2026-10-15 · chapter Yang–Mills and the mass gapPart 12 The second road: from the direction of spaceentrance, the ladder of scales, map of ten chapters · Kiichi, Shiori and Rin

Entrance — space, not time

The road of Part 10 regarded one direction of the lattice as "time", and approached the mass gap from the gap of the transfer matrix and from the time over which fluctuations are smoothed out (the diffusive-relaxation-time assumption, formerly Assumption H). Where it stopped is summarised in 10-15.

In Part 12 we change direction. Of the four directions of the four-dimensional lattice, we wind two (directions 1 and 2) into a torus $T^2$ with small period $L$, and regard the other two (directions 3 and 4) as a large plane. We ask whether a Wilson loop $W(C)$ drawn in the plane decreases as $\langle W(C)\rangle\le e^{-\sigma A}$ in the area $A$ it encloses (the area law). Where the time road looked at "the gap", what this road looks at is "how vortices are counted in the plane".

We put 't Hooft's twist (12-01) on the wound $T^2$. It is known in the physics literature that if one shrinks to a periodic $T^2$ without twist, a family of flat connections (zero modes) remains, and there is no prospect of joining the small $T^2$ to large four dimensions without a phase transition Cited (the title of §2.1 of Tanizaki–Ünsal 2022 is "compactification to periodic $T^2$ and the absence of adiabatic continuity"). For a twisted $T^2$, the same paper states the conjecture that the structure of the weakly coupled vacuum on a small $T^2\times\mathbb R^2$ is joined without a phase transition to the strongly coupled region of $\mathbb R^4$ Cited. This conjecture remains a conjecture, and this book also treats it as one.

If one starts from the lattice, builds the four-dimensional theory and tries to show confinement or a mass gap, one hits two walls of different nature.

1. The ultraviolet wall. One needs uniform bounds at all scales that do not break down as the lattice spacing $a$ approaches 0. Thanks to asymptotic freedom (4-06) the coupling becomes weak at short distances, and one can consider a multiscale analysis that controls each scale in perturbative language. For four-dimensional Yang–Mills the closest to this is Bałaban's series; what it reached and what remains are in 6-07. In this book we treat this as an assumption named "control of the ultraviolet" Assumption.
2. The crossover band. One must cross, without relying on perturbation theory, the intermediate scale at which the coupling passes from weak (short distances) to strong (long distances), around $1/\Lambda$ in physical length ($\Lambda$ is the scale born of dimensional transmutation, 4-07). On each side there are tools (perturbation theory and semiclassics on the weak-coupling side, the strong-coupling expansion on the strong-coupling side, 5-02). That no tool works inside the band is where both the road of Part 10 of this book and this road stop (Part 13).

If one heads straight for four dimensions, the two walls appear at the same time. This is because one has to manage, within a single bound, both making the lattice spacing small (the ultraviolet) and looking at long distances in a large box (crossing the band).

The skeleton of this road is that going by way of a small twisted $T^2$ lets us separate the two walls into different stages. There are two reasons.

Reason 1: the twist removes the zero modes, so one can proceed at weak coupling up to the size of the $T^2$ (an estimate). On a twisted $T^2$, the configuration with action 0 is fixed to be unique up to gauge (12-01 Fact 1), and the lattice Laplacian of the adjoint fluctuations around it has a positive gap $4\sin^2(\pi/2N)$ (12-01 Fact 2, $N=L/a$) Proved here. In physical units, when $N$ is large this gap is of order $(\pi/L)^2$, and all classical fluctuations have masses of order $1/L$. The running of the coupling then stops at the size $L$ of the $T^2$, and if $L\Lambda\ll1$ the effective coupling $g^2(L)$ is expected to stay small Reading (a physical estimate, on the assumption of control of the ultraviolet). The band does not appear in this stage.

Reason 2: the surface of a vortex winding the $T^2$ closes locally in the plane. A centre vortex (7-02) is a two-dimensional surface in four dimensions, and in four dimensions the vortex surface linked with a Wilson loop in the plane is an extended object whose cost grows with area. In the work of this book, every road that tried to cross the band while staying in four dimensions stopped at one point: "a topological quantity (whether a vortex links a loop) cannot be compared monotonically by local operations" (map in 13-08) Reading. In $\mathbb R^2\times T^2$, the surface $\{p\}\times T^2$ above a point $p$ of the plane is a closed surface of area $L^2$, and in the plane it behaves like a point. Adding or removing one vortex sheet becomes an operation inside a bounded region of the plane Proved here (that the surface lies in a bounded region is a fact from the definition). This locality supports all of the containers (12-03), slab confinement (12-04) and the insertion identities (12-05).

In this way the problem splits in two.

• The intermediate goal (on a small $T^2$): on $\mathbb R^2\times$ a small twisted $T^2$, show the gap and the area law uniformly in the lattice spacing $a$. The only wall that remains is the ultraviolet (though this is a conditional research plan, placing assumptions named control of the ultraviolet, uniform acceptance of insertions and frame decorrelation, and not a theorem) Assumption.
• The last stage (widening the $T^2$): that the strength of the area law stays uniform even as $L$ is made larger, beyond $1/\Lambda$. The only wall that remains is the band (12-09) Assumption.

Splitting does not make it easier. The band remains whole in the last stage. What the splitting gains is that all the stages before the band can be arranged in a form that can be checked independently of the band, and that the form of the inequality the band demands can be narrowed to one (Part 13).

Figure 2 takes the physical length $L$ (the side of the small $T^2$, or the scale being looked at) vertically, and lays out, from the lattice spacing $a$ at the bottom to infinity at the top, which tool works at which scale. It is a figure for seeing in one picture the shape of the problem: there are tools at the bottom (weak coupling) and at the top (strong coupling), and only the band in the middle is empty. The number and title of the figure match the entrance of Part 13 (Figure 2 "the ladder of scales" of 13-00). Figure 1 (the flow of the logic) is in 9-06.

Figure 2 in words is as follows. From the bottom: from the lattice spacing $a$ up to the base-point length $L_0$ is the ultraviolet stage (multiscale analysis; an assumption in this book); $L_0$ is the base point of the small twisted $T^2$ (where the tools of Part 12 work directly); from $L_0$ to $1/\Lambda$ is the stage of the dilute gas of fractional instantons (a semiclassical estimate, with string tension of order $\Lambda^2(\Lambda L)^{5/3}$); around $1/\Lambda$ is the crossover band (where the renormalisation comparison is known only at the one-loop level and numerically); and above that is the confinement region (where strong-coupling tools are expected to work).

Of the rows of Table 1 at $L_0$, the upper two are statements with the lattice spacing fixed, and differ in strength from the third row, "uniform in $a$". This difference (making things uniform in the lattice spacing) is the ultraviolet wall, and each chapter of Part 12 writes in the same row on which side each statement lies.

Part 12 consists of the following ten chapters. The labels are the main labels of the rows in each chapter; the finer division is in each chapter's table of five grades. The "Conditions" column gives the conditions that always attach to the chapter's main statements.

We recommend reading in numerical order. A reader in a hurry can follow the skeleton of the road by reading 12-01 §01–§02 (the setting), 12-02 §01–§03 (fractional instantons and the dilute gas), 12-04 (slab confinement) and 12-09 (where it stops), in that order.

• Group and coupling. Throughout Part 12 the group is fixed to $SU(2)$. The Wilson action is $\beta\sum_p\bigl(1-\tfrac12\operatorname{tr}U_p\bigr)$ in the convention of 5-01, with $\beta=4/g^2$.
• $N$ and $M$. The $N$ of Part 12 is not the degree of the group but the number of lattice points along a side of the small $T^2$ ($L=Na$). We write $M$ for the number of lattice points along a side of the plane. The lattice is $(\mathbb Z/N)^2\times(\mathbb Z/M)^2$.
• Band and window. The subject of this part and of Part 13 is "the crossover band" (the band for short). It is a different thing from the "windows" of Parts 10 and 11 (the site window, the window in $\hbar$, the time window), and that the words differ carries no meaning beyond that (note in 13-00).
• Names of assumptions. We call assumptions by names, not symbols: control of the ultraviolet, uniform acceptance of insertions, frame decorrelation, the uniform slab tension, the uniform slab gap. The index is collected in A-06.
• Labels. The same five as in Part 10: Proved here (a paper proof placed in the text), Cited (known, checked in the original source), Computer-assisted (a computer-assisted bound), Reading (a numerical reading; physical estimates are marked "physical estimate"), Assumption (an assumption, or a row for which only a strand of proof exists; the evidence column says where the strand is). How to read them is in 10-00.

To Part 10 (the first road). Part 10 is the road towards the mass gap from the direction of time, and it built the diffusive-relaxation-time assumption (formerly Assumption H) in the model of the first step. The entrance is 10-00, where it stopped is 10-15, and the considerations beyond are in Part 11 (11-00). The two roads aim to show different statements (Part 10 the gap, Part 12 first the area law).

To Part 13 (the band). Both roads stop just before a statement of the type "the same constant at all scales". Part 13 (13-00) treats the correct form of that statement (13-01), its reduction to one inequality (13-02), and the same band seen from the side of time (13-07). The last chapter of Part 12 (12-09) is the bridge to Part 13.

To the table of stages. The table of the stages a proof must contain (the container, the ultraviolet, the gap on a small $T^2$, confinement on a small $T^2$, the band, $\mathbb R^4$, general groups) is in 9-06. Part 12 fills the two "small $T^2$" stages of that table and the entrance to the "band" stage.

(a) Weak coupling. "The coupling stays weak up to the size of the $T^2$" is a physical estimate on the assumption of control of the ultraviolet, not a theorem. Bounds uniform in the lattice spacing appear nowhere in this book.

(b) Adiabatic continuity. That the small $T^2$ and $\mathbb R^4$ are joined without a phase transition is a conjecture, and this book does not use it. What it uses is only the form "if they are joined, what must be shown".

(c) Confinement and the gap. The goal of this road is the area law, and there is no known road from the area law to the mass gap. The two are different statements (9-06, 14-01).

(d) Figure 2. Figure 2 is a map of where the tools sit, and the match of scale and coupling (which $L$ corresponds to which $\beta$) is an estimate from asymptotic freedom. The numbers in the figure summarise the labelled rows of Table 1.

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

[01] CIT PER    no continuity on a periodic T²                     physical argument
[02] CIT ADI    continuity from twisted T² × R² to R⁴ (conj.)      conjecture
[03] PRV TW     unique vacuum absorbing twist, gap 4sin²(π/2N)     SU(2), N fixed, n₁₂ = 1
[04] PRV LOC    surface {p} × T² lies in a bounded region of plane
[05] ASM UV     control of the ultraviolet                         a to L₀
[06] RDG WEAK   weak coupling up to size of T² ⟸ [03][05]          physical estimate, LΛ ≪ 1
[07] RDG STOP   the one point where 4D crossing roads stopped      within roads tried
[08] ASM SPLIT  splits into intermediate goal and last stage ⟸ [04][06][07]
[09] MIX LAD    Figure 2, Table 1 ⟸ [05][06][08]

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 only nodes with proofs placed are [03] (proved in 12-01) and [04] (definition). The conjecture [02] enters as a premise of no node. The assumption row [05] does not contain the quantities of the conclusion (gap, area law) and states only the form of a multiscale bound. The nodes on the conclusion side of this chapter, [06] and [08], are "Reading" and "Assumption", and there is neither a node of confinement nor a node of the mass gap.

What this chapter can say

Sources and reproduction

Words used in this chapter — the second road, the small twisted $T^2$, the ultraviolet wall, the crossover band, control of the ultraviolet, the intermediate goal, the ladder of scales, the base point $L_0$, the dilute gas of fractional instantons, the renormalisation comparison. Their meanings are in A-02 Glossary.

Next chapter to read: 12-01 The twisted T² and 't Hooft flux — the setting, sectors, the flux free energy, and three elementary facts.

← 11-A1 Table of notation for Part 11Contents12-01 The twisted T² and 't Hooft flux →