2026-10-01 · chapter Yang–Mills and the mass gapPart 5 Lattice gauge theoryBakry–Émery

Detour: Bakry–Émery and one-link integration — a criterion that gets a gap from curvature, and an example of integrating first to move the constant

The $\beta$ of this chapter is in the $SU(2)$ convention $\beta=4/g^2$, inversely proportional to the square of the coupling $g$ (5-01). It plays the role of the inverse temperature of statistical mechanics (5-02).

Elements of $SU(2)$ are quaternions of length 1, and the group is the unit sphere $S^3$ (1-02). The inner product is $\langle u,v\rangle=\tfrac12\operatorname{tr}(uv^\dagger)$.

The key to the Langevin road of 5-04 §08 is the Bakry–Émery criterion, which says "if the roundness (curvature) of the group beats the bumps of the action, a Poincaré inequality holds with a box-independent constant". This chapter looks at it concretely on a single link (§02, §03) and on the whole lattice (§04–§07).

What one gets is decay of correlations in lattice units at fixed lattice spacing (4-11), the shore opposite what is wanted in the continuum limit, "the gap in lattice units closing at the right rate" (5-09).

In 5-04 §08 we used only the conclusion of the criterion. This chapter adds three things: the line of proof of the criterion (§02), a comparison with the true gap on a single link (§03), and that integrating part of the links first lowers the bending from first to second order in $\beta$ and moves the constant of the threshold (§05–§07).

The shortest way to read is to see first, in Figure 1, "where the criterion becomes conservative", then the threshold of the line $2-c\beta$ in §04, and finally the threshold of the parabola $2-27\beta^2$ in §05–§07. The breakdown in §06 and the detour in §08 can be skipped without losing the thread.

In 1-10 §08 we saw, for $e^{-V}dx/Z$ on the line, "if $V''\ge\kappa\gt0$ then $C_P\le1/\kappa$". On a group the roundness of the space itself is added to this.

As a metaphor, two people who set off due north from the equator meet at the North Pole. Precisely, the Ricci curvature $\mathrm{Ric}(v,v)$ is the sum, over each direction perpendicular to $v$, of how much a bundle of geodesics leaving in the direction of the unit vector $v$ draws together; on the $n$-dimensional sphere of radius 1 it is $(n-1)|v|^2$ Known. Geodesics on the group are $\tau\mapsto Ue^{\tau X}$ ($X$ is a pure imaginary quaternion, an element of the Lie algebra; 1-03), and we write $\mathrm{Hess}\,V(X,X)$ for the second derivative along them. It corresponds to $V''$ on the line.

The right-hand side is the Poincaré version of the criterion of Bakry–Émery (1985) ($\mu=e^{-V}\mathrm{vol}/Z$, with the condition at every point) Known. A logarithmic Sobolev inequality also follows. No dimension enters the constant, so if the lower bound on curvature does not depend on the box, neither does the Poincaré constant.

The line of proof is the same as in 1-10 §08. With $Lf=-\Delta f+\nabla V\cdot\nabla f$ ($\Delta$ is the Laplacian on the group, different from the $\Delta$ of the mass gap) and $Lf=\lambda f$, integrating by parts twice (the integrated Bochner formula) gives Known

($\int|\nabla f|^2d\mu=\lambda\int f^2d\mu$), so $\lambda\ge K$. (3) is the multidimensional form of $\int|g'|^2+\int V''g^2=\lambda\int g^2$ in 1-10: $V''$ is replaced by $\mathrm{Hess}\,V$, and the roundness of the space is added as $\mathrm{Ric}$. Example: on uniform $S^3$ the criterion says "gap $\ge2$", while the truth is 3 (first-order spherical harmonics; the first row of Table 1). It is conservative to the extent that it does not use the dimension.

Holding fixed everything but a single link $u$, the weight comes from the six plaquettes containing $u$. The product of the remaining three links of a plaquette without $u$ (a U shape) is called a staple. Using the sum $M=\sum_iV_i$ of the six staples $V_1,\dots,V_6$ with orientations aligned, the weight is $e^{\beta\langle u,M\rangle}$ (the same set-up as 10-02 Roads tried on the four-dimensional lattice, and walls; the distribution of the heat-bath method; 5-07). Taking the direction of $M$ as the North Pole and $\theta$ as the angle from the North Pole, the Hessian of the height $\cos\theta$ is $-\cos\theta$ times the metric, so

Known. All the criterion can say is a gap $\ge2-\kappa$ for $\kappa\lt2$, but the true gap grows with $\kappa$ (the measure concentrated near the North Pole is close to a Gaussian of variance $1/\kappa$). The criterion is tripped up by the negative curvature near the South Pole, where the measure carries almost no weight.

ComputedThe numbers in this figure were computed on this machine (the gaps are 41 points of the calculation of Table 1, the rest from formulas on the spot). If JavaScript is disabled, Table 1 holds the content.

The gap is set by $l=1$ (fluctuations tilting perpendicular to $M$). The criterion is a correct but crude tool.

Note. $\lambda_1=1/C_P$ is the gap in Langevin time, different from the gap of the transfer matrix (5-03) (8-03).

The configuration space of the whole lattice is a product of $S^3$, and the Ricci curvature stays 2 on each link. The criterion asks for the following.

The lower bound has the form $-c\beta$. Shen–Zhu–Zhu (2023), with a plaquette-by-plaquette estimate, showed for $SU(N)$ under $|\beta_{\mathrm{SZZ}}|\lt1/(16(d-1))$ uniqueness of the infinite-volume measure and exponential decay of correlations (abstract: “the infinite volume measure has a strictly positive mass gap”) Known. The convention for $SU(2)$ is $\beta=4\beta_{\mathrm{SZZ}}$, so in four dimensions this is $\beta\lt1/12=0.0833$, and in the notation of this chapter $K=2-24\beta$ ($\beta\lt1/12$ is the same as in 5-04 Table 4).

On the other hand, the $\mathbb Z_2$ configuration with every plaquette $-1$ gives $\lambda_{\min}(\mathrm{Hess}\,S)=-16\beta$ (elementary) Paper. The numerical search of the paper found nothing below it, but this is not a proof (paper §1, §8). The ceiling $1/8$ comes from the elementary side alone: for this configuration $K\le2-16\beta$, so as long as the criterion is used without integrating, it tops out at $\beta\lt1/8$.

The paper on this site integrates part of the links exactly before applying the criterion. The set $I^*$ to be integrated is required to satisfy the following.

No two links of the family lie on the same plaquette, so each link can be integrated independently; and since every plaquette contains a link of the family, not a single bare plaquette remains in the action after integration. The density of the family is $1/4$, and four dimensions has a family of period 2 (counting and figures in 10-02 Roads tried on the four-dimensional lattice, and walls; why only up to four dimensions in §08).

The integral over one link comes out in Catalan numbers. Under the Haar measure $\operatorname{tr}u=2\cos\theta$ has the semicircle distribution (1-08 (4)), with even moments $\mathbb E[(2\cos\theta)^{2k}]=\mathrm{Cat}_k$ (the detour in 1-08). Expanding the exponential and using $\mathrm{Cat}_k/(2k)!=1/(k!(k+1)!)$ gives

($I_1$ is a modified Bessel function) Known. In the paper this is proved in Lean from the left invariance of the Haar measure alone Lean (IsHaarS3.one_link; the only assumptions are the Haar properties). The marginal density of the remaining links is $e^{-\Phi}$, a sum over the links $s$ of the family ("stars").

For derivatives we use $F_1(x)=\sum_{k\ge0}x^k/(k!\,(k+2)!)=F_0'(x)$ and $F_2=F_1'$. By the chain rule

so $G'\le\beta^2/8$ is the same as $2F_1\le F_0$, and $G''\le0$ the same as the Turán-type inequality $F_1^2\ge F_0F_2$.

The $O(\beta)$ term disappears, and the bending starts at $O(\beta^2)$ (loops of length 6 appear with coupling $\beta^2/4$; 10-02 §02).

The central theorem of the paper is a lower bound on the second derivative of $\Phi$ along geodesics. It holds for every $\beta$ and every configuration, with no smallness assumed Lean (StaplePerturbation.hess_gauge. The assumptions are that the period $L$ is even and $L\ge3$ (that is, an even number 4 or more; at $L=2$ opposite staples share links and the counting changes), $\|U_e\|=1$, and $X_e$ pure imaginary. That the marginal density is $e^{-\Phi}$ is WilsonOneLink.integral_star_links).

Joining them takes one line. With $t=|M_s|^2$, the second derivative of a star term is

and thanks to $G''\le0$ the $(t')^2$ term does not get in the way. Summing over stars counts each link 6 times, giving $-\frac{\beta^2}8\cdot36\cdot6=-27\beta^2$. $G'\le\beta^2/8$ is the claim that "the fluctuations of a tilted Haar measure do not exceed those of the Haar measure", and this is specific to $SU(2)$ (§09).

Example. The star ratio $t''/(36\sum\|X\|^2)$ has maximum 0.371 over 200,000 random samples, and equality occurs for the configuration with $s=(+,+,+,-,-,-)$ and the three directions aligned to $s_i\,i$ (1.000000) Computed. The slack lies on the side of joining the stars (§09).

Combining (9) with $\mathrm{Ric}=2$, the curvature of the marginal measure of the remaining links is at least $K'=2-27\beta^2$, and the threshold is

This is $4\sqrt{2/3}=3.27$ times $1/12$, and 2.18 times the ceiling $1/8$ Computed. For $d$ where a family exists, the threshold is $\sqrt{2/3}/(d-1)$, and the ratio 3.27 does not depend on $d$ Paper (Lean covers only $d=4$).

ComputedThe numbers in this figure were computed on this machine (the formulas of Table 3). If JavaScript is disabled, Table 3 holds the content.

The reach of the cluster expansion of Osterwalder–Seiler (1978), the strong-coupling classic, cannot be compared numerically in this chapter (source table: bibliographic data only). The crude Kotecký–Preiss estimate of paper §9 gives $\beta\lesssim0.014$ (Mayer form) to $0.06$ (character expansion), the former of the same order as 5-04 Table 3. On the other hand, a necessary condition for the character expansion to converge absolutely allows $\beta$ up to 0.4–0.8, and a careful expansion may reach further. So the paper does not claim to have widened the strong-coupling range, and says only that "within the Bakry–Émery method the constant improves by a factor of 3.27". The same dynamical method has also been used for the area law (Cao–Nissim–Sheffield) Known.

The four limits of paper §8 are different faces of the fact that "a local curvature method can move the constant but not the shore".

(a) 27 and 21. An explicit configuration gives exactly $\lambda_{\min}(\mathrm{Hess}\,\Phi)=-21\beta^2$ (exact computation by integer arithmetic in $\mathbb Z[e^{i\pi/3}]$ in paper §8; not machine-checked, not recomputed on this machine). The true value lies between 21 and 27, and the ceiling is $\sqrt{2/21}=0.3086$.

(b) Large $\beta$. When $\beta$ is large, integrating first makes things worse. $\lambda_{\min}$ at $\beta=1$ is still $-12.1$ against $-16$ (with against without integration), but at $\beta=2$ it reverses to $-43.1\beta^2$ (without integration $-32\beta$). At either value $K\lt0$, and the criterion cannot be used (numerical calculations of the paper; not recomputed on this machine). As in §03, the criterion demands curvature at every configuration, and $\mathrm{Ric}=2$ does not move with $\beta$, so the threshold does not exceed $O(1)$.

(c) It cannot be repeated. Integrating one more link makes the weight no longer of Bessel type Paper.

(d) $SU(2)$ only. "Tilting does not increase fluctuations" is already false for $U(2)$ with $M=\mathrm{diag}(s,0)$. What the paper measured directly is mainly the star constant for $U(N)$. The threshold measured for $N=2,3,4$ is nearly constant ($\beta_{\mathrm{SZZ}}\approx0.093$), but for large $N$ it is estimated as $\beta^*\sim0.0962/\sqrt N$ (paper §8), falling below the $N$-independent threshold of Shen–Zhu–Zhu for $N\gtrsim21$. In the 't Hooft limit the gain disappears (numerical calculations and asymptotic estimates of the paper; not recomputed on this machine).

What is Lean. The Lean theorems are (7), the marginal density $e^{-\Phi}$, the three factors, (9), and $d\le4$ for families. $\mathrm{Ric}=2$ and the criterion are standard facts Known, and the step to exponential decay of correlations and uniqueness (a transfer of the argument of Shen–Zhu–Zhu) is an outline with constants Paper. That is, the lattice-unit mass gap at $\beta\lt0.2722$ is not a Lean theorem. Within the range the paper searched the same construction was not found, but the paper writes that it may be known in the context of variance reduction for Monte Carlo methods.

The curvature method works uniformly in a world with $C_P=O(1)$, where the correlation length is a few lattice spacings, the side opposite what is wanted in the continuum limit (10-01). However much this estimate is improved, it comes no closer to the mass gap of the problem statement.

What this chapter can say

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Next chapter to read: 6-01 What constructive field theory is (between the two shores). The families are used again in 10-02. For Lean, the guide and the list.

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