2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second roadthe slab-confinement theorem, group averaging, thin vortices · Kiichi, Shiori and Rin

Slab confinement

1. Two-dimensional lattice gauge theory confines at every coupling (Wilson loops decrease exponentially in the area). This is because the plaquettes behave independently of each other (6-02).
2. Seen from afar, the lattice of a slab $T^2\times\mathbb R^2$ with two small closed directions is a two-dimensional system. So, with the size of the T² held fixed, one can expect it too to confine at every coupling. This chapter states that on the lattice, with a proof Proved here.
3. The tool is the sector-summed ensemble of 12-03. For each plaquette $p$ of the plane, consider the sheet $S_p$ of the $N^2$ (3,4) plaquettes at all points of the T² standing directly above it. Flipping the signs $z$ on $S_p$ all at once preserves the coclosedness condition and leaves the links unchanged. Flipping at a $p$ inside the loop changes the sign of the Wilson loop, and the weight changes by at most a factor $e^{\pm2\beta N^2}$.
4. Averaging over all combinations of flips ($2^{M^2}$ of them), the Wilson loop is bounded by a product of factors $(1-r_p)/(1+r_p)$ over the inner plaquettes, and each factor is at most $\tanh(\beta N^2)$. Hence $|\langle W\rangle|\le\tanh(\beta N^2)^{A}$ ($A$ is the area of the loop).
5. This lower bound is superexponentially weak in $N$ (at $\beta=2.5$, $N=4$, $\sigma\ge2e^{-80}$), and becomes 0 in the continuum limit. This is because the cost $2\beta N^2$ of flipping a thin vortex sheet at once grows with the number of points of the T² lattice. The cost of a physical vortex (a fractional instanton) is of order $\pi^2\beta$ and does not depend on $N$, so the road of doing the same thing with thick vortices is the next two chapters (12-05, 12-06).

The lattice is $\Lambda=(\mathbb Z/N)^2\times(\mathbb Z/M)^2$. Directions 1 and 2 form the T² (side $N$), and directions 3 and 4 the plane (cut off with period $M$). The group is $SU(2)$, the action is the Wilson action $\tfrac\beta2\sum_p\mathrm{tr}\,U_p$, and the twist of the T² ($n_{12}$ and so on) is fixed arbitrarily. The loop $C$ lies in the (3,4) plane at a point $y_0$ of the T², and encloses a region $S(C)$ of area $A$ (number of plaquettes) in the plane.

Proved here (proof in §03). The $\sigma$ of (2) is the string tension in lattice units, and the conditions are "$SU(2)$, Wilson action, $N$ fixed, lattice spacing fixed, every $\beta$". The right-hand inequality of (2) follows from setting $u=e^{-2x}$ and $-\ln\tanh x=\ln(1+u)-\ln(1-u)=2(u+u^3/3+\cdots)\ge2u$.

Step 1 (the sector-summed ensemble). (1) of 12-03 §02: for coclosed $z\in\{\pm1\}^{\text{plaquettes}}$ and $U$ the weight is $w(U,z)=\exp(\tfrac\beta2\sum_pz_p\mathrm{tr}\,U_p)$, and the measure is Haar × counting. Every $z$ that can be written $z=z_0\cdot\delta\omega$ ($\omega\in\{\pm1\}^{\text{links}}$, $(\delta\omega)_p=\prod_{b\in\partial p}\omega_b$) gives the same integral as $z_0$ by the invariance $U_b\mapsto\omega_bU_b$ of the Haar measure, so the sum over $z$ in one cohomology class is a constant multiple of the $SU(2)$ partition function of the twisted sector that the class represents Proved here. The sector is fixed by the class of $z$, and the indicator function of a sector is a function of $z$. The gauge-invariant Wilson loop is $W(C)=\tfrac12\mathrm{tr}\,U_C\prod_{p\in S(C)}z_p$; under the same substitution $\prod_{p\in S(C)}(\delta\omega)_p=\prod_{b\in C}\omega_b$ cancels against the sign on the side of $\mathrm{tr}\,U_C$, so within a sector it has the same expectation as $\tfrac12\mathrm{tr}\,U_C$ (up to a sign depending on the choice of representative of the sector).

Step 2 (vortex sheets are coclosed). For each plaquette $p$ of the plane set $S_p:=\{\,p\text{'s overlying (3,4) plaquettes}\,\}\times T^2$ ($N^2$ plaquettes). Among three-dimensional cubes, only those of orientation (1,3,4) and (2,3,4) have (3,4) faces; for example, the (3,4) faces of the (1,3,4) cube that extends in direction 1 from a point $y$ of the T² and spans the plaquette $p'$ of the plane are the two copies of $p'$ at $y$ and $y+e_1$. If $p'=p$ both lie in $S_p$; if $p'\ne p$ neither does. Cubes of type (1,2,3) and (1,2,4) have no (3,4) faces. Hence every cube contains 0 or 2 faces of $S_p$, and flipping the $z$ on $S_p$ preserves the coclosedness condition Proved here. If $p\ne q$, $S_p$ and $S_q$ have no plaquette in common.

Step 3 (the group action). Let $P$ be the set of all plaquettes of the plane ($|P|=M^2$), and assign to a subset $g\subseteq P$ the map $z\mapsto z\cdot\chi_{S_g}$ (flipping on $S_g=\bigcup_{p\in g}S_p$). This is an action of the group $G=\{\pm1\}^P$, a bijection on coclosed $z$, and since it does not change $U$ it preserves the Haar × counting measure.

• The Wilson loop: $S(C)$ consists of $A$ plaquettes in the plane at $y_0$, and $S_p$ contains exactly one plaquette of the plane at $y_0$, the one at the position of $p$. Hence $W(g\cdot x)=W(x)\prod_{p\in g}\varepsilon_p$, with $\varepsilon_p=-1$ ($p\in S(C)$) and $+1$ (otherwise).
• The weight: flipping the $z$ on $S_p$ changes the action by $-\beta\sum_{q\in S_p}z_q\mathrm{tr}\,U_q$, so setting $r_p(x):=\exp(-\beta\sum_{q\in S_p}z_q\mathrm{tr}\,U_q)$, from $|\mathrm{tr}\,U_q|\le2$ we get $r_p\in[e^{-2\beta N^2},e^{2\beta N^2}]$. $r_p$ is determined only by the values on $S_p$, and the sheets do not meet, so $w(g\cdot x)=w(x)\prod_{p\in g}r_p(x)$.

Step 4 (averaging). Let $\mathbb E$ be the expectation of the sector-summed ensemble, and $\sum_x$ the sum over $z$ together with the Haar integral over $U$. For each $g$, relabelling the variables gives $\sum_xW(x)w(x)=\sum_xW(g\cdot x)w(g\cdot x)=\sum_xW(x)w(x)\prod_{p\in g}\varepsilon_pr_p(x)$. Averaging over $g$ and using $\sum_{g\subseteq P}\prod_{p\in g}\varepsilon_pr_p=\prod_{p\in P}(1+\varepsilon_pr_p)$,

The one-sheet inequality $|1-r|\le\tanh(a)\,(1+r)$ ($r\in[e^{-2a},e^{2a}]$, here $a=\beta N^2$) follows from $(1-r)/(1+r)=-\tanh(\tfrac12\ln r)$ and $|\tfrac12\ln r|\le a$. Together with $|W|\le1$, the absolute value of the right-hand side of the left formula of (3) is at most $t^A\sum_xw\prod_P(1+r_p)=t^A\,2^{M^2}\sum_xw$. Hence

At this step there is no need to split into orbits. We use only the relabelling (a bijection) for each $g$, and the exchange of the sum over $g$ with the sum over $x$.

Step 5 (projection to a fixed sector). The group $G$ changes only the parity of the (3,4) flux. $S_p$ contains one plaquette of the (3,4) plane at $y_0$, so it flips $(-1)^{n_{34}}$ (the product of $z$ over the whole (3,4) plane at $y_0$); it contains no plaquettes of other orientations, so it does not change the indicator function $\mathbf 1_o$ of the other labels such as $n_{12}$. The expectation with the sector fixed to $s$ is, as in (8) of 12-03,

$W(-1)^{n_{34}}=\tfrac12\mathrm{tr}\,U_C\prod_{p\notin S(C)}z_p$ (the complement in the plane at $y_0$, of area $M^2-A$) changes sign under flips at $p\notin S(C)$ and does not change under flips at $p\in S(C)$. $(-1)^{n_{34}}$ changes sign under flips at every $p$. $\mathbf 1_o$ is invariant under $G$, so by the same averaging as in step 4 (with the roles of $\varepsilon_p$ exchanged)

Bounding the numerator and denominator of (5) by (6) gives (1). As $M\to\infty$, $t^{M^2-A}\to0$ and $t^{M^2}\to0$ ($t\lt1$), so at limit points $|\langle W\rangle|\le t^A$, and (2) follows. ∎

Here the group acts on all plaquettes of the plane, so the problem of 12-03 §04, "the outer term does not become small", does not arise. The outer term automatically becomes small, $t^{M^2-A}$.

Lean SlabConfine (SlabConfine.group_average) (no hypotheses: $\sum_{g\subseteq P}\prod_{p\in g}\varepsilon_pr_p=\prod_{p\in P}(1+\varepsilon_pr_p)$), SlabConfine (SlabConfine.factor_split) (no hypotheses: the splitting into inner and outer), SlabConfine (SlabConfine.tanh_bound) (hypotheses: $t\ge0$, $r_p\gt0$, $|1-r_p|\le t(1+r_p)$), SlabConfine (SlabConfine.one_sheet) (hypotheses: $a\ge0$, $e^{-2a}\le r\le e^{2a}$. Conclusion: $|1-r|\le\tanh a\cdot(1+r)$), SlabAverage (SlabAverage.average_bound) (hypotheses: the set of configurations is finite, $w\gt0$, $|W|\le1$, the action of each $g\subseteq P$ is a bijection, $r_p\gt0$, $|1-r_p|\le t(1+r_p)$, $W(gx)w(gx)=W(x)w(x)\prod_{p\in g}\varepsilon_pr_p(x)$, $w(gx)=w(x)\prod_{p\in g}r_p(x)$. Conclusion: $|\sum Ww|\le t^{|P\cap A|}\sum w$). These are the finite-sum version of step 4. Replacing the sums by Haar integrals, and the coclosed $z$ and the topology of sectors (steps 1, 2 and 5), are outside Lean and rest on the argument in the text.

Carrying out the same averaging only inside one container, it can be written in the language of the biases of 12-03. Take as container the product of a square $Q$ of side $\ell$ in the plane ($\ell^2$ plaquettes) with the T², and let $\sigma_Q$ be the product of $z$ over the surface spanned by $Q$. Flips by $g\subseteq Q$ change only the $z$ inside the container, so they do not change the frame $\Phi$, and they multiply $\sigma_Q$ by $\prod_{p\in g}(-1)$. On the orbit $\{g\cdot x:g\subseteq Q\}$ of one $x$, the weight has the product form $w(x)\prod_{p\in g}r_p(x)$, so conditioning on the orbit makes the components of $g$ independent, and

Proved here. Conditioning on the orbit is finer than conditioning on the frame, so

Proved here. That it can be bounded by the maximum over frames is because the cost of flipping a thin sheet, $|\ln r_p|\le2\beta N^2$, is uniform and independent of the frame.

Extremely weak. The lower bound of (2) becomes superexponentially small in $N$. At $\beta=2.5$, $N=4$, $2\beta N^2=80$ and $\sigma\ge2e^{-80}\approx3.6\times10^{-35}$. The lower bound $2e^{-2\beta N^2}/a^2$ on the physical string tension $\sigma_{\rm phys}=\sigma/a^2$ goes to 0 if the physical size $L=Na$ of the T² is held fixed and $a\to0$ ($N=L/a\to\infty$, and $\beta$ increases by asymptotic freedom) Proved here. In the continuum limit, the lower bound on the tension in this theorem is 0.

The reason is the cost of thin vortices. The cost of a flip, $2\beta N^2$, is the action of turning over at once the $N^2$ plaquettes of the lattice sheet $S_p$, one lattice unit thick, and it grows with the number of points of the T² lattice. On the other hand, what physics regards as the cost of a vortex is of the order of the action $4\pi^2/g^2=\pi^2\beta$ ($\beta=4/g^2$) of a smooth fractional instanton winding the T², which does not depend on $N$ (12-02) Reading (physics). The same group-averaging argument goes through with $t=\tanh(F/2)$ whenever the ratio of weights $r_p$ of a flip lies in $[e^{-F},e^{F}]$, so if the flip of a thin sheet is replaced by "a measure-preserving map that inserts a thick vortex" and $F$ can be bounded by a quantity independent of $N$, a lower bound uniform in the lattice spacing can be expected Assumption (where the evidence is: the form (7) of this chapter, the insertion identities of 12-05, and the insertion driven by dynamics of 12-06). That needs a uniform bound on the cost of insertion (control of the ultraviolet), and this book does not show it.

(8) also shows that arguments using $\sup_\Phi$ get superexponentially worse in $N$. To get uniformity in the problem of widening the T² (the crossover band), only the averaged form over frames can be used, not the maximum (one of the reasons why Lemma 5 of 12-03 takes the average).

Applying steps 1–5 directly to the two-dimensional lattice $(\mathbb Z/M)^2$, there are no cubes, so the coclosedness condition is empty, and the vortex sheet is $S_p=\{p\}$ (a single plaquette, the formula of Theorem S with $N=1$). So the same proof gives $|\langle W\rangle|\le\tanh(\beta)^A$ for two-dimensional $SU(2)$. The true value in two dimensions is $\langle W\rangle=a_1^A$, $a_1=I_2(\beta)/I_1(\beta)$, so the theorem contains $I_2(\beta)/I_1(\beta)\le\tanh\beta$ for every $\beta$.

• Leading terms at large $\beta$: from $I_\nu(\beta)\sim e^\beta(2\pi\beta)^{-1/2}(1-\tfrac{4\nu^2-1}{8\beta})$ we get $a_1\approx1-\tfrac3{2\beta}$, while $\tanh\beta\approx1-2e^{-2\beta}$. The lower bound is exponentially smaller than the true tension $\approx\tfrac3{2\beta}$, and there is no contradiction Proved here.
• In numbers: the true $\sigma=-\ln a_1$ is $1.4263,\ 0.8367,\ 0.4185,\ 0.1994$ at $\beta=1,2,4,8$, and the lower bound of the theorem $-\ln\tanh\beta$ is $0.2723,\ 0.0366,\ 0.00067,\ 2.25\times10^{-7}$ Reading (floating point).
• This lower bound agrees exactly with "the maximum form for containers of side 1", $\sigma_{\max}(1)$, in Table 1 of 12-03. This is because in a container of side 1 only the $z$ of one plaquette is inside, $\mathbb E[z_p\,|\,V]=\tanh(\tfrac\beta2\mathrm{tr}\,V)$, and its maximum is $\tanh\beta$ Proved here (12-03 §07). The slab theorem has the same form as the bound "containers one unit thick, bounded by the maximum over frames".

Measurements of Wilson loops on four-dimensional slabs (Creutz ratios, comparison with and without twist) are placed in the chapter of the measured record, 12-08.

The method of Theorem S (averaging, as a group, the signs on central sheets winding the compact directions) is of the same lineage as the vortex free-energy arguments of Mack–Petkova and Tomboulis–Yaffe, and amounts to bounding directly by the action the free energy of a container $\{p\}\times T^2$ one unit thick Cited (the correspondence of lineage is by comparison with the framework of Kovács–Tomboulis hep-lat/0108017 §2–§3). Since the idea is classical, the same bound may already exist in the literature, or as folklore. Two comparisons remain.

• Arguments of Tomboulis–Yaffe type (Commun. Math. Phys. 100 (1985) 313): inequalities that bound a Wilson loop by a ratio of twisted partition functions (as restated in Kanazawa arXiv:0808.3442). We have not checked whether a bound of the same form on a slab is written out explicitly.
• Borgs (Nucl. Phys. B 261 (1985), the area law of spatial Wilson loops in lattice gauge theory at high temperature): recorded as work that showed, by a cluster expansion, an area law for spatial Wilson loops at sufficiently high temperature. This book has not checked the text of the original source (bibliographic data from secondary records). The expected differences from Theorem S are three — non-abelian $SU(2)$, every $\beta$ with $N$ fixed, and an explicit and elementary bound — but the comparison is not done.

Within this book, Theorem S reappears in the next chapter as a special case (thin insertion) of the insertion identities (12-05 §06).

(a) The continuum limit. The lower bound of (2) becomes 0 as $a\to0$. Theorem S says nothing about a tension uniform in the lattice spacing.

(b) Four-dimensional space. Theorem S is a statement with the size $N$ of the T² fixed, and the lower bound vanishes as $N\to\infty$ (approaching four-dimensional space). It says nothing about confinement in four dimensions. The stage of widening the T² is the crossover band (12-09, 13-01).

(c) Improvement to thick vortices. "Bounding $F$ independently of $N$ with a map of thick vortices" in §04 is a provisional strand; there is no proof, including of the uniform bound on the cost of insertion.

(d) Earlier literature. The comparison with the two items of §06 is not done. No novelty is claimed for the results of this chapter.

(e) Lean. The machine-checking column is only the algebra of finite sums (the group-average identity, the tanh bound, the one-sheet inequality). The Haar measure of $SU(2)$, the coclosed $z$, and the topology of sectors and the projection (steps 1, 2 and 5) are outside Lean.

(f) Existence of the limit. The existence of the limiting measure as $M\to\infty$ is not treated in this chapter; the statement is an upper bound at all limit points.

The bundles in the machine-checking column are those confirmed, by re-checking the main text, to contain no sorry and to use axioms forming a subset of the three standard ones. What is guaranteed is the algebra of sums over a finite set of configurations; steps 1, 2 and 5 and the Haar integral are outside Lean (10-13).

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

[01] PRV SUM    summed = sum of sector partition fns; W matches   SU(2), finite lattice
[02] PRV SHEET  vortex sheets S_p coclosed, disjoint              slab
[03] PRV ACT    W times ε_p, weight times r_p, |ln r_p| ≤ 2βN²
[04] PRV AVG    group average (3)(4) ⟸ [02][03]                   Lean average_bound etc.
[05] PRV PROJ   to a fixed sector (5)(6) ⟸ [01][04]
[06] PRV THM    Theorem S (1)(2) ⟸ [05]                           N fixed, a fixed, every β
[07] PRV BOX    container form (7)(8) ⟸ [03]
[08] PRV ZERO   lower bound 0 in continuum limit ⟸ [06]           L = Na fixed
[09] RDG FI     cost of physical vortex π²β (physics)             dilute, semiclassical
[10] ASM THICK  uniform bound with thick vortices ⟸ [07][09]      under UV control
[11] PRV TWO    to two dimensions; matches maximum form ⟸ [06]
[12] RDG NUM    numbers of §05 ⟸ [11]                             floating point
[13] CIT FAM    lineage of vortex free energy

The way to check is to see by eye that every number to the right of "⟸" is smaller than the number on the left. There are no circles. The premises of Theorem S ([06]) are only [01]–[05], and the assumption row [10] enters as a premise of no row. The conclusion of [10] is "a lower bound uniform in the lattice spacing", and does not contain the quantity of [06] (a lower bound with $N$ fixed).

What this chapter can say

Sources and reproduction

Words used in this chapter — slab, the vortex sheet $S_p$, the group of flips, group averaging, the cost of thin vortices, the bias of a container $\theta_Q$, the slab-confinement theorem (Theorem S). Their meanings are in A-02 Glossary (the words of Part 12 are to be added).

Next chapter to read: 12-05 Vortex-insertion identities — if the flip of a thin sheet is replaced by a general "bijection that flips the sector", what can be said unconditionally.

← 12-03 Containers — conditioning on the frame makes things independentContents12-05 Vortex-insertion identities →