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

Lattice gauge theory with finite groups — exponential decay at weak coupling is a theorem, but it lies on the side opposite the continuum limit

Replacing the group on the edges by a finite group $G$ leaves the definitions of the lattice, plaquettes, gauge transformations and Wilson loops unchanged (5-01 §02). The integral over each edge becomes an average over $|G|$ elements. The action uses the character $\chi=\operatorname{tr}\rho$ of a faithful unitary representation $\rho$ (of dimension $d$; $\rho(g)\ne I$ if $g\ne1$), and we write $\sigma_p$ for the product once around a plaquette $p$.

(Adhikari–Cao (1.2)(1.3)) Known. $\chi(g)$ is a sum of $d$ eigenvalues of absolute value 1, so each term is 0 or more, and it is 0 only when $\sigma_p=1$.

Example. For $\mathbb Z_2=\{\pm1\}$ the weight is proportional to $e^{\beta\sum_p\sigma_p}$: Ising lattice gauge theory, which Chatterjee (2020) calls the first lattice gauge theory, defined by Wegner (1971) Known (2-09). $\mathbb Z_N$ is $U(1)$ with angles restricted to multiples of $2\pi/N$. For the finite subgroups of $SU(2)$, $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$ (the unit quaternions) and $\bar T$, $\bar O$, $\bar I$ (the rotation groups of the regular tetrahedron, octahedron and icosahedron, subgroups of $SO(3)$, pulled back to $SU(2)$; of orders 24, 48, 120; 1-12 §08), $\chi=\operatorname{tr}$, and relative to Wilson's convention $\beta_W=2\beta$.

A finite group has no Lie algebra, and no elements close to the identity. There is no classical continuum limit starting from the $U=e^{iaA}$ of 5-01 §05; we read it as a lattice model with the same gauge structure.

When $\beta$ is large, the weight strongly favours configurations in which every plaquette is the identity. The only configurations on a box with this property are pure gauge ($\sigma_{(x,y)}=h_xh_y^{-1}$), and up to gauge there is just one, "every edge the identity". The weak-coupling theory is small fluctuations around it, and we say that "the edges freeze".

From here on we write $D$ for the dimension of the lattice, and keep $d$ for the dimension of the representation. The cheapest fluctuation is to change a single edge to $g\ne1$. On a $D$-dimensional lattice that edge lies in $2(D-1)$ plaquettes, and each of their products is conjugate to $g^{\pm1}$. Since $\chi(g^{-1})=\overline{\chi(g)}$,

$\Delta_G\gt0$ is equivalent to $\rho$ being faithful (Cao §1.2) Known. Example: for four-dimensional $\mathbb Z_2$, $\Delta_G=2$, and the weight of this defect is $e^{-12\beta}$. This 12 appears in the theorem of §05.

For $U(1)$, changing an edge to $e^{i\theta}$ costs $\approx(D-1)\theta^2$, which can be as small as one likes, so long waves (photons) are cheap to make (5-06 §03). $\Delta_G$ is a different thing from the mass gap of 4-11.

For $\mathbb Z_2$, taking a surface (a set of plaquettes) $P$ with boundary $\gamma$, the inner edges appear twice each and cancel (a discrete Stokes; the existence of $P$ is Lemma 3.1 of Chatterjee 2020) Known.

Here we use the dual lattice. It is the lattice whose vertices are the centres of the $D$-dimensional cubes, and the $k$-dimensional cells of the original lattice (edges, faces, cubes, …) correspond one to one with the $D-k$-dimensional cells of the dual lattice. In Lemma 3.3 of the same paper, the negative plaquettes split into "vortices", and the dual of each vortex is closed. The dual of a plaquette ($k=2$) is a $(D-2)$-dimensional cell, so vortices are points in two dimensions, closed loops in three, and closed surfaces in four Known. Only vortices linked with $\gamma$ change the sign (Lemma 3.2), and if their number $N$ is Poisson-distributed with mean $\mu$ then $\mathbb E[(-1)^N]=e^{-2\mu}$ (1-08, 7-02 §06).

Example. In two dimensions with free boundary the plaquettes are independent and $\mathbb E\sigma_p=\tanh\beta$, so for a loop of area $A$

Known. In two dimensions even finite groups have the area law at every $\beta$ (for general groups, 6-02 §03). In three and four dimensions at large $\beta$ the vortices are small, only nearby ones link with $\gamma$, and their number is proportional to the perimeter (the perimeter law) Physics. The area law at small $\beta$ is a theorem of the strong-coupling expansion (5-02 §03) Known.

Theorem 4.1 of Chatterjee 2020 writes the partition function of four-dimensional $\mathbb Z_2$ as the same theory on the dual lattice, moving the coupling to a $\lambda$ that exchanges strong and weak Known.

The number in (5) is Computed. Duality maps transition points to transition points, so if there is exactly one transition it falls at the self-dual point Known. According to the abstract of Wegner (1971), the dual of the Ising model on the simple cubic lattice is a model of four-spin products (plaquettes), that is, three-dimensional $\mathbb Z_2$ gauge theory Known.

A first-order transition is one in which a quantity such as the mean plaquette jumps at one point of $\beta$. Disagreement (hysteresis) between runs raising $\beta$ and runs lowering it is a sign of it. In four dimensions Creutz–Jacobs–Rebbi (1979) reported a first-order transition numerically, and reported that for $\mathbb Z_N$ with $N\ge5$ there are two transitions that are not first order, one of which moves to $\beta\to\infty$ with $N$ Physics.

On this machine too we ran the heat-bath method on a periodic $8^4$ lattice (the run raising $\beta$ from a random start, the run lowering it from all edges $+1$, averaging the second half of 200 sweeps at each $\beta$). The leading term $1-8e^{-12\beta}$ comes from the probability $\approx4e^{-12\beta}$ that a defect of §02 sits on one of the four edges.

For $\beta=0.43$–$0.45$ the two runs disagree (by 0.41 at 0.44), and that interval straddles the self-dual point Computed. These are short runs on a small lattice, with no power to decide the order of the transition (first order, or a transition without a jump) (5-07; the same comparison for $U(1)$ and $SU(2)$ is in 5-06 §04).

Theorem 1.1 of Chatterjee (2020): let $\ell$ be the number of edges of $\gamma$, and $\ell_0$ the number of corner edges (edges lying in the same plaquette as another edge of $\gamma$; 8 for a rectangle). Then for $\beta_0$ and above, with universal constants $C_1,C_2$, Known

The paper reads it as “Nontrivial behavior happens if and only if ℓ is like a constant multiple of $e^{12\beta}$”. Each edge becomes the centre of a defect of §02 with probability $\approx e^{-12\beta}$ and changes the sign once; if their number is Poisson, this is the leading term of (6) (§03) Physics.

Cao (2020) extended this to any finite group. Its Theorem 1.2.1 is about a single self-avoiding loop $\gamma$, with conditions $\beta\ge(1000+14\log|G|)/\Delta_G$ and $\|A_\beta\|_{\rm op}\lt1$. Under them the leading term is

where $\lambda_i$ are the eigenvalues of the matrix $A_\beta$, the average of the $\rho(g)$ carried by defects weighted by the defect weights, and $\|A_\beta\|_{\rm op}$ is its operator norm (the largest factor by which it stretches a unit vector) Known. Because of the latter condition, $\mathbb Z_2$ ($A_\beta=-1$, norm 1) does not fall directly under this theorem (Cao's Remark 1.2.3) Known. Even so, putting $r_\beta=e^{-12\beta}$ and $\lambda=-1$ for $\mathbb Z_2$ makes the form of the leading term agree with (6) Computed. The leading term is a function of the perimeter alone: for the same perimeter, long and thin or square, it has the same value. In two dimensions, (4) was a function of the area alone.

ComputedThe numbers in this figure were computed on this machine ((4)(6) computed on the spot). Example: at $\beta=0.7$, for 1×11, 3×9, 6×6 the two-dimensional values are $3.9\times10^{-3}$, $1.2\times10^{-6}$, $1.3\times10^{-8}$, and the four-dimensional value is 0.989 for all. Theorem 1.1 guarantees only sufficiently large $\beta$ ($\beta_0$ and above; the value is not stated explicitly in the paper). 0.4407 is a guide to the transition, and below it (dashed) the leading term does not fit even in form.

At $\beta=0.40$ the values divided by the area line up, and at $\beta=0.50$ those divided by the perimeter Computed. The latter are about 1.5 times the $2e^{-6}=0.00496$ of (6); just above the transition, corrections from corners and large vortices matter.

That separated observables become nearly independent (decay of correlations) had earlier results for abelian finite groups at weak coupling ($\mathbb Z_2$ in Chatterjee 2020 §5, and the results of Seiler and Forsström listed in Adhikari–Cao §1.1), and Adhikari–Cao (Ann. Probab. 2025) showed it in four dimensions including the non-abelian case. Let $B_1,B_2$ be rectangles separated by at least $L$ in the $\ell^\infty$ distance (the largest difference of coordinates), and $|B|$ the number of plaquettes contained in a rectangle $B$. $f_i$ is a function of the holonomies of loops in $B_i$ (the product $\sigma_\gamma$ of the edge elements along the loop; 5-01 §02) that is conjugation-invariant in each argument ($f(h^{-1}gh)=f(g)$). This includes any function of Wilson loops. $\|f\|_\infty=\max|f|$, and $\operatorname{Cov}$ is the covariance (1-08). If $\beta\ge(114+4\log|G|)/\Delta_G$, then in every finite box Known

It is an estimate independent of the box. The constant grows exponentially in the size of the supports of the observables ($B_1,B_2$), but does not depend on $L$. The authors write that it “can be interpreted as showing the existence of a mass gap in the setting of finite (non-Abelian) gauge groups at large β” (for small $\beta$, 5-04 §07). The proof uses the fact that a map exchanging two independent configurations on a region preserves the measure, and bounds the covariance by "the probability that defects connect $B_1$ to $B_2$".

The condition and the rate are set by $\Delta_G$ ($1-\cos(2\pi/N)$ for $\mathbb Z_N$, $2-2\cos\theta_{\min}$ for subgroups of $SU(2)$).

ComputedThe numbers in this figure were computed on this machine (the groups were built by closing under products, and the eigenvalue angles and $\Delta_G$ counted; the same as Table 3).

The finer the group, the smaller $\Delta_G$, and the condition grows as $1/\Delta_G$ ($\approx2\pi^2/N^2$ for $\mathbb Z_N$). Cao 2020 calls its own condition (7) “very loose” (the threshold of Adhikari–Cao is about an order of magnitude smaller).

Detour: non-abelian defects become links

The constant in (8) depends on the size of the supports of the observables but not on $L$, so (8) gives an upper bound on the correlation length (4-11 §07).

Known (a consequence of (8), with the definition of 4-11 §07)

At $\beta$ exactly at the condition, the upper bound is 0.0171 lattice spacings for $\mathbb Z_2$ and 0.0150 for $\bar I$. The perimeter $1/\bigl(r_\beta(1-\lambda)\bigr)$ at which the leading term of (7) moves away from 1 is, at $\beta$ exactly at Cao's condition, $10^{2721.1}$ for $\bar T$ and $10^{2780.1}$ for $\bar I$ Computed. The conditions are loose, so what matters is the direction: a finite group at weak coupling is a frozen lattice whose correlations do not reach even one lattice spacing.

Problem 5.1 of 5-09 §05 (Chatterjee 2018, arXiv:1803.01950) asks, for four-dimensional non-abelian compact groups, for $\xi(\beta)\lt\infty$ at each $\beta$, and $\xi\to\infty$ as $\beta\to\infty$. Finite groups satisfy the first part at large $\beta$, and go to $\xi\to0$, the exact opposite of the second Known. This is "Type 1" of 5-09 §08, and the physical mass is infinite.

Theories with finite groups are cited as examples of "a gap, but no confinement". This "gap" has three meanings.

Theorem 2.4 of Chatterjee (2021) takes the group $G$ to be a closed connected subgroup of $U(n)$ and derives the area law from (b) for representations acting non-trivially on the centre Known. Non-trivial finite groups are not connected, so they do not literally fall under this assumption. Even so, p.8 of the same paper cites theories with finite groups as examples “believed to be ‘gapped’ ... not confining at large β”, and explains why this does not contradict Theorem 2.4: the decay there is in a weaker sense than Definition 2.3 (truncated two-point correlations under certain boundary conditions) Known. The paper goes on to say that for some theories it has been rigorously shown that there are several Gibbs measures. Read with the proviso "if the proof goes through without connectedness", (b) does not hold for non-confining finite groups.

The errors of (6)(7) do not go to 0 as the loop grows, so that the string tension is 0 at fixed $\beta$ is not a claim of the theorems cited here (we have not checked, in this chapter, a result that states the perimeter law directly for abelian finite groups). It is the standard physical view (consistent with Table 2) Physics. Confinement and the gap are different statements (7-02 §02).

According to the abstract of Petcher–Weingarten (1980), $\bar T$, $\bar O$ and $\bar I$ all have first-order transitions, and as the order is raised the critical coupling goes to 0. $\bar I$ agreed with the numbers for $SU(2)$ over a wide range of couplings Physics.

We make a rough guide. For $SU(2)$ at $\beta_W=2.4$ in 5-02, $\langle2-\operatorname{tr}U_p\rangle=0.739\gt\Delta_{\bar I}=0.382$. From the leading weak-coupling term $\langle\tfrac12\operatorname{tr}U_p\rangle\approx1-3/(4\beta_W)$ (5-06 §04), $\langle2-\operatorname{tr}U_p\rangle\approx3/(2\beta_W)$ equals $\Delta_G$ at

Computed. It is the crude estimate that for $\beta_W$ larger than this the fluctuations of $SU(2)$ are finer than the steps of the finite group, not the value of a transition point Physics.

The point is the direction of the constants. Bringing the subgroups closer to $SU(2)$ gives $\Delta_G\to0$, and both the condition $\propto1/\Delta_G$ and the rate $(\beta/2)\Delta_G$ degenerate. What is needed for $SU(2)$ is that the correlation length be finite in a regime where the action has no gap, and moreover diverge as $\beta\to\infty$. That mass gap is thought to come from the coupling growing stronger with distance (5-06 §07, 4-07) Physics.

What this chapter can say

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Next chapter to read: 5-11 Detour: Bakry–Émery and one-link integration (seeing, on a single link, how theorems come out on the strong-coupling side).

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