \Lambda(r)=\sup\left\{\sup_{\lVert h\rVert=1}L_h(b):b\in\mathcal{R}_r\right\}, Λ ( r ) = sup { ∥ h ∥ = 1 sup L h ( b ) : b ∈ R r } , where R r \mathcal{R}_r R r diag ( a 1 , a 2 , x 3 + i y 3 ) ∈ r conv T \operatorname{diag}(a_1,a_2,x_3+iy_3)\in r\operatorname{conv}T diag ( a 1 , a 2 , x 3 + i y 3 ) ∈ r conv T a 1 ≥ a 2 ≥ ∣ x 3 + i y 3 ∣ a_1\ge a_2\ge\lvert x_3+iy_3\rvert a 1 ≥ a 2 ≥ ∣ x 3 + i y 3 ∣ y 3 ≥ 0 y_3\ge0 y 3 ≥ 0 [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] [0,r]^2\times[-r,r]\times[0,r] [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] ( a 1 , a 2 , x 3 , y 3 ) (a_1,a_2,x_3,y_3) ( a 1 , a 2 , x 3 , y 3 )
Exact moments and interpolation
Lemma 7.6. For D , D ′ ∈ C 3 × 3 D,D' \in\mathbb{C}^{3\times3} D , D ′ ∈ C 3 × 3 E H \mathbb{E}_{H} E H S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
Z ( D , D ′ ) = E H exp ( 1 2 Tr ( U D ) + 1 2 Tr ( U ‾ D ′ ) ) . Z(D,D')=\mathbb{E}_{H}\exp\left(\frac{1}{2}\operatorname{Tr}(UD)+\frac{1}{2}\operatorname{Tr}(\overline{U}D')\right). Z ( D , D ′ ) = E H exp ( 2 1 Tr ( U D ) + 2 1 Tr ( U D ′ ) ) . Then
Z ( D , D ′ ) = ∑ p ≥ q ≥ 0 s ( p , q ) ( D D ′ T ) G p , q ( det D ) G p , q ( det D ′ ) d p , q . Z(D,D')=\sum_{p\geq q\geq0}\frac{s_{(p,q)}(DD'^{\mathsf{T}})G_{p,q}(\det D)G_{p,q}(\det D')}{d_{p,q}}. Z ( D , D ′ ) = p ≥ q ≥ 0 ∑ d p , q s ( p , q ) ( D D ′ T ) G p , q ( det D ) G p , q ( det D ′ ) . with G p , q ( z ) = ∑ k ≥ 0 c ( p + k , q + k , k ) z k G_{p,q}(z)=\sum_{k\geq0}c_{(p+k,q+k,k)}z^k G p , q ( z ) = ∑ k ≥ 0 c ( p + k , q + k , k ) z k c λ = 2 − ∣ λ ∣ / ∏ hooks ( λ ) c_{\lambda}=2^{-|\lambda|}/\prod\operatorname{hooks}(\lambda) c λ = 2 − ∣ λ ∣ / ∏ hooks ( λ ) d p , q = 1 2 ( p − q + 1 ) ( q + 1 ) ( p + 2 ) d_{p,q}=\frac{1}{2}(p-q+1)(q+1)(p+2) d p , q = 2 1 ( p − q + 1 ) ( q + 1 ) ( p + 2 ) s ( p , q ) s_{(p,q)} s ( p , q ) j j j U U U k k k U ‾ \overline{U} U
E H [ u α u ‾ β e Re Tr ( U b ) ] = 2 ∣ α ∣ + ∣ β ∣ ∂ D α ∂ D ′ β Z ( b , b ‾ ) , \mathbb{E}_{H}\left[u^{\alpha}\overline{u}^{\beta}e^{\operatorname{Re}\operatorname{Tr}(Ub)}\right]=2^{|\alpha|+|\beta|}\partial_{D}^{\alpha}\partial_{D'}^{\beta}Z(b,\overline{b}), E H [ u α u β e Re Tr ( U b ) ] = 2 ∣ α ∣ + ∣ β ∣ ∂ D α ∂ D ′ β Z ( b , b ) , the entry u i j u_{ij} u ij D j i D_{ji} D j i
Proof. We use ( Tr X ) n = ∑ λ ⊢ n f λ s λ ( X ) (\operatorname{Tr}X)^n=\sum_{\lambda\vdash n}f^\lambda s_{\lambda}(X) ( Tr X ) n = ∑ λ ⊢ n f λ s λ ( X ) f λ f^\lambda f λ λ \lambda λ [32 Cor. 7.21.6] . Expanding the exponential gives
e 1 2 Tr X = ∑ n ≥ 0 ( Tr X ) n 2 n n ! = ∑ λ c λ s λ ( X ) , e^{\frac{1}{2}\operatorname{Tr}X}=\sum_{n\geq0}\frac{(\operatorname{Tr}X)^n}{2^n n!}=\sum_{\lambda}c_{\lambda}s_{\lambda}(X), e 2 1 Tr X = n ≥ 0 ∑ 2 n n ! ( Tr X ) n = λ ∑ c λ s λ ( X ) , the sum over partitions with at most three parts. For U ∈ S U ( 3 ) U\in\mathrm{SU}(3) U ∈ SU ( 3 ) s λ + ( k , k , k ) ( U D ) = ( det D ) k s λ ( U D ) s_{\lambda+(k,k,k)}(UD)=(\det D)^k s_{\lambda}(UD) s λ + ( k , k , k ) ( U D ) = ( det D ) k s λ ( U D )
e 1 2 Tr ( U D ) = ∑ p ≥ q G p , q ( det D ) χ p , q ( U D ) , e^{\frac{1}{2}\operatorname{Tr}(UD)}=\sum_{p\geq q}G_{p,q}(\det D)\chi_{p,q}(UD), e 2 1 Tr ( U D ) = p ≥ q ∑ G p , q ( det D ) χ p , q ( U D ) , where χ p , q \chi_{p,q} χ p , q ( p , q , 0 ) (p,q,0) ( p , q , 0 ) d p , q d_{p,q} d p , q U ‾ = ( U − 1 ) T \overline{U}=(U^{-1})^{\mathsf{T}} U = ( U − 1 ) T χ p , q ( U ‾ D ′ ) = χ p , q ( U − 1 D ′ T ) \chi_{p,q}(\overline{U}D')=\chi_{p,q}(U^{-1}D'^{\mathsf{T}}) χ p , q ( U D ′ ) = χ p , q ( U − 1 D ′ T ) [12 Ch. 2] ,
E H [ χ ρ ( U A ) χ ρ ′ ( U − 1 B ) ] = δ ρ ρ ′ χ ρ ( A B ) d ρ \mathbb{E}_{H}\left[\chi_{\rho}(UA)\chi_{\rho'}(U^{-1}B)\right]=\delta_{\rho\rho'}\frac{\chi_{\rho}(AB)}{d_{\rho}} E H [ χ ρ ( U A ) χ ρ ′ ( U − 1 B ) ] = δ ρ ρ ′ d ρ χ ρ ( A B ) gives the expansion. The bound below gives absolute convergence. Finally Re Tr ( U b ) = 1 2 Tr ( U b ) + 1 2 Tr ( U ‾ b ‾ ) \operatorname{Re}\operatorname{Tr}(Ub)=\frac{1}{2}\operatorname{Tr}(Ub)+\frac{1}{2}\operatorname{Tr}(\overline{U}\,\overline{b}) Re Tr ( U b ) = 2 1 Tr ( U b ) + 2 1 Tr ( U b )
The computation keeps p ≤ 34 p\leq34 p ≤ 34 k ≤ 11 k\leq11 k ≤ 11 ∣ s λ ( X ) ∣ ≤ dim ( λ ) ∥ X ∥ ∣ λ ∣ |s_{\lambda}(X)|\leq\dim(\lambda)\|X\|^{|\lambda|} ∣ s λ ( X ) ∣ ≤ dim ( λ ) ∥ X ∥ ∣ λ ∣ ∑ λ ⊢ n f λ dim λ = 3 n \sum_{\lambda\vdash n}f^{\lambda}\dim\lambda=3^n ∑ λ ⊢ n f λ dim λ = 3 n ∑ n > 34 ( 3 2 ∥ D ∥ ) n / n ! \sum_{n>34}( \frac{3}{2}\|D\|)^n/n! ∑ n > 34 ( 2 3 ∥ D ∥ ) n / n ! b b b
Lemma 7.7 (Interpolation on a box). Let Q = ∏ k [ l k , l k + w k ] Q=\prod_k[l_k,l_k+w_k] Q = ∏ k [ l k , l k + w k ] ( a 1 , a 2 , x 3 , y 3 ) (a_1,a_2,x_3,y_3) ( a 1 , a 2 , x 3 , y 3 ) b i b_i b i ω i ( b ) ≥ 0 \omega_i(b)\geq0 ω i ( b ) ≥ 0 ∑ i ω i ( b ) = 1 \sum_i\omega_i(b)=1 ∑ i ω i ( b ) = 1 ς = w 1 2 + w 2 2 + max ( w 3 2 , w 4 2 ) \varsigma=w_1^2+w_2^2+\max(w_3^2,w_4^2) ς = w 1 2 + w 2 2 + max ( w 3 2 , w 4 2 ) b ∈ Q b\in Q b ∈ Q Z ( b ) ≥ e − ς / 8 ∑ i ω i ( b ) Z ( b i ) Z(b)\geq e^{-\varsigma/8}\sum_i\omega_i(b)Z(b_i) Z ( b ) ≥ e − ς /8 ∑ i ω i ( b ) Z ( b i ) ν b ( f ) ≤ e ς / 8 max i ν b i ( f ) \nu_b(f)\leq e^{\varsigma/8}\max_i\nu_{b_i}(f) ν b ( f ) ≤ e ς /8 max i ν b i ( f ) f ≥ 0 f\geq0 f ≥ 0 b b b Y b = ( A + ∑ k ( b k − β k ) B k ) 2 Y_b=(\mathcal{A}+\sum_k(b_k-\beta_k)\mathcal{B}_k)^2 Y b = ( A + ∑ k ( b k − β k ) B k ) 2 A , B k \mathcal{A},\mathcal{B}_k A , B k U U U β \beta β
ν b ( Y b ) ≤ e ς / 8 ( max i ν b i ( Y b i ) + ∑ k w k 2 4 max i Z ( b i ) ν b i ( B k 2 ) min i Z ( b i ) ) . \nu_b(Y_b)\leq e^{\varsigma/8}\left(\max_i\nu_{b_i}(Y_{b_i})+\sum_k\frac{w_k^2}{4}\frac{\max_i Z(b_i)\nu_{b_i}(\mathcal{B}_k^2)}{\min_i Z(b_i)}\right). ν b ( Y b ) ≤ e ς /8 ( i max ν b i ( Y b i ) + k ∑ 4 w k 2 min i Z ( b i ) max i Z ( b i ) ν b i ( B k 2 ) ) . Proof. Write E ( b ) = e Re Tr ( U b ) = ∏ k e c k b k E(b)=e^{\operatorname{Re}\operatorname{Tr}(Ub)}=\prod_k e^{c_kb_k} E ( b ) = e Re Tr ( U b ) = ∏ k e c k b k
c = ( Re u 11 , Re u 22 , Re u 33 , − Im u 33 ) , c=(\operatorname{Re}u_{11},\operatorname{Re}u_{22},\operatorname{Re}u_{33},-\operatorname{Im}u_{33}), c = ( Re u 11 , Re u 22 , Re u 33 , − Im u 33 ) , so that ∣ c k ∣ ≤ 1 |c_k|\leq1 ∣ c k ∣ ≤ 1 c 3 2 + c 4 2 ≤ 1 c_3^2+c_4^2\leq1 c 3 2 + c 4 2 ≤ 1 e c k x e^{c_kx} e c k x [ l k , l k + w k ] [l_k,l_k+w_k] [ l k , l k + w k ] [21 ] for a two-point variable, below e c k x e c k 2 w k 2 / 8 e^{c_kx}e^{c_k^2w_k^2/8} e c k x e c k 2 w k 2 /8
E ( b ) ≤ ∑ i ω i ( b ) E ( b i ) ≤ E ( b ) e ∑ k c k 2 w k 2 / 8 ≤ E ( b ) e ς / 8 . (12) E(b)\leq\sum_i\omega_i(b)E(b_i)\leq E(b)e^{\sum_k c_k^2w_k^2/8}\leq E(b)e^{\varsigma/8}. \tag*{(12)} E ( b ) ≤ i ∑ ω i ( b ) E ( b i ) ≤ E ( b ) e ∑ k c k 2 w k 2 /8 ≤ E ( b ) e ς /8 . (12) Integrating the right inequality gives (i), and (ii) follows from the left one and (i). For (iii) let F ( b ) = Y b E ( b ) \mathcal{F}(b)=Y_bE(b) F ( b ) = Y b E ( b ) I k I_k I k k k k F − I 1 I 2 I 3 I 4 F = ∑ k I 1 ⋯ I k − 1 ( F − I k F ) \mathcal{F}-I_1I_2I_3I_4\mathcal{F}=\sum_k I_1\cdots I_{k-1}(\mathcal{F}-I_k\mathcal{F}) F − I 1 I 2 I 3 I 4 F = ∑ k I 1 ⋯ I k − 1 ( F − I k F ) k k k F \mathcal{F} F ℓ 2 e c k x \ell^2e^{c_kx} ℓ 2 e c k x ℓ \ell ℓ B k \mathcal{B}_k B k ∂ k 2 F = ( ( c k ℓ + 2 B k ) 2 − 2 B k 2 ) E ≥ − 2 B k 2 E \partial_k^2\mathcal{F}=((c_k\ell+2\mathcal{B}_k)^2-2\mathcal{B}_k^2)E\ge-2\mathcal{B}_k^2E ∂ k 2 F = (( c k ℓ + 2 B k ) 2 − 2 B k 2 ) E ≥ − 2 B k 2 E w k 2 / 8 w_k^2/8 w k 2 /8
F − I k F ≤ 2 B k 2 ∫ G E ≤ w k 2 4 B k 2 ∑ i ω i ( k ) E ( b i ) \mathcal{F}-I_k\mathcal{F}\le2\mathcal{B}_k^2\int GE\le\frac{w_k^2}{4}\mathcal{B}_k^2\sum_i\omega_i^{(k)}E(b_i) F − I k F ≤ 2 B k 2 ∫ GE ≤ 4 w k 2 B k 2 i ∑ ω i ( k ) E ( b i ) by the left inequality of (12) , with weights ω i ( k ) ≥ 0 \omega_i^{(k)}\ge0 ω i ( k ) ≥ 0 U U U I j I_j I j U U U Z ( b ) Z(b) Z ( b )
The cover and the theorem
Fix a box Q Q Q C C C κ \kappa κ Var ( X h ) ≤ ν b [ ( X h − κ ) 2 ] \operatorname{Var}(X_h)\le\nu_b[(X_h-\kappa)^2] Var ( X h ) ≤ ν b [( X h − κ ) 2 ] κ = ⟨ κ ( b ) , h ⟩ \kappa=\langle\kappa(b),h\rangle κ = ⟨ κ ( b ) , h ⟩ κ ( b ) \kappa(b) κ ( b ) b b b (11) , X h − κ = A + ∑ k ( b k − β k ) B k X_h-\kappa=\mathcal{A}+\sum_k(b_k-\beta_k)\mathcal{B}_k X h − κ = A + ∑ k ( b k − β k ) B k β \beta β Q Q Q B k = ⟨ ∇ φ F k , ∇ q h ⟩ − ⟨ ∂ k κ , h ⟩ \mathcal{B}_k=\langle\nabla\varphi_{F_k},\nabla q_h\rangle-\langle\partial_k\kappa,h\rangle B k = ⟨ ∇ φ F k , ∇ q h ⟩ − ⟨ ∂ k κ , h ⟩ b ∈ Q ∩ R r b\in Q\cap\mathcal{R}_r b ∈ Q ∩ R r
sup h L h ( b ) ≤ ε s / 8 max i λ A ( b i ) + ε s / 8 γ ( r ) ( max i λ V ( b i ) + e N min i Z ( b i ) ) (13) \sup_h L_h(b)\le\sqrt{\varepsilon^{s/8}\max_i\lambda_A(b_i)}
+\sqrt{\frac{\varepsilon^{s/8}}{\gamma(r)}\left(\max_i\lambda_V(b_i)+\frac{e_N}{\min_i Z(b_i)}\right)} \tag*{(13)} h sup L h ( b ) ≤ ε s /8 i max λ A ( b i ) + γ ( r ) ε s /8 ( i max λ V ( b i ) + min i Z ( b i ) e N ) (13) where e N = ∑ k w k 2 4 max i Z ( b i ) λ k ( b i ) e_N=\sum_k\frac{w_k^2}{4}\max_i Z(b_i)\lambda_k(b_i) e N = ∑ k 4 w k 2 max i Z ( b i ) λ k ( b i ) λ A \lambda_A λ A λ V \lambda_V λ V λ k \lambda_k λ k h ↦ ν b i ∣ ∇ q h ∣ 2 h\mapsto\nu_{b_i}|\nabla q_h|^2 h ↦ ν b i ∣∇ q h ∣ 2 h ↦ ν b i [ ( X h − κ ) 2 ] h\mapsto\nu_{b_i}[(X_h-\kappa)^2] h ↦ ν b i [( X h − κ ) 2 ] b = b i b=b_i b = b i h ↦ ν b i [ B 2 ] h\mapsto\nu_{b_i}[\mathcal{B}^2] h ↦ ν b i [ B 2 ] R 18 \mathbb{R}^{18} R 18 ν b i \nu_{b_i} ν b i § 16 ); each largest eigenvalue is bounded above by an L D L T LDL^{\mathsf{T}} L D L T t I tI t I [19 , 23 ] .
The matrix C C C b = β b=\beta b = β u = q h u=q_h u = q h C C C θ > 0 \theta>0 θ > 0 ν ∣ ∇ q ∣ 2 + θ Var ( X ) / γ \nu|\nabla q|^2+\theta\operatorname{Var}(X)/\gamma ν ∣∇ q ∣ 2 + θ Var ( X ) / γ h h h θ \theta θ C C C 2 − 30 2^{-30} 2 − 30
A box is discarded when it contains no point with a 1 ≥ a 2 ≥ ∣ z 3 ∣ a_1\ge a_2\ge|z_3| a 1 ≥ a 2 ≥ ∣ z 3 ∣ ℓ ( b ) = Re Tr ( c ∗ b ) \ell(b)=\operatorname{Re}\operatorname{Tr}(c^*b) ℓ ( b ) = Re Tr ( c ∗ b ) min Q ℓ > r max T ℓ \min_Q\ell>r\max_{\mathcal{T}}\ell min Q ℓ > r max T ℓ r conv T r\operatorname{conv}\mathcal{T} r conv T R r \mathcal{R}_r R r
Proposition 7.8. 2 r Λ ( r ) ≤ 0.999887 2r\Lambda(r)\le0.999887 2 r Λ ( r ) ≤ 0.999887 r = 3 2 r=\frac{3}{2} r = 2 3
Proof. The box [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] [0,r]^2\times[-r,r]\times[0,r] [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] r / 8 r/8 r /8 B Q B_Q B Q (13) , with γ ( r ) = 187 128 \gamma(r)=\frac{187}{128} γ ( r ) = 128 187 2 r B Q < 1 2rB_Q<1 2 r B Q < 1 R r \mathcal{R}_r R r [ 1.3125 , 1.5 ] × [ 0.9375 , 1.125 ] × [ 0.5625 , 0.75 ] × [ 0 , 0.1875 ] [1.3125,1.5]\times[0.9375,1.125]\times[0.5625,0.75]\times[0,0.1875] [ 1.3125 , 1.5 ] × [ 0.9375 , 1.125 ] × [ 0.5625 , 0.75 ] × [ 0 , 0.1875 ] Figure 3 shows the cover, and Appendix § 16 the error analysis. □
Figure 3. The 2341 certified boxes at r = 3 2 r = \frac{3}{2} r = 2 3 ( a 1 , a 2 ) (a_1,a_2) ( a 1 , a 2 ) diag ( a 1 , a 2 , x 3 + i y 3 ) \operatorname{diag}(a_1,a_2,x_3+\mathrm{i}y_3) diag ( a 1 , a 2 , x 3 + i y 3 ) 2 r sup h L h 2r\sup_h L_h 2 r sup h L h r / 8 r/8 r /8 ∣ x 3 + i y 3 ∣ \lvert x_3+\mathrm{i}y_3\rvert ∣ x 3 + i y 3 ∣ x 3 x_3 x 3 R r \mathcal{R}_r R r
The same cover at a larger r r r Λ ( r ) \Lambda(r) Λ ( r ) 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r )
Proposition 7.9. 2 r Λ ( r ) ≤ 1.029967 2r\Lambda(r)\le1.029967 2 r Λ ( r ) ≤ 1.029967 r = 38 25 r=\frac{38}{25} r = 25 38
Proof. As in the proof of Proposition 7.8, with γ ( r ) = 4643 3200 \gamma(r)=\frac{4643}{3200} γ ( r ) = 3200 4643 r / 8 r/8 r /8 R r \mathcal{R}_r R r
Proof of Theorem 1.5. Proposition 7.8 gives
Λ ( 3 2 ) ≤ 0.999887 3 < 0.333296. \Lambda\left(\frac{3}{2}\right) \le\frac{0.999887}{3} < 0.333296. Λ ( 2 3 ) ≤ 3 0.999887 < 0.333296. For β W ≤ 3 4 \beta_W \le\frac{3}{4} β W ≤ 4 3 2 β W ≤ 3 2 2\beta_W \le\frac{3}{2} 2 β W ≤ 2 3 (8) with the monotonicity of Λ \Lambda Λ
α ≤ 4 β W Λ ( 3 2 ) < 1.333183 β W ≤ 3 4 ⋅ 1.333183 < 1. \alpha\le4\beta_W\Lambda\left(\frac{3}{2}\right) < 1.333183\beta_W \le\frac{3}{4}\cdot1.333183 < 1. α ≤ 4 β W Λ ( 2 3 ) < 1.333183 β W ≤ 4 3 ⋅ 1.333183 < 1. Proposition 3.2 applies with the Hilbert–Schmidt metric. □
SU(3) with plaquette blocks
In this section G = S U ( 3 ) G = \mathrm{SU}(3) G = SU ( 3 ) β ≥ 0 \beta\ge0 β ≥ 0 β W = 9 β \beta_W = 9\beta β W = 9 β N β = β W / 3 N\beta= \beta_W/3 N β = β W /3 § 8 : d d d K = conv S U ( 3 ) K = \operatorname{conv}\mathrm{SU}(3) K = conv SU ( 3 ) ν c ( d U ) ∝ e Re Tr ( U c ) d U \nu_c(dU) \propto e^{\operatorname{Re}\operatorname{Tr}(U_c)}dU ν c ( d U ) ∝ e Re Tr ( U c ) d U φ h ( U ) = Re Tr ( U h ) \varphi_h(U) = \operatorname{Re}\operatorname{Tr}(Uh) φ h ( U ) = Re Tr ( U h ) Λ ( r ) \Lambda(r) Λ ( r ) (8) . Put Z ( c ) = ∫ e Re Tr ( U c ) d U Z(c) = \int e^{\operatorname{Re}\operatorname{Tr}(U_c)}dU Z ( c ) = ∫ e Re Tr ( U c ) d U g = log Z g = \log Z g = log Z C 3 × 3 \mathbb{C}^{3 \times3} C 3 × 3 R 18 \mathbb{R}^{18} R 18 ⟨ x , y ⟩ = Re Tr ( x y ∗ ) \langle x,y\rangle= \operatorname{Re}\operatorname{Tr}(xy^*) ⟨ x , y ⟩ = Re Tr ( x y ∗ ) g g g c c c m ( c ) ∗ m(c)^* m ( c ) ∗ m ( c ) = ν c ( U ) ∈ K m(c) = \nu_c(U) \in K m ( c ) = ν c ( U ) ∈ K Hess g ( c ) ( h , h ) = Var ν c ( φ h ) \operatorname{Hess} g(c)(h,h) = \operatorname{Var}_{\nu_c}(\varphi_h) Hess g ( c ) ( h , h ) = Var ν c ( φ h ) Z ( V c W ) = Z ( c ) Z(VcW) = Z(c) Z ( V c W ) = Z ( c ) V , W ∈ S U ( 3 ) V,W \in\mathrm{SU}(3) V , W ∈ SU ( 3 ) g g g
The blocks are those of Lemma 5.1, which concerns the line field only and holds for every group. For a block plaquette with active links e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 Re Tr Q p = Re Tr ( y 1 y 2 y 3 ) \operatorname{Re}\operatorname{Tr} Q_p = \operatorname{Re}\operatorname{Tr}(y_1y_2y_3) Re Tr Q p = Re Tr ( y 1 y 2 y 3 ) y i = Q e i y_i = Q_{e_i} y i = Q e i Q e i − 1 Q_{e_i}^{-1} Q e i − 1
μ b ( d y ) ∝ exp ( ∑ i Re Tr ( y i b i ) + N β Re Tr ( y 1 y 2 y 3 ) ) d y . (14) \mu_b(dy) \propto\exp\left(\sum_i \operatorname{Re}\operatorname{Tr}(y_i b_i) + N\beta\operatorname{Re}\operatorname{Tr}(y_1y_2y_3)\right)dy. \tag*{(14)} μ b ( d y ) ∝ exp ( i ∑ Re Tr ( y i b i ) + N β Re Tr ( y 1 y 2 y 3 ) ) d y . (14) where b i b_i b i N β N\beta N β y i y_i y i b i ∈ 5 N β K b_i \in5N\beta K b i ∈ 5 N β K ∥ b i ∥ op ≤ 5 N β \lVert b_i\rVert_{\operatorname{op}} \le5N\beta ∥ b i ∥ op ≤ 5 N β S U ( 3 ) 3 \mathrm{SU}(3)^3 SU ( 3 ) 3 d ( y , y ′ ) = ∑ i ∥ y i − y i ′ ∥ H S d(y,y') = \sum_i \lVert y_i-y_i'\rVert_{\mathrm{HS}} d ( y , y ′ ) = ∑ i ∥ y i − y i ′ ∥ HS
The sensitivity of one block
Let r = 2 β W = 6 N β r=2\beta W=6N\beta r = 2 β W = 6 N β λ ∗ \lambda_* λ ∗ τ \tau τ g g g r K rK r K M M M r K rK r K ∣ D 3 g ( c ) ( u , v , v ) ∣ \lvert D^3g(c)(u,v,v)\rvert ∣ D 3 g ( c ) ( u , v , v )∣ u , v u,v u , v γ = γ ( r ) \gamma=\gamma(r) γ = γ ( r ) Lemma 7.3 . Put
w = 3 2 ( N β ) 2 ( τ − λ ∗ ) , s = Λ ( r ) + ( e w − 1 ) e w τ γ , w=\frac{3}{2}(N\beta)^2(\tau-\lambda_*),\qquad s=\Lambda(r)+(e^w-1)e^w\sqrt{\frac{\tau}{\gamma}}, w = 2 3 ( N β ) 2 ( τ − λ ∗ ) , s = Λ ( r ) + ( e w − 1 ) e w γ τ , C = 5 ( N β ) 2 ( τ s + 3 N β M e w γ ) , (15) C=5(N\beta)^2\left(\tau s+\frac{\sqrt{3}N\beta M e^w}{\gamma}\right), \tag*{(15)} C = 5 ( N β ) 2 ( τ s + γ 3 N β M e w ) , (15) K 3 = ( 1 + N β Λ ( r ) ) s 1 − C , α 3 = 10 N β K 3 , K_3=\frac{(1+N\beta\Lambda(r))s}{1-C},\qquad\alpha_3=10N\beta K_3, K 3 = 1 − C ( 1 + N β Λ ( r )) s , α 3 = 10 N β K 3 , assuming C < 1 C<1 C < 1
Lemma 8.1. Let a , c 0 ∈ 5 N β K a,c_0\in5N\beta K a , c 0 ∈ 5 N β K
κ ( d x ) ∝ exp ( Re Tr ( x a ) + g ( c 0 + N β x ) ) d x on S U ( 3 ) . \kappa(dx)\propto\exp(\operatorname{Re}\operatorname{Tr}(xa)+g(c_0+N\beta x))dx\quad\text{on }\mathrm{SU}(3). κ ( d x ) ∝ exp ( Re Tr ( x a ) + g ( c 0 + N β x )) d x on SU ( 3 ) . For every 1 1 1 f f f h ∈ C 3 × 3 h\in\mathbb{C}^{3\times3} h ∈ C 3 × 3 Cov κ ( f , ϕ h ) ≤ s ∥ h ∥ H S \operatorname{Cov}_{\kappa}(f,\phi_h)\le s\lVert h\rVert_{\mathrm{HS}} Cov κ ( f , ϕ h ) ≤ s ∥ h ∥ HS c 0 c_0 c 0 k k k 5 N β K 5N\beta K 5 N β K
∣ d d t κ ( f ) ∣ ≤ N β ( τ s + 3 N β M e w γ ) ∥ k ∥ H S . \left\lvert\frac{d}{dt}\kappa(f)\right\rvert\le N\beta\left(\tau s+\frac{\sqrt{3}N\beta M e^w}{\gamma}\right)\lVert k\rVert_{\mathrm{HS}}. d t d κ ( f ) ≤ N β ( τ s + γ 3 N β M e w ) ∥ k ∥ HS . Proof. (i) The segment from c 0 c_0 c 0 c 0 + N β x c_0+N\beta x c 0 + N β x 6 N β K = r K 6N\beta K=rK 6 N β K = r K
g ( c 0 + N β x ) = g ( c 0 ) + N β Re Tr ( x m ( c 0 ) ) + R ( x ) g(c_0+N\beta x)=g(c_0)+N\beta\operatorname{Re}\operatorname{Tr}(xm(c_0))+R(x) g ( c 0 + N β x ) = g ( c 0 ) + N β Re Tr ( x m ( c 0 )) + R ( x ) with R ( x ) = 1 2 ( N β ) 2 Hess g ( ξ ) ( x , x ) R(x)=\frac{1}{2}(N\beta)^2\operatorname{Hess}g(\xi)(x,x) R ( x ) = 2 1 ( N β ) 2 Hess g ( ξ ) ( x , x ) ∥ x ∥ H S 2 = 3 \lVert x\rVert_{\mathrm{HS}}^2=3 ∥ x ∥ HS 2 = 3 R ∈ [ 3 2 ( N β ) 2 λ ∗ , 3 2 ( N β ) 2 τ ] R\in[\frac{3}{2}(N\beta)^2\lambda_*,\frac{3}{2}(N\beta)^2\tau] R ∈ [ 2 3 ( N β ) 2 λ ∗ , 2 3 ( N β ) 2 τ ] osc R ≤ w \operatorname{osc}R\le w osc R ≤ w κ \kappa κ e R ν / ν ( e R ) e^R\nu/\nu(e^R) e R ν / ν ( e R ) ν = ν c ′ \nu=\nu_{c'} ν = ν c ′ c ′ = a + N β m ( c 0 ) ∈ r K c'=a+N\beta m(c_0)\in rK c ′ = a + N β m ( c 0 ) ∈ r K Λ \Lambda Λ Cov ν ( f , ϕ h ) ≤ Λ ( r ) ∥ h ∥ \operatorname{Cov}_{\nu}(f,\phi_h)\le\Lambda(r)\lVert h\rVert Cov ν ( f , ϕ h ) ≤ Λ ( r ) ∥ h ∥ f f f 1 1 1 d d d sd ν f ≤ γ − 1 / 2 \operatorname{sd}_{\nu}f\le\gamma^{-1/2} sd ν f ≤ γ − 1/2 Lemma 7.3 , and sd ν ϕ h ≤ τ ∥ h ∥ \operatorname{sd}_{\nu}\phi_h\le\sqrt{\tau}\lVert h\rVert sd ν ϕ h ≤ τ ∥ h ∥ Lemma 5.2 gives (i).
(ii) The derivative is Cov κ ( f , ⟨ ∇ g ( c 0 + N β x ) , k ⟩ ) \operatorname{Cov}_{\kappa}(f,\langle\nabla g(c_0+N\beta x),k\rangle) Cov κ ( f , ⟨ ∇ g ( c 0 + N β x ) , k ⟩) ∇ g ( c 0 + N β u ) = ∇ g ( c 0 ) + N β Hess g ( c 0 ) u + N ( u ) \nabla g(c_0+N\beta u)=\nabla g(c_0)+N\beta\operatorname{Hess}g(c_0)u+N(u) ∇ g ( c 0 + N β u ) = ∇ g ( c 0 ) + N β Hess g ( c 0 ) u + N ( u ) ϕ h \phi_h ϕ h ∥ h ∥ ≤ N β τ ∥ k ∥ \lVert h\rVert\le N\beta\tau\lVert k\rVert ∥ h ∥ ≤ N β τ ∥ k ∥ N β τ s ∥ k ∥ N\beta\tau s\lVert k\rVert N β τ s ∥ k ∥ c 0 c_0 c 0 N β M ∥ u ∥ ≤ 3 N β M N\beta M\lVert u\rVert\le\sqrt{3}N\beta M N β M ∥ u ∥ ≤ 3 N β M x ↦ ⟨ N ( x ) , k ⟩ x\mapsto\langle N(x),k\rangle x ↦ ⟨ N ( x ) , k ⟩ 3 ( N β ) 2 M ∥ k ∥ \sqrt{3}(N\beta)^2M\lVert k\rVert 3 ( N β ) 2 M ∥ k ∥ d d d κ \kappa κ ν \nu ν w w w e − w γ e^{-w}\gamma e − w γ [20 ] , and the covariance of two Lipschitz functions under κ \kappa κ e w / γ e^w/\gamma e w / γ
:::
Lemma 8.2. Let C < 1 C<1 C < 1 b , b ′ b,b' b , b ′ 5 N β K 5N\beta K 5 N β K b i b_i b i W 1 ( μ b , μ b ′ ) ≤ K 3 ∥ b i − b i ′ ∥ H S W_1(\mu_b,\mu_{b'})\le K_3\lVert b_i-b_i'\rVert_{\mathrm{HS}} W 1 ( μ b , μ b ′ ) ≤ K 3 ∥ b i − b i ′ ∥ HS d d d
Proof. By the cyclic symmetry of Re Tr ( y 1 y 2 y 3 ) \operatorname{Re}\operatorname{Tr}(y_1y_2y_3) Re Tr ( y 1 y 2 y 3 ) i = 1 i=1 i = 1 y 1 , y 3 y_1,y_3 y 1 , y 3 y 2 y_2 y 2 ν c \nu_c ν c c = b 2 + N β y 3 y 1 ∈ r K c=b_2+N\beta y_3y_1\in rK c = b 2 + N β y 3 y 1 ∈ r K y 1 y_1 y 1 η \eta η d d d c c c N β η N\beta\eta N β η ν c \nu_c ν c N β Λ ( r ) η N\beta\Lambda(r)\eta N β Λ ( r ) η W 1 W_1 W 1 (8) ; so F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] F(y_1,y_3)=\mathbb{E}[f\mid y_1,y_3] F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] ( 1 + N β Λ ( r ) ) (1+N\beta\Lambda(r)) ( 1 + N β Λ ( r )) ( y 1 , y 3 ) (y_1,y_3) ( y 1 , y 3 ) e Re Tr ( y 1 b 1 ) + Re Tr ( y 3 b 3 ) Z ( b 2 + N β y 3 y 1 ) e^{\operatorname{Re}\operatorname{Tr}(y_1b_1)+\operatorname{Re}\operatorname{Tr}(y_3b_3)}Z(b_2+N\beta y_3y_1) e Re Tr ( y 1 b 1 ) + Re Tr ( y 3 b 3 ) Z ( b 2 + N β y 3 y 1 ) Z ( b 2 + N β y 3 y 1 ) = Z ( y 3 − 1 b 2 + N β y 1 ) = Z ( b 2 y 1 − 1 + N β y 3 ) Z(b_2+N\beta y_3y_1)=Z(y_3^{-1}b_2+N\beta y_1)=Z(b_2y_1^{-1}+N\beta y_3) Z ( b 2 + N β y 3 y 1 ) = Z ( y 3 − 1 b 2 + N β y 1 ) = Z ( b 2 y 1 − 1 + N β y 3 ) Lemma 8.1 , with c 0 = y 3 − 1 b 2 c_0=y_3^{-1}b_2 c 0 = y 3 − 1 b 2 c 0 = b 2 y 1 − 1 c_0=b_2y_1^{-1} c 0 = b 2 y 1 − 1 y 3 y_3 y 3 y 3 ′ y_3' y 3 ′ y 3 − 1 b 2 y_3^{-1}b_2 y 3 − 1 b 2 5 N β K 5N\beta K 5 N β K ∥ b 2 ∥ o p ∥ y 3 − y 3 ′ ∥ ≤ 5 N β ∥ y 3 − y 3 ′ ∥ \lVert b_2\rVert_{\mathrm{op}}\lVert y_3-y_3'\rVert\le5N\beta\lVert y_3-y_3'\rVert ∥ b 2 ∥ op ∥ y 3 − y 3 ′ ∥ ≤ 5 N β ∥ y 3 − y 3 ′ ∥ Lemma 5.4 now gives d 1 ≤ s ϵ + C d 3 d_1\le s\epsilon+Cd_3 d 1 ≤ sϵ + C d 3 d 3 ≤ C d 1 d_3\le Cd_1 d 3 ≤ C d 1 ∣ μ b ( f ) − μ b ′ ( f ) ∣ ≤ ( 1 + N β Λ ( r ) ) ( d 1 + d 3 ) ≤ K 3 ϵ \lvert\mu_b(f)-\mu_{b'}(f)\rvert\le(1+N\beta\Lambda(r))(d_1+d_3)\le K_3\epsilon ∣ μ b ( f ) − μ b ′ ( f )∣ ≤ ( 1 + N β Λ ( r )) ( d 1 + d 3 ) ≤ K 3 ϵ
The constants and the proof of Theorem 1.6
Lemma 8.3. At r = 3 2 r=\frac{3}{2} r = 2 3 g g g r K rK r K [ 0.056015 , 0.247316 ] [0.056015,0.247316] [ 0.056015 , 0.247316 ] ∣ D 3 g ( c ) ( u , v , v ) ∣ ≤ 0.182126 \lvert D^3g(c)(u,v,v)\rvert\le0.182126 ∣ D 3 g ( c ) ( u , v , v )∣ ≤ 0.182126 c ∈ r K c\in rK c ∈ r K u , v u,v u , v r = 38 25 r=\frac{38}{25} r = 25 38 [ 0.054978 , 0.248280 ] [0.054978,0.248280] [ 0.054978 , 0.248280 ] 0.183546 0.183546 0.183546
Proof. For V , W ∈ S U ( 3 ) V,W\in\mathrm{SU}(3) V , W ∈ SU ( 3 ) g ( V c W ) = g ( c ) g(VcW)=g(c) g ( V c W ) = g ( c ) x ↦ V x W x\mapsto VxW x ↦ V x W Hess g \operatorname{Hess}g Hess g D 3 g D^{3}g D 3 g R r \mathcal{R}_{r} R r g g g φ \varphi φ ν c \nu_{c} ν c [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] [0,r]^{2}\times[-r,r]\times[0,r] [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] r / 8 r/8 r /8 r = 38 25 r=\frac{38}{25} r = 25 38 § 8.7 . On each box Q Q Q φ \varphi φ b i b_{i} b i b 0 b_{0} b 0 k = m ( b 0 ) k=m(b_{0}) k = m ( b 0 ) φ \varphi φ
Upper bound. For b ∈ Q b\in Q b ∈ Q h h h Var ν b ( φ h ) ≤ ν b [ ( φ h − ⟨ k , h ⟩ ) 2 ] ≤ e ς / 8 max i ν b i [ ( φ h − ⟨ k , h ⟩ ) 2 ] \operatorname{Var}_{\nu_{b}}(\varphi_{h})\leq\nu_{b}[(\varphi_{h}-\langle k,h\rangle)^{2}]\leq e^{\varsigma/8}\max_{i}\nu_{b_{i}}[(\varphi_{h}-\langle k,h\rangle)^{2}] Var ν b ( φ h ) ≤ ν b [( φ h − ⟨ k , h ⟩ ) 2 ] ≤ e ς /8 max i ν b i [( φ h − ⟨ k , h ⟩ ) 2 ] Lemma 7.7 (ii).
Lower bound. Integrating the right inequality of (12) against f ≥ 0 f\geq0 f ≥ 0 Z ( b ) ≤ ∑ i ω i Z ( b i ) Z(b)\leq\sum_{i}\omega_{i}Z(b_{i}) Z ( b ) ≤ ∑ i ω i Z ( b i ) ν b ( f ) ≥ e − ς / 8 min i ν b i ( f ) \nu_{b}(f)\geq e^{-\varsigma/8}\min_{i}\nu_{b_{i}}(f) ν b ( f ) ≥ e − ς /8 min i ν b i ( f ) f = ( φ h − ν b φ h ) 2 f=(\varphi_{h}-\nu_{b}\varphi_{h})^{2} f = ( φ h − ν b φ h ) 2 Var ν b ( φ h ) ≥ e − ς / 8 min i Var ν b i ( φ h ) \operatorname{Var}_{\nu_{b}}(\varphi_{h})\geq e^{-\varsigma/8}\min_{i}\operatorname{Var}_{\nu_{b_{i}}}(\varphi_{h}) Var ν b ( φ h ) ≥ e − ς /8 min i Var ν b i ( φ h )
Third cumulant. Put δ = m ( b ) − k \delta=m(b)-k δ = m ( b ) − k ∥ δ ∥ ≤ τ ρ + ϵ \|\delta\|\leq\tau\rho+\epsilon ∥ δ ∥ ≤ τ ρ + ϵ ρ \rho ρ b 0 b_{0} b 0 Q Q Q ϵ \epsilon ϵ k k k
D 3 g ( b ) ( u , v , v ) = T b ( u , v , v ) − δ u Var ( φ v ) − 2 δ v Cov ( φ u , φ v ) − δ u δ v 2 , D^{3}g(b)(u,v,v)=T_{b}(u,v,v)-\delta_{u}\operatorname{Var}(\varphi_{v})
-2\delta_{v}\operatorname{Cov}(\varphi_{u},\varphi_{v})-\delta_{u}\delta_{v}^{2}, D 3 g ( b ) ( u , v , v ) = T b ( u , v , v ) − δ u Var ( φ v ) − 2 δ v Cov ( φ u , φ v ) − δ u δ v 2 , with T b = ν b [ ( φ − k ) ⊗ 3 ] T_{b}=\nu_{b}[(\varphi-k)^{\otimes3}] T b = ν b [( φ − k ) ⊗ 3 ] δ u = ⟨ δ , u ⟩ \delta_{u}=\langle\delta,u\rangle δ u = ⟨ δ , u ⟩ ν b \nu_{b} ν b G G G G = G + − G − G=G^{+}-G^{-} G = G + − G − G ± G^{\pm} G ± ∣ ν b ( G ) ∣ ≤ ( max i ∣ ν b i ( G ) ∣ + ϵ max i ν b i ∣ G ∣ ) / ( 1 − ϵ ) |\nu_{b}(G)|\leq(\max_{i}|\nu_{b_{i}}(G)|+\epsilon\max_{i}\nu_{b_{i}}|G|)/(1-\epsilon) ∣ ν b ( G ) ∣ ≤ ( max i ∣ ν b i ( G ) ∣ + ϵ max i ν b i ∣ G ∣ ) / ( 1 − ϵ ) ϵ = 1 − e − ς / 8 \epsilon=1-e^{-\varsigma/8} ϵ = 1 − e − ς /8 G = ( φ u − k u ) ( φ v − k v ) 2 G=(\varphi_{u}-k_{u})(\varphi_{v}-k_{v})^{2} G = ( φ u − k u ) ( φ v − k v ) 2 ∣ ν b i ( G ) ∣ ≤ ∥ T b i ∥ |\nu_{b_{i}}(G)|\leq\|T_{b_{i}}\| ∣ ν b i ( G ) ∣ ≤ ∥ T b i ∥ 18 × 324 18\times324 18 × 324 T b i T_{b_{i}} T b i ν b i ∣ G ∣ ≤ 2 3 ν b i [ ( φ v − k v ) 2 ] \nu_{b_{i}}|G|\leq2\sqrt{3}\,\nu_{b_{i}}[(\varphi_{v}-k_{v})^{2}] ν b i ∣ G ∣ ≤ 2 3 ν b i [( φ v − k v ) 2 ] ∥ U − k ∥ H S ≤ 2 3 \|U-k\|_{\mathrm{HS}}\leq2\sqrt{3} ∥ U − k ∥ HS ≤ 2 3 § 8.7 , and the maxima over these boxes are the numbers stated (Appendix C).
Proof of Theorem 1.6. Let first β W ≤ 3 4 \beta_{W}\leq\frac{3}{4} β W ≤ 4 3 r = 2 β W ≤ 3 2 r=2\beta_{W}\leq\frac{3}{2} r = 2 β W ≤ 2 3 Lemmas 8.1 and 8.2 lies in 3 2 K \frac{3}{2}K 2 3 K Lemma 8.3 at 3 2 \frac{3}{2} 2 3 Λ ( 3 2 ) ≤ 0.999887 3 \Lambda(\frac{3}{2})\leq\frac{0.999887}{3} Λ ( 2 3 ) ≤ 3 0.999887 Proposition 7.8 and γ ( 3 2 ) = 187 128 \gamma(\frac{3}{2})=\frac{187}{128} γ ( 2 3 ) = 128 187 (15) is increasing in β W \beta_{W} β W 3 4 < β W ≤ 19 25 \frac{3}{4}<\beta_{W}\leq\frac{19}{25} 4 3 < β W ≤ 25 19 38 25 K \frac{38}{25}K 25 38 K Lemma 8.3 at 38 25 \frac{38}{25} 25 38 Λ ( 38 25 ) ≤ 1.029967 3.04 \Lambda(\frac{38}{25})\leq\frac{1.029967}{3.04} Λ ( 25 38 ) ≤ 3.04 1.029967 Proposition 7.9 and γ ( 38 25 ) = 4643 3200 \gamma(\frac{38}{25})=\frac{4643}{3200} γ ( 25 38 ) = 3200 4643 j j j η \eta η η \eta η b i b_{i} b i N β η N\beta\eta N β η Lemma 8.2 gives ∑ B C B j ≤ 10 N β K 3 = α 3 ( β W ) \sum_{B}C_{Bj}\leq10N\beta K_{3}=\alpha_{3}(\beta_{W}) ∑ B C B j ≤ 10 N β K 3 = α 3 ( β W ) β W = 3 4 \beta_{W}=\frac{3}{4} β W = 4 3 w ≤ 0.017935 w\leq0.017935 w ≤ 0.017935 s ≤ 0.34088 s\leq0.34088 s ≤ 0.34088 C ≤ 0.043520 C\leq0.043520 C ≤ 0.043520 K 3 ≤ 0.38609 K_{3}\leq0.38609 K 3 ≤ 0.38609 α 3 ≤ 0.96521 \alpha_{3}\leq0.96521 α 3 ≤ 0.96521 β W = 0.72 \beta_{W}=0.72 β W = 0.72 α 3 ≤ 0.91815 \alpha_{3}\leq0.91815 α 3 ≤ 0.91815 β W = 0.76 \beta_{W}=0.76 β W = 0.76 w ≤ 0.018609 w\leq0.018609 w ≤ 0.018609 s ≤ 0.34673 s\leq0.34673 s ≤ 0.34673 C ≤ 0.045770 C\leq0.045770 C ≤ 0.045770 K 3 ≤ 0.39454 K_{3}\leq0.39454 K 3 ≤ 0.39454 α 3 ≤ 0.99950 \alpha_{3}\leq0.99950 α 3 ≤ 0.99950 Proposition 5.6 , whose proof does not use the group, applies with the Hilbert–Schmidt metric.
Remark 8.4. Theorem 1.5 ends where 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r ) 12 10 \frac{12}{10} 10 12 β W = 3 4 \beta_{W}=\frac{3}{4} β W = 4 3 1.1 ⋅ 10 − 4 1.1\cdot10^{-4} 1.1 ⋅ 1 0 − 4 0.0177 0.0177 0.0177 β W ≥ 0.715 \beta_{W}\geq0.715 β W ≥ 0.715 10 12 \frac{10}{12} 12 10 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r ) 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r ) Proposition 7.9 certifies 2 r Λ ( r ) ≤ 1.03 2r\Lambda(r)\leq1.03 2 r Λ ( r ) ≤ 1.03 r = 1.52 r=1.52 r = 1.52 β W = 0.76 \beta_{W}=0.76 β W = 0.76 R r \mathcal{R}_{r} R r β W ≈ 0.78 \beta_{W}\approx0.78 β W ≈ 0.78
Where the methods stop
S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) Theorem 1.2 closes the single-link route in every forest gauge at β W = 1 3 \beta_{W}=\frac{1}{3} β W = 3 1 [16 ] , pass it: plaquette blocks reach
Figure 4. The floating-point value of 2 r sup h 2r\sup_h 2 r sup h Lemma 7.4 on the face b = diag ( r , ρ , ρ ) b=\operatorname{diag}(r,\rho,\rho) b = diag ( r , ρ , ρ ) ρ / r \rho/r ρ / r r = 1.44 r=1.44 r = 1.44 1.50 1.50 1.50 1.54 1.54 1.54 u = s φ h u=s\varphi_h u = s φ h s s s u = q h u=q_h u = q h C C C r = 1.44 r=1.44 r = 1.44 r = 1.50 r=1.50 r = 1.50 Proposition 7.8 certifies the bound on all of R r \mathcal{R}_r R r r = 1.54 r=1.54 r = 1.54
β W = 0.345 \beta_W=0.345 β W = 0.345 Theorem 1.3 ), and the sharper floating-point bound of Remark 5.7 stops near 0.357 0.357 0.357
Curvature. Remark 6.4 locates the end of the curvature method in this gauge between ∣ β ∣ = 1 29 |\beta|=\frac{1}{29} ∣ β ∣ = 29 1 1 26.25 \frac{1}{26.25} 26.25 1
SU(3). The route of § 8 bounds the H − 1 ( ν b ) H^{-1}(\nu_b) H − 1 ( ν b ) φ h − ν b φ h \varphi_h-\nu_b\varphi_h φ h − ν b φ h L h ( b ) L_h(b) L h ( b ) r K rK r K r K rK r K 0.79 0.79 0.79 r = 1.25 r=1.25 r = 1.25 0.89 0.89 0.89 r = 1.4 r=1.4 r = 1.4 0.97 0.97 0.97 r = 1.5 r=1.5 r = 1.5 1.009 1.009 1.009 r = 1.55 r=1.55 r = 1.55 β W ≈ 0.77 \beta_W\approx0.77 β W ≈ 0.77 u = s φ h u=s\varphi_h u = s φ h Lemma 7.4 stops earlier: in floating point its largest values on R r \mathcal{R}_r R r diag ( r , ρ , ρ ) \operatorname{diag}(r,\rho,\rho) diag ( r , ρ , ρ ) ρ \rho ρ 0.75 r 0.75r 0.75 r 0.8 r 0.8r 0.8 r 2 r 2r 2 r 0.9458 0.9458 0.9458 r = 1.36 r=1.36 r = 1.36 0.9767 0.9767 0.9767 r = 1.40 r=1.40 r = 1.40 1.0080 1.0080 1.0080 r = 1.44 r=1.44 r = 1.44 β W ≈ 0.715 \beta_W\approx0.715 β W ≈ 0.715 u = q h u=q_h u = q h 0.9305 0.9305 0.9305 r = 1.44 r=1.44 r = 1.44 0.9729 0.9729 0.9729 r = 1.50 r=1.50 r = 1.50 1.0018 1.0018 1.0018 ({xref}Figure 4), close to the Galerkin values. {xref}Lemma 7.3 therefore costs little at these couplings, and most of the distance between the floating-point value at $r=1.5$ and the certified $0.999887$ of {xref}Proposition 7.8 is the interpolation over boxes. Between $\beta_W=\frac{3}{4}$ and $0.77$ there is room only for smaller boxes. The plaquette blocks of {xref}Theorem 1.6 use $\Lambda(2\beta_W)$ as well, with a little more room: they reach $0.76$ with the certified cover of {xref}Proposition 7.9 and near $0.78$ in floating point ({xref}Remark 8.4`); past that one needs larger blocks, whose sensitivity does not go through the one-link modulus.
The four open problems. Our earlier paper [9 ] listed four problems between the strong-coupling results and the Clay problem [22 ] : (1) uniqueness and a gap at intermediate fixed coupling; (2) the crossover from strong coupling to Bałaban’s weak-coupling regime; (3) the continuum limit with the Osterwalder–Schrader axioms; and (4) a gap in physical units. The present paper moves the boundary of (1): for SU(2) from β W = 2 9 \beta_W=\frac{2}{9} β W = 9 2 0.345 0.345 0.345 9 32 \frac{9}{32} 32 9 0.76 0.76 0.76 3 4 \frac{3}{4} 4 3 0.0177 0.0177 0.0177 N N N β W ≈ 2.2 \beta_W \approx2.2 β W ≈ 2.2 β W ≈ 6 \beta_W \approx6 β W ≈ 6 [10 ] none of them can by itself produce a continuum mass gap; that requires control of the gap in physical units along the renormalisation group flow [3 , 11 ] .
Declarations
Data availability. The analytic proofs are complete as written, except the evaluations in Propositions 7.8 and 7.9 and Lemma 8.3 and the numerical bounds of Lemma 5.5 , which are computations with rigorous error bounds, and the finite check of Lemma 5.1 . The archive creelie/4Y-M (release v3.4.0, https://github.com/creelie/4y-m/releases/tag/v3.4.0 ) contains the programs of that computation, in Python with python-flint (FLINT and Arb) and a C kernel for products of Taylor polynomials, with their outputs and the list of certified boxes; the programs and outputs of the earlier certificate at β W = 45 64 \beta_W = \frac{45}{64} β W = 64 45 Lemmas 6.2 and 7.2 and of (10) ; numerical checks of Lemma 6.1 on tori; the searches of Remark 6.4 and § 10 ; a Lean 4 certificate of the line field (Lemmas 2.1 and 2.2 for the table of Appendix A, by decide); the programs of Appendix C for the plaquette blocks, with the floating-point search of Remark 5.7; and the source of the preprint version of this paper. Deep Bhattacharjee carried out the computations and the Lean, Python and shell checks with AI assistance from Claude (Anthropic). The first composite behind Theorem 1.1 was assembled by Anaxia, the reasoning engine of the BLLM system, in release v3.0.5 [7 ] , as described in the footnote to that theorem; release v3.1.0 [8 ] records the engine’s Yang–Mills runs in its failure log, whose last Yang–Mills entry is the change from single links to blocks of several links; the plaquette blocks of § 5 , the bound of Lemma 5.4 , their SU(3) version in § 9 and the certified constants were found and checked by the authors. Both authors approved the final text.
Funding. None.
Competing interests. None.
Appendix A. The line field
Table 2 lists, for each residue class of x x x σ ( x ) \sigma(x) σ ( x ) π x ( μ ) = σ ( x + e μ ) \pi_x(\mu) = \sigma(x + e_\mu) π x ( μ ) = σ ( x + e μ ) π x \pi_x π x σ ( x ) \sigma(x) σ ( x ) Lemma 2.1 .
Appendix B. The comparison lemma
Let ( G , δ ) (G,\delta) ( G , δ ) D D D Ω = G A \Omega= G^A Ω = G A C e e ′ C_{ee'} C e e ′ § 3 , with row and column sums at most α < 1 \alpha< 1 α < 1 C e e ′ = 0 C_{ee'} = 0 C e e ′ = 0 e , e ′ e,e' e , e ′ ∣ e − e ′ ∣ ≤ 1 \lvert e-e'\rvert\le1 ∣ e − e ′ ∣ ≤ 1 λ > 0 \lambda> 0 λ > 0 θ = α e λ < 1 \theta= \alpha e^\lambda< 1 θ = α e λ < 1
Lemma B.1. Let S ⊂ A S \subset A S ⊂ A μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 Ω \Omega Ω γ e F \gamma_e^F γ e F e ∉ S e \notin S e ∈ / S T ⊂ A ∖ S T \subset A \setminus S T ⊂ A ∖ S w b = e − λ dist ( b , T ) w_b = e^{-\lambda\operatorname{dist}(b,T)} w b = e − λ dist ( b , T ) ∑ z ∈ S w z < ∞ \sum_{z\in S} w_z < \infty ∑ z ∈ S w z < ∞ μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 ρ ( b ) = E δ ( σ b , σ b ′ ) \rho(b) = \mathbb{E}\,\delta(\sigma_b,\sigma'_b) ρ ( b ) = E δ ( σ b , σ b ′ )
∑ b ∉ S w b ρ ( b ) ≤ θ D 1 − θ ∑ z ∈ S w z . \sum_{b\notin S} w_b\rho(b) \le\frac{\theta D}{1-\theta}\sum_{z\in S} w_z. b ∈ / S ∑ w b ρ ( b ) ≤ 1 − θ θ D z ∈ S ∑ w z . If S = ∅ S = \varnothing S = ∅ μ 1 = μ 2 \mu_1 = \mu_2 μ 1 = μ 2
Proof. The couplings form a weakly compact set and Ψ ( π ) = ∑ b w b E π δ ( σ b , σ b ′ ) \Psi(\pi) = \sum_b w_b\mathbb{E}_\pi\delta(\sigma_b,\sigma'_b) Ψ ( π ) = ∑ b w b E π δ ( σ b , σ b ′ ) ∑ b w b < ∞ \sum_b w_b < \infty ∑ b w b < ∞ π ∗ \pi^* π ∗ e ∉ S e \notin S e ∈ / S ( σ e , σ e ′ ) (\sigma_e,\sigma'_e) ( σ e , σ e ′ ) γ e F ( ⋅ ∣ σ ) \gamma_e^F(\mathord{\cdot}\mid\sigma) γ e F ( ⋅ ∣ σ ) γ e F ( ⋅ ∣ σ ′ ) \gamma_e^F(\mathord{\cdot}\mid\sigma') γ e F ( ⋅ ∣ σ ′ ) [33 ] , Cor. 5.22]. Both μ i \mu_i μ i γ F \gamma^F γ F W 1 ≤ ∑ e ′ C e e ′ δ ( σ e ′ , σ e ′ ′ ) W_1 \le\sum_{e'} C_{ee'}\delta(\sigma_{e'},\sigma'_{e'}) W 1 ≤ ∑ e ′ C e e ′ δ ( σ e ′ , σ e ′ ′ ) ϱ ( e ) ≤ ∑ e ′ C e e ′ ϱ ( e ′ ) \varrho(e) \le\sum_{e'} C_{ee'}\varrho(e') ϱ ( e ) ≤ ∑ e ′ C e e ′ ϱ ( e ′ ) w e w_e w e e ∉ S e\notin S e ∈ / S C e e ′ ≠ 0 C_{ee'}\ne0 C e e ′ = 0 w e ≤ e λ w e ′ w_e\le e^{\lambda}w_{e'} w e ≤ e λ w e ′
∑ e ∉ S w e ϱ ( e ) ≤ θ [ ∑ e ′ ∉ S w e ′ ϱ ( e ′ ) + ∑ z ∈ S w z ϱ ( z ) ] . \sum_{e\notin S} w_e\varrho(e) \le\theta\left[\sum_{e'\notin S}w_{e'}\varrho(e')+\sum_{z\in S}w_z\varrho(z)\right]. e ∈ / S ∑ w e ϱ ( e ) ≤ θ e ′ ∈ / S ∑ w e ′ ϱ ( e ′ ) + z ∈ S ∑ w z ϱ ( z ) . all sums are finite, and ϱ ≤ D \varrho\le D ϱ ≤ D S S S S = ∅ S=\varnothing S = ∅ ϱ ≡ 0 \varrho\equiv0 ϱ ≡ 0
Corollary B.2. Let μ \mu μ γ F \gamma^F γ F F F F S F S_F S F L F L_F L F H H H S H S_H S H S F S_F S F
∣ Cov μ ( F , H ) ∣ ≤ ∥ H ∥ ∞ L F θ D 1 − θ ∣ S H ∣ e − λ dist ( S F , S H ) . \left|\operatorname{Cov}_{\mu}(F,H)\right|\le\|H\|_{\infty}L_F\frac{\theta D}{1-\theta}|S_H|e^{-\lambda\operatorname{dist}(S_F,S_H)}. ∣ Cov μ ( F , H ) ∣ ≤ ∥ H ∥ ∞ L F 1 − θ θ D ∣ S H ∣ e − λ dist ( S F , S H ) . Proof. Condition on the links of S H S_H S H [24 Ch. 3] ; the conditional measures are invariant under γ F \gamma^F γ F e ∉ S H e\notin S_H e ∈ / S H S = S H S=S_H S = S H T = S F T=S_F T = S F [9 Cor. 29] . □
Lemma B.3. Let the blocks partition A A A C B j C_{Bj} C B j ∑ B C B j ≤ α < 1 \sum_B C_{Bj}\le\alpha<1 ∑ B C B j ≤ α < 1 j j j C B j = 0 C_{Bj}=0 C B j = 0 B B B j j j λ > 0 \lambda>0 λ > 0 θ = α e 2 λ < 1 \theta=\alpha e^{2\lambda}<1 θ = α e 2 λ < 1 S ⊂ A S\subset A S ⊂ A μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 γ B \gamma^B γ B B ⊄ S B\not\subset S B ⊂ S T T T w b w_b w b ∑ z ∈ S w z < ∞ \sum_{z\in S}w_z<\infty ∑ z ∈ S w z < ∞ ∑ b ∉ S w b ϱ ( b ) ≤ θ D 1 − θ ∑ z ∈ S w z \sum_{b\notin S}w_b\varrho(b)\le\frac{\theta D}{1-\theta}\sum_{z\in S}w_z ∑ b ∈ / S w b ϱ ( b ) ≤ 1 − θ θ D ∑ z ∈ S w z μ 1 = μ 2 \mu_1=\mu_2 μ 1 = μ 2 S = ∅ S=\varnothing S = ∅
Proof. Take a minimiser π ∗ \pi^* π ∗ Ψ \Psi Ψ B ⊄ S B\not\subset S B ⊂ S ( σ B , σ B ′ ) (\sigma_B,\sigma'_B) ( σ B , σ B ′ ) d d d d d d ∑ j C B j ϱ ( j ) \sum_j C_{Bj}\varrho(j) ∑ j C B j ϱ ( j ) w ˉ B = max i ∈ B w i \bar w_B=\max_{i\in B}w_i w ˉ B = max i ∈ B w i
∑ i ∈ B w i ϱ ( i ) ≤ w ˉ B ∑ j C B j ϱ ( j ) . \sum_{i\in B}w_i\varrho(i)\le\bar w_B\sum_j C_{Bj}\varrho(j). i ∈ B ∑ w i ϱ ( i ) ≤ w ˉ B j ∑ C B j ϱ ( j ) . and w ˉ B ≤ e 2 λ w j \bar w_B\le e^{2\lambda}w_j w ˉ B ≤ e 2 λ w j C B j ≠ 0 C_{Bj}\ne0 C B j = 0 S S S
∑ b ∉ S w b ϱ ( b ) ≤ θ [ ∑ j ∉ S w j ϱ ( j ) + ∑ z ∈ S w z ϱ ( z ) ] , \sum_{b\notin S}w_b\varrho(b)\le\theta\left[\sum_{j\notin S}w_j\varrho(j)+\sum_{z\in S}w_z\varrho(z)\right], b ∈ / S ∑ w b ϱ ( b ) ≤ θ j ∈ / S ∑ w j ϱ ( j ) + z ∈ S ∑ w z ϱ ( z ) , which gives the claim as in Lemma B.1. □
Appendix C. The plaquette blocks
Table 3 lists the block plaquettes of § 5 modulo 2 Z 4 2\mathbb{Z}^{4} 2 Z 4 x x x { μ , ν } \{\mu,\nu\} { μ , ν } x x x { μ , ν } \{\mu,\nu\} { μ , ν } { 0 , 1 } \{0,1\} { 0 , 1 } { 2 , 3 } \{2,3\} { 2 , 3 }
The program cover.py in the folder strong-coupling/gauge-fixed/blocks/ of the archive finds this set by an exact-cover search on the torus of side two and checks Lemma 5.1 on the tori of side four and six. The program constants.py evaluates the bounds of Lemma 5.5 in ball arithmetic with python-flint at 120 bits: the series of Z 0 Z_{0} Z 0 s = r 2 s=r^{2} s = r 2 s ≤ 5 s\leq5 s ≤ 5 ( 5 / 4 ) 40 40 3 / ( 40 ! 41 ! ) (5/4)^{40}40^{3}/(40!41!) ( 5/4 ) 40 4 0 3 / ( 40 ! 41 !) alpha.py evaluates the constants of (5.2) and α \alpha α β W \beta_{W} β W
For SU(3) the folder strong-coupling/gauge-fixed/blocks/su3/ contains consts.py, which evaluates the bounds of Lemma 8.3 on the 841 boxes of side 3 / 16 3/16 3/16 r = 3 / 2 r=3/2 r = 3/2 19 / 100 19/100 19/100 r = 38 / 25 r=38/25 r = 38/25 § 16 , and alpha3.py, which reads these bounds and the cover of Proposition 7.8 or Proposition 7.9 and evaluates the constants of (8.2) and α 3 \alpha_{3} α 3 β W = 3 / 4 \beta_{W}=3/4 β W = 3/4 Proposition 7.9 is computed by the program of Appendix § 16 with -r 38/25 -thr 1.03.
Appendix D. The blocks of Lemma 6.2
Diagonal blocks. Let w α = e i ϕ w_{\alpha}=e^{i\phi} w α = e i ϕ f p = − cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 f_{p}=-\cos\phi\,(y_{1}+y_{2}-y_{3}-y_{4})^{2} f p = − cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 c p = 2 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 ) c_{p}=2(y_{1}y_{4}-y_{1}y_{2}+y_{2}y_{3}-y_{3}y_{4}) c p = 2 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 )
F = ∑ i α i y i 2 + cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 − 9 5 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 ) \mathcal{F}=\sum_{i}\alpha_{i}y_{i}^{2}+\cos\phi\,(y_{1}+y_{2}-y_{3}-y_{4})^{2}
-\frac{9}{5}(y_{1}y_{4}-y_{1}y_{2}+y_{2}y_{3}-y_{3}y_{4}) F = i ∑ α i y i 2 + cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 − 5 9 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 ) with y k = 0 y_{k}=0 y k = 0 α k ± 1 = 21 10 \alpha_{k\pm1}=\frac{21}{10} α k ± 1 = 10 21 α k + 2 = 5 4 \alpha_{k+2}=\frac{5}{4} α k + 2 = 4 5 cos ϕ \cos\phi cos ϕ cos ϕ = ± 1 \cos\phi=\pm1 cos ϕ = ± 1 k = 1 k=1 k = 1 cos ϕ = 1 \cos\phi=1 cos ϕ = 1 ( y 2 , y 3 , y 4 ) (y_{2},y_{3},y_{4}) ( y 2 , y 3 , y 4 )
( 31 10 − 19 10 − 1 − 19 10 9 4 19 10 − 1 19 10 31 10 ) , \begin{pmatrix}
\frac{31}{10} & -\frac{19}{10} & -1 \\
-\frac{19}{10} & \frac{9}{4} & \frac{19}{10} \\
-1 & \frac{19}{10} & \frac{31}{10}
\end{pmatrix}, 10 31 − 10 19 − 1 − 10 19 4 9 10 19 − 1 10 19 10 31 , and Table 4 lists the leading principal minors for every position; all are positive.
Table 4. Leading principal minors of the diagonal blocks of Lemma 6.2 .
Pair blocks. Let w a = e i ( χ + ϑ ) w_a=e^{i(\chi+\vartheta)} w a = e i ( χ + ϑ ) w b = e i ( χ − ϑ ) w_b=e^{i(\chi-\vartheta)} w b = e i ( χ − ϑ ) f p f_p f p w a + w b ‾ = 2 cos χ e i ϑ w_a+\overline{w_b}=2\cos\chi\,e^{i\vartheta} w a + w b = 2 cos χ e i ϑ c p c_p c p w a w b ‾ = e 2 i ϑ w_a\overline{w_b}=e^{2i\vartheta} w a w b = e 2 i ϑ M ( m , ϑ ) M(m,\vartheta) M ( m , ϑ ) F \mathcal{F} F m = cos χ m=\cos\chi m = cos χ m m m M ( − 1 , ϑ ) = M ( 1 , ϑ + π ) M(-1,\vartheta)=M(1,\vartheta+\pi) M ( − 1 , ϑ ) = M ( 1 , ϑ + π ) m = 1 m=1 m = 1 e 2 e_2 e 2 ( z 1 , z 3 , z 4 ) (z_1,z_3,z_4) ( z 1 , z 3 , z 4 ) c = cos ϑ c=\cos\vartheta c = cos ϑ
M = ( 21 5 + 2 c − 2 e i ϑ − 9 5 − 2 e i ϑ − 2 e − i ϑ 21 5 + 2 c 9 5 + 2 e − i ϑ − 9 5 − 2 e − i ϑ 9 5 + 2 e i ϑ 5 2 + 2 c ) . M=\begin{pmatrix}
\frac{21}{5}+2c & -2e^{i\vartheta} & -\frac{9}{5}-2e^{i\vartheta}\\
-2e^{-i\vartheta} & \frac{21}{5}+2c & \frac{9}{5}+2e^{-i\vartheta}\\
-\frac{9}{5}-2e^{-i\vartheta} & \frac{9}{5}+2e^{i\vartheta} & \frac{5}{2}+2c
\end{pmatrix}. M = 5 21 + 2 c − 2 e − i ϑ − 5 9 − 2 e − i ϑ − 2 e i ϑ 5 21 + 2 c 5 9 + 2 e i ϑ − 5 9 − 2 e i ϑ 5 9 + 2 e − i ϑ 2 5 + 2 c . With u = cos 2 ( ϑ / 2 ) ∈ [ 0 , 1 ] u=\cos^2(\vartheta/2)\in[0,1] u = cos 2 ( ϑ /2 ) ∈ [ 0 , 1 ]
20 u + 11 5 , ( 20 u + 1 ) ( 20 u + 21 ) 25 , 400 u ( 40 u 2 − 23 u + 4 ) + 21 250 , \frac{20u+11}{5},\qquad\frac{(20u+1)(20u+21)}{25},\qquad\frac{400u(40u^2-23u+4)+21}{250}, 5 20 u + 11 , 25 ( 20 u + 1 ) ( 20 u + 21 ) , 250 400 u ( 40 u 2 − 23 u + 4 ) + 21 , and 40 u 2 − 23 u + 4 40u^2-23u+4 40 u 2 − 23 u + 4 529 − 640 529-640 529 − 640 20 u + 11 5 \frac{20u+11}{5} 5 20 u + 11 8 u + 1 2 \frac{8u+1}{2} 2 8 u + 1 800 u 2 − 180 u + 53 50 \frac{800u^2-180u+53}{50} 50 800 u 2 − 180 u + 53 32400 − 169600 32400-169600 32400 − 169600
The inequality is not tight. In floating point the least eigenvalue of the whole form over W ∈ U ( N ) W\in U(N) W ∈ U ( N ) N ≤ 4 N\leq4 N ≤ 4 0.0149 0.0149 0.0149 21 10 , 5 4 , 9 10 \frac{21}{10},\frac{5}{4},\frac{9}{10} 10 21 , 4 5 , 10 9 29 2 \frac{29}{2} 2 29
The certified computation
Truncation. For a corner b b b Lemma 7.6 are derivatives of Z Z Z ( b , b ‾ ) (b,\overline{b}) ( b , b ) E H [ A t ( U D ) B t ( U D ‾ ′ ) ] E_H[A_t(U D)B_t(U\overline{D}^{\prime})] E H [ A t ( U D ) B t ( U D ′ )] A t A_t A t ∑ λ c λ s λ ( U D ) \sum_\lambda c_\lambda s_\lambda(UD) ∑ λ c λ s λ ( U D ) λ 1 − λ 3 ≤ 34 \lambda_1-\lambda_3\leq34 λ 1 − λ 3 ≤ 34 λ 3 ≤ 11 \lambda_3\leq11 λ 3 ≤ 11 B t B_t B t ∣ λ ∣ > 34 |\lambda|>34 ∣ λ ∣ > 34 3 ⋅ 12 > 34 3\cdot12>34 3 ⋅ 12 > 34 T ( ∥ D ∥ ) = ∑ n > 34 ( 3 2 ∥ D ∥ ) n / n ! T(\lVert D\rVert)=\sum_{n>34}(\frac{3}{2}\lVert D\rVert)^n/n! T (∥ D ∥) = ∑ n > 34 ( 2 3 ∥ D ∥ ) n / n ! ∣ D i j − b i j ∣ ≤ ρ |D_{ij}-b_{ij}|\leq\rho ∣ D ij − b ij ∣ ≤ ρ ∥ D ∥ ≤ ∥ b ∥ + 3 ρ \lVert D\rVert\leq\lVert b\rVert+3\rho ∥ D ∥ ≤ ∥ b ∥ + 3 ρ α \alpha α α ! ρ − ∣ α ∣ T ( ∥ b ∥ + 3 ρ ) \alpha!\rho^{-|\alpha|}T(\lVert b\rVert+3\rho) α ! ρ − ∣ α ∣ T (∥ b ∥ + 3 ρ ) 2 − ∣ α ∣ e v / 2 2^{-|\alpha|}e^{v/2} 2 − ∣ α ∣ e v /2 v = ∑ k ∣ b k ∣ v=\sum_k|b_k| v = ∑ k ∣ b k ∣ ∣ Tr ( U b ) ∣ ≤ ∑ k ∣ b k ∣ |\operatorname{Tr}(Ub)|\leq\sum_k|b_k| ∣ Tr ( U b ) ∣ ≤ ∑ k ∣ b k ∣ b b b ϵ α = 2 ∣ α ∣ α ! ρ − ∣ α ∣ T ( ∥ b ∥ + 3 ρ ) \epsilon_\alpha=2^{|\alpha|}\alpha!\rho^{-|\alpha|}T(\lVert b\rVert+3\rho) ϵ α = 2 ∣ α ∣ α ! ρ − ∣ α ∣ T (∥ b ∥ + 3 ρ ) ϵ α e v / 2 + ( e v / 2 + ϵ α ) ϵ β \epsilon_\alpha e^{v/2}+(e^{v/2}+\epsilon_\alpha)\epsilon_\beta ϵ α e v /2 + ( e v /2 + ϵ α ) ϵ β ρ \rho ρ ρ − 6 T \rho^{-6}T ρ − 6 T 1.7 × 10 − 11 1.7\times10^{-11} 1.7 × 1 0 − 11
Weight zero. Conjugation by the diagonal torus T T T Re Tr ( U b ) \operatorname{Re}\operatorname{Tr}(Ub) Re Tr ( U b ) b b b u i j u_{ij} u ij t i t j − 1 t_i t_j^{-1} t i t j − 1 T T T 0 0 0 Tr ( X ) \operatorname{Tr}(X) Tr ( X ) e 2 ( X ) e_2(X) e 2 ( X ) det X \det X det X X = D D ′ T X=DD^{\prime\mathsf{T}} X = D D ′ T det D \det D det D det D ′ \det D' det D ′
Rounding. The truncated series is a Taylor polynomial of total degree six in the 36 real directions of ( D − b , D ′ − b ˉ ) (D-b,D'-\bar b) ( D − b , D ′ − b ˉ ) n n n γ n = n u / ( 1 − n u ) \gamma_n=nu/(1-nu) γ n = n u / ( 1 − n u ) u = 2 − 53 u=2^{-53} u = 2 − 53 [19 Sec. 3.1] . With C C C 2 − 30 Z 2^{-30}\mathbb{Z} 2 − 30 Z 2 31 q h 2^{31}q_h 2 31 q h χ α ( 2 31 q h ) \chi_\alpha(2^{31}q_h) χ α ( 2 31 q h ) 12 ⋅ 2 31 ( φ h + Δ q h ) 12\cdot2^{31}(\varphi_h+\Delta q_h) 12 ⋅ 2 31 ( φ h + Δ q h ) 12 ⋅ 2 31 ⟨ ∇ φ F r , ∇ q h ⟩ 12\cdot2^{31}\langle\nabla\varphi_{F_r},\nabla q_h\rangle 12 ⋅ 2 31 ⟨ ∇ φ F r , ∇ q h ⟩ h h h Z [ i ] \mathbb{Z}[i] Z [ i ] 2 50 2^{50} 2 50 Z Z Z [23 ] .
Separation. The functional c c c 72 × 72 72 \times72 72 × 72 T T T max T ℓ \max_T \ell max T ℓ 512 × 512 512 \times512 512 × 512 1 2 H δ 2 / 2 \frac{1}{2}H\delta^2/2 2 1 H δ 2 /2 δ = 2 π / 512 \delta= 2\pi/512 δ = 2 π /512 H = ∣ c 1 ∣ + ∣ c 2 ∣ + 2 ∣ c 3 ∣ H = \lvert c_1\rvert+ \lvert c_2\rvert+ 2\lvert c_3\rvert H = ∣ c 1 ∣ + ∣ c 2 ∣ + 2 ∣ c 3 ∣ 10 − 12 10^{-12} 1 0 − 12
Running time. The moments at one corner take about one and a half seconds and the bound on one box about two and a half seconds on one core. The cover runs in about an hour on four cores and writes every certified box with its θ \theta θ
Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India; formerly, University of Calcutta, Senate House, 87/1 College Street, Kolkata 700073, West Bengal, India
Email address: itsdeep@live.com, d.bhattacharjee@erl-forschung.de
URL: orcid.org/0000-0003-0466-750X
Formerly, National Taiwan University, Taipei 106319, Taiwan
Email address: rai.bhattacharya.in@gmail.com
URL: orcid.org/0009-0002-2254-3914
