2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe lateral-widening inequality, the margin 5/3, U(1) and ℤ₂ in contrast · Kiichi, Shiori and Rin

The only place where non-abelian enters

The notation is as in 13-01 §01 and 13-02 §01. $\theta(R;T^2_L)$ is the sector bias, in a typical frame, of the container $D\times T^2_L$ of size $R$ in the planar directions (area $A_D$), with the twist $n_{12}=1$ fixed on the transverse torus $T^2_L$ (area $A_T=L^2$). Theorem 1 of 13-02 derived the correct form of the band, $\ln(1/\theta(R,R))\ge cR^2$, from three assumptions. The third of them was the lateral-widening inequality (strong form)

If we widen the transverse $T^2$ while keeping the container fixed, which way does the sector bias $\theta$ move? The smaller $\theta$ is, the greater the disorder of the sector (the more vortices), so the question is whether widening the transverse directions makes vortices costlier ($\theta$ moves towards 1) or not costlier. Even if they become costlier, (1) allows it only up to the factor $s^2$ of the number of positions ($s=L'/L$).

We write the answer of this chapter first. In an abelian theory the flux of a vortex can only spread out uniformly over the whole transverse cross-section, so the wider the transverse directions, the costlier the vortex, and (1) fails. In $SU(2)$ a vortex can localise in a self-dual lump of size about $L$, and its action does not depend on the size, so classically the cost of widening the transverse directions is 0. The remaining quantum effects lean, in the dilute regime, to the side of asymptotic freedom, and (1) is expected to hold with margin $5/3$.

First we set out one elementary fact about the classical action.

Lemma 1 Proved here Let $\Omega$ be a planar region of area $|\Omega|$, and $f$ a square-integrable real function on $\Omega$ with $\int_\Omega f=\Phi$. Then

with equality only when $f$ is almost everywhere equal to the constant $\Phi/|\Omega|$ on $\Omega$.

Proof (2) follows from the Cauchy–Schwarz inequality $\bigl(\int_\Omega f\cdot1\bigr)^2\le\int_\Omega f^2\cdot\int_\Omega1$. Equality holds only when $f$ and $1$ are proportional, that is, only when $f$ is constant. ∎

The action of an abelian gauge field, $\frac1{2e^2}\int F_{\mu\nu}F_{\mu\nu}$, is the square of the field strength: a linear field theory. Consider a configuration in which the same flux $\Phi$ passes through each cross-section $D\times\{y\}$ ($y\in T^2_L$) of the container $D\times T^2_L$. Using Lemma 1 on each cross-section and integrating over the transverse $T^2_L$, we get

Proved here (under the condition that the flux through each cross-section is the same $\Phi$; a lower bound on the classical action). Equality holds for the configuration in which the flux spreads out uniformly over the whole container, and localising the flux only increases the action. The right-hand side is proportional to the transverse area $A_T$, so if we fix the container and widen the transverse directions by a factor $s$, the minimal action of the flux $\pi$ (corresponding to the sector $\sigma_D=-1$) becomes $s^2$ times larger. An abelian vortex becomes costlier when the transverse directions are widened: that is the answer at the classical level.

The count at the quantum level is as in 13-01 §03 Reading (physics). Seen at long distance on the slab $T^2_L\times\mathbb R^2$, U(1) in the Coulomb phase is a two-dimensional U(1) (coupling $e^2/A_T$), and the partition function with flux $\Phi$ is the Poisson sum over wrappings $k$, $Z(\Phi)\propto\sum_ke^{ik\Phi}e^{-e^2A_Dk^2/(2A_T)}$. This is the sum over $k$ of the classical action $\frac{A_T}{2e^2A_D}(\Phi+2\pi k)^2$ of (3), rewritten with the Poisson summation formula, and when $q:=e^{-e^2A_D/(2A_T)}$ is small

If the transverse directions are widened by a factor $s$, $A_T$ becomes $s^2$ times larger and $q$ becomes $q^{1/s^2}$, so $\theta_{\rm new}\approx2(\theta_{\rm old}/2)^{1/s^2}$. The ratio is

and it cannot be bounded by any constant $Cs^2$. For U(1), the lateral-widening inequality (1) fails Reading (physics; one line of algebra once (4) is granted). The weak form ((C′) of 13-02) also fails, because the slab tension goes to 0 like $e^2/L^2$ Reading (physics).

The four-dimensional $\mathbb Z_2$ theory is, at weak coupling (large $\beta$), in a non-confining phase (5-10). There the flux through the container creates a wall of area $A_T$ spanning the whole transverse $T^2$, and its tension does not depend on $A_D$. Widening the transverse directions increases the cost of the wall in proportion to $A_T$, and (1) fails Reading (physics). What the two abelian theories have in common is that the cost of the flux increases with the transverse area: in U(1) because of linearity (Lemma 1), in $\mathbb Z_2$ because of the area of the wall.

In $SU(2)$ things change. On $T^2_L\times\mathbb R^2$ with the twist $n_{12}=1$, the topological charge $Q$ can take half-integer values ('t Hooft's twist, 12-01 Cited). And the action has the Bogomolny bound

(3-05 Cited), with equality for self-dual (or anti-self-dual) configurations. A configuration with $Q=\tfrac12$ — a fractional instanton — has action at least $4\pi^2/g^2$, and plays the role of a vortex that flips the planar sector (12-02).

The right-hand side of (6) depends neither on the size $L$ of $T^2$ nor on the size $R$ of the container. This is because the classical action of four-dimensional Yang–Mills does not change under a change of scale (the coupling $g$ is dimensionless, 11-07 §01). The estimate of the original paper is that the minimal-action configuration with $Q=\tfrac12$ is a self-dual lump localised in a region of size about $L$, whose action equals the bound $4\pi^2/g^2$ and does not depend on the ratio of the transverse and longitudinal sizes Reading (physics). On the lattice, it has been measured that the action of cooled fractional instantons approaches $\pi^2\beta$ from below (with $\beta=4/g^2$, $4\pi^2/g^2=\pi^2\beta$; the values obtained by cooling are upper bounds on the minimal action on the lattice) (the table in 12-02).

Here the difference from Lemma 1 of §02 becomes clear.

• Abelian: the action is a quadratic form, so the minimal action for a given flux is attained by the configuration that spreads the flux out uniformly, and its value is proportional to the transverse area $A_T$ ((3)). Localising does not make it cheaper; one pays the cost over the whole region it is spread over.
• Non-abelian: the self-duality equations are non-linear, and a solution of finite action $4\pi^2/g^2$ can be localised to a size of about $L$. As long as the container is large enough to hold the lump ($R\ge L'$), widening the transverse directions from $L$ to $L'$ only requires re-spreading the lump, and the classical cost is 0.

This is not a difference of local kinematics, but a difference in the existence of solutions of the self-duality equations — in non-linearity. Since at the classical level the cost of lateral widening is 0, the sign of (1) is fixed by quantum effects: the number of positions of the vortices, and the running of the coupling. That is the count of the next section.

The count for a dilute gas. For $L\Lambda\ll1$ ($\Lambda$ is the scale of dimensional transmutation, 4-07), the coupling $g(L)$ is small, and the vortices (lumps with $Q=\pm\tfrac12$) can be treated as a dilute gas, far apart from one another. For $SU(N)$, Tanizaki–Ünsal derived the ground-state energy and the string tension from the facts that on a small twisted $T^2$ the centre vortices behave like points on the plane $\mathbb R^2$ and that the weight of each is $Ke^{-S_I/N}$ ($S_I=8\pi^2/g^2$, with $K$ the prefactor per unit area):

(Tanizaki–Ünsal, eqs. (2.32) and (2.37) Cited, dilute regime, one loop). The $\theta$ here is the vacuum angle, a symbol different from this book's sector bias $\theta$. The exponent $5/3$ comes out as follows: from the one-loop running of the coupling $8\pi^2/g^2(L)=\frac{11N}3\ln\frac1{L\Lambda}$ (4-06 Cited) we get $e^{-S_I/N}=(L\Lambda)^{11/3}$, and from the size $L$ of a lump the number of positions per unit area is $1/L^2$, so $(L\Lambda)^{11/3}/L^2=\Lambda^2(\Lambda L)^{5/3}$.

We move on to the sector bias. In $SU(2)$, let $\zeta$ be the density per unit area of vortices and antivortices together, and approximate the positions of the vortices by independent Poisson points Reading (physics; the dilute-gas approximation). The sector $\sigma_D$ of the container is $(-1)^n$ for the number $n$ of vortices inside $D$. For $n$ in a Poisson distribution with mean $m$,

so Proved here (an elementary identity), putting $m=\zeta A_D$,

Reading (physics; dilute regime, one loop). Set beside (4), the power in $L$ of $\ln(1/\theta)$ with the container fixed is as follows.

The power $5/3$ for $SU(2)$ is the difference between the coefficient of asymptotic freedom $11/3$ and the $2$ of the number of positions of a lump in the plane (what this part has been calling "the transverse two-dimensional entropy", Figure 2 of 13-00). The power $-2$ for U(1) can be read as the form in which there is no asymptotic-freedom term and only the term of the number of positions remains. The margin $5/3\gt0$ by which asymptotic freedom beats the transverse two-dimensional entropy is the sign of the lateral-widening inequality itself Reading (physics).

From (9), the ratio when the transverse directions are widened by a factor $s$ in $SU(2)$ in the dilute regime is

($\zeta\propto L^{5/3}$ is increasing), and (1) holds with $C=1$, without using the factor $s^2$ Reading (physics). But (9) and (10) can be used only in the dilute regime $L\Lambda\ll1$. In the confining regime ($L\Lambda\gg1$) one expects $\ln(1/\theta)\approx\sigma A_D-\ln(A_T/\xi^2)$ ($\xi$ is the correlation length); the dependence on $A_T$ is then only the logarithm of the entropy of the position of the vortex sheet, and (1) is expected to hold within the factor $s^2$ Reading (physics). There is no estimate that passes non-perturbatively between the two regimes, $L\sim1/\Lambda$ — the band.

We set out the components of the chain of Theorem 1 of 13-02 according to how they behave when the group is changed.

• The nesting in the planar directions ((2) of 13-02) is a statement that comes from the Markov property — that conditioning on the frame makes the insides of the containers independent — and from the container identity of Mack–Petkova, and it does not use the non-commutativity of the group Cited.
• (A) The small-T² base point is an estimate that holds for U(1) too. Putting $R=L$ in (4), $\theta(L,L)\approx2e^{-e^2/2}$ is a constant less than 1 independent of $L$ (13-01 §03) Reading (physics).
• (B) Frame decorrelation, if $L$ is fixed, is replaced under reflection positivity in z by the chessboard density $\kappa_{\rm cb}$ (13-05), and its structure is the same for $\mathbb Z_2$ and U(1) too (13-04). There is no estimate that U(1) fails in the form uniform in $L$ (the uniform slab gap); U(1) is expected to fail only on the side of the uniform slab tension Reading (physics, 13-05 §05).
• (C) The lateral-widening inequality fails for U(1) and $\mathbb Z_2$ (§02), and is expected to hold with margin $5/3$ for $SU(2)$ in the dilute regime (§04).

Proposition 2 Proved here If, in some theory, the conclusion of Theorem 1 of 13-02 ($\ln(1/\theta(R_k,R_k))\ge\gamma R_k^2$ on a dyadic sequence of scales, and a tension on the slab independent of $k$) is false, then at least one of (A), (B), (C) is false in that theory.

Proof The contrapositive of Theorem 1. ∎

For U(1) at weak coupling, the tension on the slab is estimated to go to 0 like $e^2/L^2$ (13-01 §03), and the conclusion of Theorem 1 is false. By Proposition 2 one of the three assumptions is false, and in the list above (A) holds and (C) fails. Within this reduction, the only assumption that is necessarily used to exclude U(1) and $\mathbb Z_2$ is the lateral-widening inequality, and that is where non-abelian dynamics enters Reading (physics; resting on estimates of the behaviour of U(1), $\mathbb Z_2$ and $SU(2)$).

Two qualifications go with this way of speaking.

• It depends on the reduction. "The only place" is a way of speaking within the chain of 13-02. In another reduction that splits the band into "β-monotonicity" and "the renormalisation comparison" (13-06), asymptotic freedom enters through the shift of the coupling in the comparison, $\beta'\lt\beta$. It is natural to read this as the same physics entering a differently shaped inequality in each reduction, but the precise correspondence between the two reductions has not been shown.
• The estimates of the sign are only at the two ends. The margin $5/3$ is a count in the dilute regime ($L\Lambda\ll1$) at one loop, and the "only a logarithm" of the confining regime is also an estimate. There is no tool that shows the lateral-widening inequality in the band $L\sim1/\Lambda$. The difficulty of the band is concentrated in this one interval of scales, within this one inequality.

These are values of the vortex free energy $\Delta F=F_v$ (the ratio to the partition function with the twist flux $n_{34}$ inserted, $\theta=\tanh(\Delta F/2)$) when the transverse $T^2$ is widened to $N_T\times N_T$, in the global form in which the container is the whole of the planar directions ($T^2_M$ with period $M$) ((8) of 13-01), measured by replica exchange and Bennett's acceptance-ratio method Reading. The lattice is $N_T^2\times M^2$, with twist $n_{12}=1$ and the Wilson action. The errors are the estimates given by the source, and the source does not state the number of seeds.

The reading is as follows Reading.

• For $U(1)$, at both values of $M$, $\Delta F$ increases when the transverse directions are widened (the vortex becomes costlier). At $M=8$, $\theta$ is $0.39,\ 0.78,\ 0.99$ at $N_T=4,6,8$, and $\ln(1/\theta)$ goes $0.93\to0.012$ for $N_T=4\to8$ (about a factor of 80 smaller). This is the direction predicted in §02.
• For $SU(2)$ the increase is much gentler, or there is none. At $M=8$, $\beta=2.5$, $\ln(1/\theta)$ goes $1.19\to0.45$ for $N_T=4\to8$ (about a factor of 2.6 smaller), and the ratio of $\theta$, $0.64/0.31\approx2$, is within the $4C$ that the strong form allows at $s=2$. At $M=12$ and at $\beta=2.4$, $\Delta F$ decreases for $N_T=4\to8$. The exception is $N_T=8$ at $\beta=2.5$, $M=8$ (the symmetric torus $N_T=M$), where $\Delta F$ rises from $0.33$ to $1.51$.

We write the limits with the same weight. Since this is a regime where $\theta$ is of order 1, the ratio of $\theta$ for U(1) is also held to about 2.5 because $\theta\le1$, and the discriminating power in this geometry is weak. $\beta$ takes one or two points for each theory, and $\beta=1.5$ for U(1) and $\beta=2.5$ for $SU(2)$ are not matched to the same physics. The errors are $0.04$–$0.8$. The values at $N_T=2$ are in the region of lattice artefacts (13-02 §06). These are signs of a direction, not the conclusion of a test of the lateral-widening inequality. The detailed tables of the measurements are in 12-08 and 13-09.

The dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] CIT BOG    Bogomolny bound S ≥ 8π²|Q|/g²                         continuum, classical
[02] CIT HALF   Q half-integer under the twist                        T²_L × R²
[03] CIT AF     8π²/g²(L) = (11N/3) ln(1/LΛ)                          one loop
[04] CIT TU     T_R ~ Λ²(ΛL)^(5/3) sin²(π|R|/N)                       dilute, one loop
[05] PRV CS     Lemma 1: ∫f² ≥ Φ²/|Ω|
[06] PRV ABEL   abelian action ≥ (A_T/2e²A_D)Φ² ⟸ [05]                same flux through each cross-section
[07] RDG COUL   U(1): θ ≈ 2exp(−e²A_D/2A_T)                           estimate
[08] RDG U1X    lateral widening fails for U(1) ⟸ [06][07]
[09] RDG Z2X    lateral widening fails for Z₂                         weak coupling, wall
[10] RDG LUMP   SU(2) vortex is a self-dual lump of size L ⟸ [01][02]  classical cost 0
[11] PRV POI    E[(−1)^n] = e^(−2m)
[12] RDG DIL    θ ≈ e^(−2ζA_D), ln(1/θ) ∝ A_D L^(5/3) ⟸ [03][04][11]  dilute, one loop
[13] RDG MAR    margin 5/3 = 11/3 − 2 is the sign ⟸ [08][10][12]
[14] RDG RAT    ratio ≤ 1 for dilute SU(2) ⟸ [12]
[15] RDG CONF   only a logarithm in the confining regime              estimate
[16] PRV CP     Proposition 2 (contrapositive of Theorem 1)           13-02
[17] RDG ONLY   in this reduction only lateral widening separates groups ⟸ [08][09][13][16]
[18] RDG NUM    direction of Table 2                                  small lattices, one or two β

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. The nodes that stand on paper are the four [05], [06], [11], [16], all of them elementary. The claims of the chapter, [13] and [17], are "Reading (physics)", and their premises include the estimate rows [07], [10], [12]. The numerical row [18] is not a premise of any conclusion. There is no node for the lateral-widening inequality itself. It is placed as an assumption in the ledger of 14-02.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the lateral-widening inequality, the sector bias $\theta$, the self-dual lump (fractional instanton), the dilute gas, the margin $5/3$, the transverse two-dimensional entropy, the Coulomb phase, the crossover band (the band). Their meanings are in A-02 Glossary and A-06 Index of assumptions.

Next chapter: 13-04 Structure alone cannot cross — why reflection positivity and duality alone do not give the lateral-widening inequality.

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