2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second roadthe band, lateral widening, the end of the road · Kiichi, Shiori and Rin

Where this road stops — widening the T² is the band

The road of this part went by way of a small twisted $T^2$ in order to split the four-dimensional problem in two (12-00). Since the twist removes the zero modes, the theory stays weakly coupled at scales up to the size $L$ of the $T^2$, and the vortex sheets winding the $T^2$ close locally as fractional instantons. In this way the problem splits into an intermediate goal where only the ultraviolet wall remains (the gap and confinement on a small $T^2$, stages 3 and 4 of the seven) and a stage where all other walls gather (widening the $T^2$, stage 5).

What is in hand on the side of the small $T^2$ is all either conditional or stated with the lattice spacing fixed. With the lattice spacing and the $T^2$ fixed, the area law holds at every $\beta$, but the tension is fixed by the cost of thin vortices and becomes 0 in the continuum limit Proved here (12-04). The insertion identities and averaged bounds reduce confinement to a single local quantity Proved here (12-05). Confinement and the gap on a small $T^2$ with the lattice spacing sent to 0 are a strand resting on assumptions such as control of the ultraviolet Assumption (arrows G1 and C2 of 14-01).

What the problem statement asks for is the theory on $\mathbb R^4$, so $L\to\infty$ is needed. One must widen $L$ from well below the correlation length $1/\Lambda$ to well above it, keeping a bound with the same constant at every $L$. This stage is where this road stops, and in this book we call it the crossover band (the band for short) Assumption. It is a different word from the "windows" of Parts 10 and 11 (the site window, the window in $\hbar$, the time window) (note in 13-00).

The statement to show in the band, in the language of the bias of the sector of a container $\theta(R;T^2_L)$ (container side $R$, side $L$ of the transverse $T^2$, 12-03), has the form that the bias does not grow when the transverse $T^2$ is widened (the lateral-widening inequality, 13-02). The situation differs at the two ends and in the middle.

Putting U(1) beside it as a control: the flux of $U(1)$ can only spread uniformly, and $U(1)$ on a slab becomes at long distances a two-dimensional $U(1)$ with coupling $e^2/A_T$ ($A_T$ is the area of the $T^2$), so $\ln(1/\theta)\propto A_D/A_T$, and the ratio of biases when the $T^2$ is widened (after over before) becomes arbitrarily large Reading (physics: the estimate counting of Paper 2). Weakly coupled four-dimensional $\mathbb Z_2$ breaks it in the same way. What separates SU(2) from U(1) is that non-abelian flux can localise into lumps of finite action independent of scale (13-03). The measured record (the vortex free energy when the $T^2$ is widened goes up for U(1) and not for SU(2); the slab tension grows and saturates) is in 12-08 and 13-09 Reading.

The goal of the band is not "the bias stays uniformly away from 1" ($\sup_L\theta(L;T^2_L)\le\theta_0\lt1$). This form is satisfied by four-dimensional $U(1)$ too, and the tension it gives is $\ln(1/\theta_0)/L^2\to0$ (13-01). The correct form is that the vortex free energy of the container becomes small exponentially in the area,

a local version, conditioned on the frame, of 't Hooft's criterion for confinement Assumption. (1) follows by iterating doubling, given the base point, frame decorrelation and the lateral-widening inequality (Theorem 1 of 14-01 §03 and its premises). In the language of the chessboard, the content of (1) becomes two uniformities: the uniform slab tension and the uniform slab gap (13-05).

As intermediate goals before reaching (1), two weaker statements have been named Assumption. (a) That the vortex free energy of the container approaches 0 as $L\to\infty$ (a restatement of there being no deconfined phase). (b) $\ln(1/\theta)\ge cL^2/\ln L$ (an almost-area law). Neither has been shown.

This road came as far as just before the band and stopped. What is needed in the band can be narrowed in form with the tools of this part: the correct form (1), the reduction to one inequality, where non-abelianness enters, and that structure alone cannot cross, in that order. The road from the direction of time (Parts 10 and 11) also arrives at a uniformity of the same type when its four-dimensional goal is written in the form "a constant uniform in $\beta\to\infty$" (10-15 §08). That the two roads meet at this band is treated in the entrance of Part 13.

What this chapter can say

Sources and reproduction

Words used in this chapter — the crossover band (the band), the lateral-widening inequality, the bias of the sector $\theta(R;T^2_L)$, the dilute region, the confinement region, intermediate goals. Their meanings are in A-02 Glossary.

Next chapter to read: 13-00 Entrance — confluence — that the two roads arrive at a statement of one type, "the same constant at all scales".

← 12-08 The measured recordContents13-00 Entrance — confluence →