2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second roadthe container identity, nesting, limits in four dimensions · Kiichi, Shiori and Rin

Containers — conditioning on the frame makes things independent

This section is a summary for [Y3] readers. The formulae and proofs are from §02 on.

1. The Wilson action on the lattice is an action that couples only neighbouring links (nearest-neighbour). So the links inside a region and the links outside it talk only through the links on the border. If all of the border is held fixed, the insides of separate regions become independent of each other. This is exactly the Markov property of statistical mechanics.
2. The Wilson loop $W(C)$ can be rewritten as a product of "signs of the sector" $\sigma_j=\pm1$ per small region (container) on the surface spanned by the loop (in the sector-summed ensemble, §02). If the frame (the outside of the containers) is held fixed, the $\sigma_j$ are independent of each other, so the expectation becomes a product, and taking absolute values and averaging gives $|\langle W\rangle|\le\mathbb E[\prod_j\theta_j]$, with $\theta_j=|\mathbb E[\sigma_j\,|\,\text{frame}]|$. This is Lemma 5 Cited.
3. $\theta_j$ can be written $\tanh(|\Delta F_j|/2)$. $\Delta F_j$ is the difference in free energy, inside a container with the frame fixed, between the sector with one vortex and the sector without. The more cheaply a vortex enters a container (the smaller $\Delta F_j$), the smaller $\theta_j$, and the smaller the bound on the Wilson loop.
4. This identity is the "vortex container" identity of Mack–Petkova (1980), written as it stands in equations (18)–(22) of Kovács–Tomboulis (2001). The only differences are that we wrote it with the sectors summed, and that we took the average over frames rather than the maximum. This book claims no novelty.
5. To pass to the Wilson loop in a fixed sector (one twisted sector), containers are needed not only inside the loop but also outside it (Proposition 2 of §04) Proved here.
6. If containers are nested in the directions of the plane, the bias of a large container is bounded by the average of the product of the biases of small containers (Proposition 3 of §05) Proved here. But this concerns a slab with the size of the T² held fixed, and cannot be used for widening together with the T² (the crossover band).
7. In four-dimensional space (R⁴), the length of the containers through points inside the loop grows with the loop. To keep the cost of a container constant its thickness must grow to order $\sqrt{\ln R}$, and independent containers give only up to $V(R)\gtrsim R/\ln R$ (Kovács–Tomboulis) Cited. In a slab with a small T², the length of a container stops at the area of the T², so this obstruction does not appear.
8. Two-dimensional $SU(2)$ is solvable, so one can see in numbers how sharp the bound of Lemma 5 is. Tiling with containers of side $\ell$, the string tension coming from the bound approaches the true value in the form $\sigma-\ln(16/(3\pi))/\ell^2$ (§07) Proved here + Reading.

The lattice $\Lambda$ is a finite hypercubic lattice, and the link variables are $U=(U_b)\in SU(2)^{\text{links}}$. On each plaquette $p$ we place a sign $z_p\in\{\pm1\}$, and impose $\prod_{p\in\partial c}z_p=1$ (coclosed) on every 3-dimensional cube $c$. We set the weight and the measure to be

($dU_b$ is the Haar measure). $z$ in the same cohomology class are related by central gauge transformations $U_b\mapsto\omega U_b$ ($\omega\in\{\pm1\}$), so the sum (1) equals, up to a constant factor, the sum of the $SU(2)$ partition functions of all of 't Hooft's twisted sectors (12-01) (the summed ensemble of Mack–Petkova) Cited. Which sector one is in is fixed by the cohomology class of $z$, and the indicator function of a sector is a function of $z$. In this chapter we call (1) the sector-summed ensemble, and write its expectation $\mathbb E$.

Taking a loop $C$ and one surface $S(C)$ spanning $C$ (a collection of plaquettes whose boundary is $C$),

is unchanged by central gauge transformations, and $|W(C)|\le1$. The Wilson loop with the sector fixed to one is (2) multiplied by the indicator function of the sector, i.e. conditioned (§04).

Frame and containers. We take regions $R_1,\dots,R_n$ (containers) in the lattice, separated from each other, call the variables inside a container (the inner links and the $z$ of the inner plaquettes) $x_j$, and call all other variables the frame $\Phi$. The links on the border of a container belong to the frame. Since the action is nearest-neighbour, each plaquette term and each cube constraint depends only on the inside of at most one container and on the frame. Hence the weight splits when the frame is fixed.

If the loop $C$ runs inside the frame, and we write $\sigma_j(x_j)\in\{\pm1\}$ (the sign of the sector of the container) for the product of the $z$ on the part of $S(C)$ inside the container $R_j$, then (2) splits as well.

($c(\Phi)$ is $\tfrac12\mathrm{tr}\,U_C$ times the product of the $z$ on the part of $S(C)$ in the frame.) $\sigma_j$ is unchanged by central gauge transformations inside the container (the boundary $C$ of $S(C)$ lies in the frame), so it is well defined as a quantity on the conditional distribution with the frame fixed. Moreover, fixing the frame does not determine $\sigma_j$. This is because, in the small-T² slab $(\mathbb Z/N)^2\times(\mathbb Z/M)^2$, if a container is taken as "a disc of the plane × T²", then the surface $S_p=\{p\}\times T^2$ above a plaquette $p$ of the plane inside the disc ($N^2$ plaquettes of type (3,4)) is coclosed and lies inside the container, and flipping all the $z$ on it flips only $\sigma_j$ without changing the frame Proved here (the check that it is coclosed is step 2 of 12-04 §03).

With the frame $\Phi$ fixed, we write $q_j^\pm(\Phi)=\mathbb P(\sigma_j=\pm1\,|\,\Phi)$ for the probabilities of the sectors of container $j$, and $Z_j^\pm(\Phi)=\sum_{x_j:\sigma_j=\pm1}f_j(\Phi,x_j)$ for the partition functions of the two sectors inside the container (Haar integrals for continuous variables), and set

We call $\theta_j$ the bias of container $j$, and $\Delta F_j$ the vortex free energy with the frame fixed.

Cited (the identity) + Proved here (the rewriting in the summed ensemble). Proof. By (3), the conditional distribution with the frame fixed is the product of the distributions $f_j(\Phi,\cdot)/(Z_j^++Z_j^-)$ per container, and each $\sigma_j$ is a function of the variables of a different container, so the conditional expectation splits into a product. Substituting (4) gives the first formula. Taking absolute values, using $|c|\le1$ and averaging over the frame gives the second. Finally, from $q^++q^-=1$ and $q^\pm\gt0$,

∎

Lean FrameLemma (FrameLemma.frame_bound) (hypotheses: the space of frames, the number of containers and the space inside each container are finite, the inside of each container is non-empty, $a(\varphi)\gt0$, $f_j(\varphi,y)\gt0$. Conclusion: $|\sum_\varphi\sum_x a\prod_jf_j\cdot c\prod_j\sigma_j|\le\sum_\varphi a\prod_jZ_j\cdot|c|\prod_j|\sum_yf_j\sigma_j|/Z_j$, $Z_j=\sum_yf_j$), FrameLemma (FrameLemma.sign_mean_le_one) (hypotheses: $f\gt0$, $\sigma=\pm1$. Conclusion: the absolute value of the normalised conditional mean is at most 1), FrameLemma (FrameLemma.ratio_eq_tanh) (hypotheses: $q^+,q^-\gt0$. Conclusion: (7)), Containers (Containers.container_bound) (hypotheses: finite, $a\ge0$, $|s|\le1$. Conclusion: $|\sum_\varphi\sum_xa\,s\prod_i f_i\sigma_i|\le\sum_\varphi a\prod_i|\sum_yf_i\sigma_i|$). All are algebra of finite sums; that (3)(4) follow from the Haar integral, a coclosed $z$, the topology of sectors and the nearest-neighbour action is outside Lean.

Correspondence with the original source. Kovács–Tomboulis (hep-lat/0108017, Phys. Rev. D 65 (2002) 074501), in a fixed sector, fix the links on the border of a container $\Lambda_i$ and integrate the inside, and from the partition functions $z^{(\mp)}_{\Lambda_i}$ with the signs of the couplings on a coclosed surface $V_i$ flipped (their equation (19)) write

and attribute (20) to Mack–Petkova (Ann. Phys. 125 (1980) 117, their reference [3]) Cited (the formulae of the original source checked in the text of hep-lat/0108017; the text of Mack–Petkova has not been checked in this book). There are only two differences from Lemma 5. (i) Since we wrote it with (1), with the sectors summed, the factor is $(Z^+-Z^-)/(Z^++Z^-)=\tanh(\Delta F/2)$ rather than $\tfrac12(1-z^-/z^+)$ (a difference in the normalisation of the measure). (ii) (22) takes the maximum over frames, while (6) takes the average. (ii) leaves room to use the rarity of bad frames. In the two-dimensional test of §07, the averaged form approaches the true value faster.

Lemma 5 is a statement about the sector-summed ensemble. What one wants to see in physics is the Wilson loop with the twisted sector fixed to one. In the slab $(\mathbb Z/N)^2\times(\mathbb Z/M)^2$, let the loop $C$ lie in the (3,4) plane at a point $y_0$ of the T². Writing $(-1)^{n_{34}}$ for the parity of the (3,4) flux among the labels of the sector (the product of $z$ over the whole (3,4) plane at $y_0$), and $n_{12}$ for the indicator function of the other labels ($\mathbf 1_o$ and so on), the expectation with the sector fixed to $s\in\{\pm1\}$ is

If containers are placed only inside the loop, the second term of the numerator, $\mathbb E[W(-1)^{n_{34}}\mathbf 1_o]$, is a problem. $W(-1)^{n_{34}}$ is $\tfrac12\mathrm{tr}\,U_C$ multiplied by the product of the $z$ outside $S(C)$ in the plane; it contains no signs of the inner containers, so Lemma 5 does not make it small. The remedy is to place containers outside as well.

Proved here Proof. $n_{12},n_{13},n_{14},n_{23},n_{24}$ are each the product of $z$ over the respective plane, and that plane can be retaken on a straight strip passing only through the frame (since $z$ is coclosed, the product does not depend on the position of the plane), so they are functions of the frame. Hence $\mathbf 1_o$ is a function of the frame. The (3,4) plane splits into $S(C)$ and its complement, each of which splits into a frame part and container parts, so $W=c(\Phi)\prod_j\sigma_j$, $W(-1)^{n_{34}}=c'(\Phi)\prod_k\sigma'_k$, $(-1)^{n_{34}}=c''(\Phi)\prod_j\sigma_j\prod_k\sigma'_k$ ($|c|,|c'|\le1$, $|c''|=1$). Multiplying the first formula of Lemma 5 by $\mathbf 1_o$ and averaging over the frame gives $|\mathbb E[W\mathbf 1_o]|\le\mathbb E[\mathbf 1_o\prod_j\theta_j]$, $|\mathbb E[W(-1)^{n_{34}}\mathbf 1_o]|\le\mathbb E[\mathbf 1_o\prod_k\theta'_k]$, $|\mathbb E[(-1)^{n_{34}}\mathbf 1_o]|\le\mathbb E[\mathbf 1_o\prod_j\theta_j\prod_k\theta'_k]$. The numerator of (8) is at most half the sum of the first two, and the denominator is at least $\tfrac12(\mathbb E[\mathbf 1_o]-\mathbb E[\mathbf 1_o\prod\theta\prod\theta'])$. Dividing both by $\mathbb E[\mathbf 1_o]$ gives (9). ∎

The second term of (9) goes to 0 as the size of the plane $M\to\infty$ when the biases are typically smaller than 1 in the outer containers too (when the rarity of bad frames can be used outside as well). This book does not show that Assumption. In the theorem of slab confinement (12-04), the group of flips acts on all plaquettes of the plane, so the outer term automatically becomes small, $t^{M^2-A}$, and this problem does not arise.

We divide a large container $R$ (a disc of the plane × T²) into small containers $R_1,\dots,R_k$ separated by an inner frame $\Phi_{\rm int}$. The product of $z$ over the surface spanned by $R$ is $\sigma_R=s(\Phi_{\rm int})\prod_i\sigma_i$ ($|s|=1$, the product of $z$ over the part in the frame).

Proved here Proof. Apply Lemma 5 with $\Phi_R$ and $\Phi_{\rm int}$ together as the frame ($c=s$), and average over $\Phi_{\rm int}$ with $\Phi_R$ held fixed. ∎ Lean Containers (Containers.container_bound) (hypotheses: finite, $a\ge0$, $|s|\le1$) is the finite-sum version of this algebra, the form whose description in the Lean file says it "contains the Mack–Petkova container identity and the planar doubling inequality".

How to read it. If, for containers of some size, the bias is $\theta_0\lt1$ for typical frames, one can expect that each doubling of the containers in the directions of the plane lowers the bias to order $\theta_0^4$ Reading (a physical estimate: the right-hand side of (10) is the average of a product, not the product of averages, so this does not follow directly from the inequality). Disorder propagates to large scales in the directions of the plane.

However, (10) concerns a slab with the size of the T² held fixed. What is needed when widening the T² (approaching four-dimensional space) is the behaviour of the bias $\theta(L,L)$ when the T² and the containers are doubled together, and (10) says nothing about that (when the T² changes, the very surface around which the vortex sheet winds changes). That the form needed there cannot be "$\theta(L,L)$ stays uniformly away from 1" but must be $\ln(1/\theta(L,L))\ge cL^2$ (the contrast with $U(1)$) is treated in the chapter on the crossover band, 13-01.

Lemma 5 is an identity that depends neither on the size of the lattice nor on the dimension. So, if one lines up many containers in four-dimensional space (the lattice of R⁴), does an area law follow? Kovács–Tomboulis (hep-lat/0108017 §3) show by the following counting that independent containers do not reach it Cited ((23)–(27) below are the equation numbers of the original source; checked in its text).

1. Let $d_i$ be the width in the two directions transverse to the coclosed surface $V_i$ of a container $\Lambda_i$, and $|V_i|$ its longitudinal length. Assume that a sufficiently thick container is in the regime where $-\ln(z^{(-)}/z^{(+)})\sim|V_i|\exp(-c\,d_i^2)$ for all boundary values ((23) of the original source; $c$ is the $\rho$ of the original source).
2. To keep the free-energy cost of one container below a constant $f$, one needs $d_i^2\ge c^{-1}\ln(|V_i|/f)$ ((24)). Then $\max|f_{\Lambda_i}|\lt\tfrac12(1-e^{-f})$ is constant.
3. For a rectangular loop with sides $T\times R$ ($T\gt R$), the length of the containers through points away from the perimeter is $|V_i|\sim R^{d-2}$ ($R^2$ in four dimensions). So a thickness of order $\sqrt{\ln R}$ is needed, and $\sim RT/\ln R$ containers fit inside the loop. From (22)

is all that follows, which is not a purely linear potential (an area law). The original source goes on to state that if the boundary values of neighbouring containers are also integrated, grouping containers as thick as possible, and containers with transverse area $\sim R^2$ are used, one gets $V(R)\ge\mathrm{const}\,R-\frac{\mathrm{const}}R(\ln R+\mathrm{const})$ ((27)), that is, the linear potential comes from thick vortex fluctuations of the size of the loop. It also states that the form (22) is crude, and that the inequalities coming from reflection positivity (their (11)(13), the global form of Tomboulis–Yaffe) give an area law directly, with bounds uniform in the size of the lattice Cited.

Restated in the words of this chapter: in R⁴ the length of the containers grows with the loop, so the road of placing containers of fixed length independently does not reach the area law. In a small-T² slab, the length of the container $\{\text{disc}\}\times T^2$ (the area of the vortex sheet) stops at the area of the T², so this obstruction does not appear Proved here (a rereading of the fact that $|V_i|=N^2$ becomes a constant in (24) of the original counting). The obstruction returns when the T² is widened, and that is the crossover band (13-01, 12-09).

Two-dimensional $SU(2)$ (a planar lattice, Wilson action $\tfrac\beta2\mathrm{tr}\,U_p$) is solvable because the plaquettes behave independently of each other. Here we see how close the bound of Lemma 5 is to the true value. Two dimensions has no problem of "the length of containers", so this test says nothing about the four-dimensional obstruction of §06.

Setting. Tile the inside of the loop with square containers of side $\ell$, and take the links on the borders of the containers as the frame (container area $A=\ell^2$). With the frame fixed, container $j$ sees only the holonomy $V_j\in SU(2)$ of its border. Using the characters $\chi_n(V)=\sin((n+1)\vartheta)/\sin\vartheta$ ($e^{\pm i\vartheta}$ are the eigenvalues of $V$, and $n=0,1,2,\dots$ is $2\times$ the spin of the representation) and $a_n=I_{n+1}(\beta)/I_1(\beta)$ ($I$ is the modified Bessel function), the heat-kernel expansion gives

and the holonomies of the containers are independent of each other, distributed with the denominator as density. So the bound of Lemma 5 becomes a product of factors per container,

The true value is $\langle W\rangle=a_1^{\text{area}}$, and the true string tension is $\sigma=-\ln a_1$. We write $\sigma_{\rm L5}(\ell):=-\ln B(\ell^2)/\ell^2$ for the string tension coming from (13), and $\sigma_{\max}(\ell):=-\ln\bigl(\max_V|(12)|\bigr)/\ell^2$ for that coming from the form of (22) of Kovács–Tomboulis (maximum over frames).

The limit of large containers. As $A\to\infty$, the odd sum is dominated by the $n=1$ term $2a_1^A\chi_1(V)=4a_1^A\cos\vartheta$ and the even sum by the $n=0$ term 1, so with the Haar distribution $\tfrac2\pi\sin^2\vartheta\,d\vartheta$

Proved here. The bound of Lemma 5 is sharp in its exponent, and the loss vanishes as $1/\ell^2$.

How to read it Reading. (i) The $\ell=16$ column agrees with the right-hand side of (14) to the fourth decimal place (for example, at $\beta=1$, $1.426309-\ln(1.69765)/256=1.424242$). (ii) For thin containers ($\ell=1$), the bound becomes exponentially worse when $\beta$ is large (at $\beta=8$, $0.0016$ against $0.1994$). This is the two-dimensional form of the need for thick containers. (iii) The maximum form approaches the same limit, but lags behind. Lemma 5, which takes the average, is always closer to the true value at the same $\ell$.

The maximum form for $\ell=1$ in (ii) can be written in closed form. A container of side 1 is a single plaquette, and all four of its links belong to the frame, so $\mathbb E[z_p\,|\,V]=\tanh\bigl(\tfrac\beta2\mathrm{tr}\,V\bigr)$, and the maximum is $\tanh\beta$ at $\mathrm{tr}\,V=\pm2$ Proved here. The $\sigma_{\max}(1)$ of the table agrees with $-\ln\tanh\beta$ ($0.27234,\ 0.036635,\ 0.000671$ at $\beta=1,2,4$). This is the same formula as the case $N=1$ of the slab-confinement bound $\sigma\ge-\ln\tanh(\beta N^2)$ of the next chapter (12-04 §05).

(a) Novelty. The identity of Lemma 5 is the container identity of Mack–Petkova, written in (18)–(22) of Kovács–Tomboulis. The rewriting in the sector-summed ensemble, Proposition 2 and Proposition 3 are elementary, and have not been compared with literature other than Kovács–Tomboulis. No novelty is claimed for the results of this chapter.

(b) The second term of Proposition 2. That the biases of the outer containers are typically small (that the second term goes to 0 as $M\to\infty$) is not shown.

(c) The reading of Proposition 3. "$\theta_0^4$ at each doubling" is an estimate and does not follow directly from (10). (10) says nothing about widening together with the T².

(d) The four-dimensional counting. The $R/\ln R$ of §06 is counting on top of assumption (23) of the original source (that the free energy of a thick container has the form $|V_i|e^{-cd_i^2}$ uniformly in the boundary values), and is not a four-dimensional theorem. This book shows neither (23) nor its negation.

(e) Uniform bounds on the free energy of containers. Bounding $\Delta F_j$ uniformly over good frames and uniformly in the lattice spacing is not shown anywhere in this book. That needs a multiscale analysis of the conditional measure with the frame fixed, and the variational principle alone (which gives only lower bounds on partition functions) points the wrong way.

(f) Lean. The machine-checking column is only the algebra of finite sums. That (3)(4) follow from the Haar measure, a coclosed $z$, the topology of sectors and the nearest-neighbour action is outside Lean.

(g) The two-dimensional test. Table 1 is two-dimensional numerics and says nothing about the problem of the length of containers in four dimensions.

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 (propext, Classical.choice, Quot.sound). What is guaranteed is only the algebra of sums and products of real numbers over finite sets (10-13).

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

[01] CIT SUM    sector-summed ensemble (1)                        SU(2), finite lattice
[02] PRV MKV    splits by frame (3)(4); σ_j not fixed by frame    nearest-neighbour
[03] CIT KT     Kovács–Tomboulis (18)(20)(22)                     fixed sector
[04] PRV L5     Lemma 5 (6)(7) ⟸ [01][02] (same identity as [03]) Lean FrameLemma.frame_bound etc.
[05] PRV P2     Prop. 2 (9) ⟸ [04]                                slab, inner and outer
[06] ASM OUT    second term of (9) → 0                            rarity outside
[07] PRV P3     Prop. 3 (10) ⟸ [04]                               T² fixed
[08] RDG TH4    θ₀⁴ per doubling (physics) ⟸ [07]
[09] CIT KT3    R/ln R; thick containers linear (on assn. (23))   four dimensions
[10] PRV LEN    in a slab container length stops ⟸ [09]
[11] PRV ASY    limit 16/(3π) ⟸ [04]                              two dimensions
[12] RDG TAB    Table 1 ⟸ [04][11]                                floating point
[13] PRV ONE    maximum for side 1 = tanh β ⟸ [03]                two dimensions

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. [06] is an assumption row and enters as a premise of no row. The inequality (9) of [05] holds without [06]; [06] is needed only when reading (9) as an area law. This chapter has neither a node of the continuum limit nor a node of the four-dimensional area law.

What this chapter can say

Sources and reproduction

Words used in this chapter — the sector-summed ensemble, frame, container, the sign of the sector, the bias $\theta$, the vortex free energy with the frame fixed $\Delta F$, the container identity (Lemma 5), inner and outer containers, nesting, the length and thickness of containers. Their meanings are in A-02 Glossary (the words of Part 12 are to be added).

Next chapter to read: 12-04 Slab confinement — letting the group of flips act on all plaquettes of the plane gives, in a slab with the lattice spacing and the T² fixed, an area law at every $\beta$.

← 12-02 Vortices and fractional instantonsContents12-04 Slab confinement →