2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlBałaban's series, the BBS chain, correspondence with the sources · Kiichi, Shiori and Rin

A map of the sources — Bałaban's programme, the BBS chain, two rows absent from the sources

• Bałaban's four-dimensional theorem assumes "the effective coupling stays in $(0,\gamma]$", and the theorem that removes this is unpublished. What was reached is: on a finite-volume lattice, upper and lower bounds on the effective densities and the partition function uniform in the lattice spacing, assuming (U3) (§01, §02).
• Rewriting in easier models has reached three-dimensional $\phi^4$ and three-dimensional QED, but the later stages for three-dimensional Yang–Mills and the two-dimensional sigma model are not to be found (§03).
• The one-step estimate for flat fields follows from the chain: interchange of expectation and seminorm → the definition of the norm → properties (ii)(iii)(iv); and the measure of the expectation in property (iii) does not depend on the background (§04).
• On a twisted $T^4$ with the zero modes removed, ① can be stated in five rows, and only two rows are absent from the sources: the content of (T-U3) (the one-step remainder including large fields (S)) and (T-Obs) (the source for observables) (§05, §06).

Dividing Bałaban's series on gauge theories (Commun. Math. Phys., 1983–1989) into five stages by the work each takes on gives Table 1. The bibliography is the same as in 6-07 Table 1. The only text we have checked is the opening of the three-dimensional paper and the abstract and introduction of the last paper; stages 4 and 5 and the row for 119 were confirmed from abstracts. The "work it takes on" for stages 1–3 is read from the titles, not checked against the content. The "three conditions" are (U1)(U2)(U3) of 15-00 §01.

The theorem in the introduction of the last paper is as follows Cited (6-07 §03).

'Theorem 1. If the sequence of the effective coupling constants is contained in an interval ]0, γ] with a sufficiently small positive γ, then the effective densities ρk have the form, and satisfy all the conditions and bounds, described in Sect. 2.'

The assumption is (U3) itself, and the conclusion is the form of the effective densities ((2) of 15-00) and the estimates (U1)(U2). The author says that the assumption can be removed because it follows from a basic inequality derived from Theorem 2 of an earlier paper, and continues: 'The proof of Theorem 2, which is based on second order perturbative calculations, is very awkward and long ... and has not been published yet' Cited. Within our search of the bibliography of the author's papers after 1989, we found none that contains the proof of Theorem 2 (the source table of 6-07). So the four-dimensional achievement is: on a finite-volume lattice, upper and lower bounds on the effective densities and the partition function uniform in the lattice spacing, assuming (U3) Cited (level 1 in 6-07 Table 4). No source includes estimates of correlation functions, uniqueness of the limit, the OS axioms, infinite volume or the gap in what was reached. The review (Rivasseau 2000 §II.1) summarises Bałaban's result as the ultraviolet stability of the effective action after arbitrarily many steps in finite volume, and does not mention the assumption (U3) Cited (the text of the review). This is a difference in the level of summary; the statement in this section of what was reached rests not on the review but on the introduction of the last paper.

In one line, where it stopped is the estimate of the one-step remainder that removes (U3). In Lemma F of 15-03 we see that (U3) splits into that estimate (S) and an elementary part, continuity within the domain (C).

Magnen–Rivasseau–Sénéor (1993), with a continuum momentum cutoff, stated the ultraviolet limit of the Schwinger functions for four-dimensional $SU(2)$ in the trivial topological sector, with a fixed infrared cutoff and a regularised axial gauge, but wrote 'we do not provide a detailed proof' and said the details would run to 'several hundred pages' Cited (6-07 §06).

Work rewriting the method in easier models has reached three-dimensional $\phi^4$ (Dimock I–III, up to convergence of the expansion and the stability estimates) and three-dimensional QED (Dimock's three papers arXiv:1512.04373, 1712.10029, 2009.01156: the stage of tuning the bare quantities in the bounded-field approximation to fix the endpoint of the flow; multiscale propagators and the local regularity of minimisers; and upper and lower bounds for ultraviolet stability in a fixed finite volume, with the flow fully controlled at weak coupling) Cited (the three QED papers from abstracts; the source table of 6-05, 11-07 §05). For three-dimensional Yang–Mills, no series of the same form was found within our search by author name. Transplanting it to the two-dimensional sigma model has reached the point corresponding to stage 3 of Table 1 (15-01 §04), and work corresponding to stages 4 and 5 has not yet been written even for easier models (within our search) Cited (abstracts). For fields valued in a sphere, Bałaban's series on the low-temperature expansion of the classical $N$ vector model gives a description of the complete induction including large-field regions (abstract of Comm. Math. Phys. 196 (1998) 485) Cited (abstract). This is an infrared (low-temperature) problem, not the two-dimensional ultraviolet Reading.

In the renormalisation of flat fields (scalar fields), the estimate of the expectation in one step follows by a short chain from the properties of the weight (regulator) $\tilde G$ alone. Of the four properties of 15-01 §02, the lemma uses three: (ii) (sum of fields), (iii) (closing one step with the expectation over the fluctuation) and (iv) (bounding by the weight of the next step). The text of the source is as follows Cited (Bauerschmidt–Brydges–Slade, arXiv:1907.05474v2, Appendix A §A.6.2, the text on pp. 214–215).

'Lemma A.6.1. Suppose that $\tilde G$ obeys (ii), (iii) and (iv) above. Then $\|\mathbb E_+\theta K(X)\|_{\tilde G_+}\le2^{|B(X)|}\|K(X)\|_{\tilde G}$.'

The proof is a chain of three steps Cited (the text). $\theta$ is the map that shifts the field by $\phi\mapsto\phi+\zeta$, $\mathbb E_+$ is the expectation over the fluctuation $\zeta$, and $T_\phi(h)$ is the Taylor seminorm at scale $h$ at the point $\phi$.

(1) is the interchange of expectation and seminorm ((A.6.5) of the source, which takes it from Prop. 3.19 of reference [54]), (2) is the definition of the norm ((A.6.7)), and (3) is the application of the properties ((A.6.8); the power convention in (3) is written with the notation abbreviated, and in the source it is in the form of the power $t$). Putting (2) into (1) and paying for the expectation with (3) gives the lemma.

One thing is learned from reading this chain Reading (a reading of the chain of the proof). The expectation in (iii) is the expectation of a function of the fluctuation $\zeta$ alone, and its measure does not depend on the background $\phi$. The interchange in (1) also holds with factor 1 because the law of the fluctuation does not depend on the background. For group-valued fields, a background-dependent factor $1+\eta_k$ enters here. What becomes of each step of this chain on the group — that property (ii) holds by Lemma R, and that if the law of the fluctuation does not depend on the background every step goes through and gives $\eta=0$ (Lemma H) — is in 15-07 §02 and 15-07 §05.

On a periodic $T^4$ the classical minima at the last step (flat connections) form a continuous family, and the integrals of the last few steps degenerate. We put forward, as a proposal, a statement in which this incidental difficulty is removed by 't Hooft's twist. It moves the small twisted $T^2$ of the spatial road (12-01) to the whole four-dimensional torus.

Setting. $G=SU(2)$, a four-dimensional torus of side $L_0$, blocking factor 2, the number of points on a side of the last lattice $M$ (fixed), the number of steps $n$, and lattice spacing $a_n=L_0/(M\cdot2^n)$. Put the twist $n_{12}=1$ into Wilson's action (multiply one stack of plaquettes in the $(1,2)$ plane by the central element $z=-1$):

Define the blocking transformation $T$ by an averaging that is gauge covariant and commutes with central shifts, and set $\rho_0=e^{-S_{\rm tw}}$, $\rho_{k+1}=T\rho_k$.

Statement T⁴-UV (proposal, unproved). There are a $\gamma\gt0$ and constants $C,\kappa,c,r\gt0$ depending only on $M$ such that, for every $\gamma_0\in(0,\gamma)$ and every $n$, choosing the bare coupling $g_0=g_0(n,\gamma_0)$, the following hold Assumption.

• (T-U3) The sequence of effective couplings $g_0\lt g_1\lt\cdots\lt g_n=\gamma_0$ is defined throughout.
• (T-U1) $\rho_n$ has the form (2) of 15-00, and each $E_n(X,\cdot)$ is analytic in a complex neighbourhood of radius $r$ in the background-field variables, with $|E_n(X)|\le C\gamma_0\,e^{-\kappa d(X)}$.
• (T-U2) $|R_n(\Omega^c)|\le\exp\bigl(-c\,p(\gamma_0)^2|\Omega^c|\bigr)$ ($p$ is the same threshold factor as in (U2) of 15-00).
• (T-Z) $e^{-CM^4}\le Z_n/Z_n^{\rm G}\le e^{CM^4}$ ($Z_n^{\rm G}$ is the sum of the Gaussian integrals around the isolated flat connections).
• (T-Obs) The three above hold with the same constants even when sources (of strength $\le s_0$) are inserted for smoothed observables whose support is larger than the scale of the last lattice $L_0/M$.

None of the constants depends on $n$ or on $a_n$. (T-U3) has the form of the conclusion of Lemma F.

Where the twist works. The flat connections on the last lattice are only those whose holonomies in directions 1 and 2 anticommute and whose holonomies in directions 3 and 4 are central elements, and up to gauge equivalence there are finitely many of them. The adjoint fluctuations around them are antiperiodic in direction 1 or 2, and there are no zero modes Assumption (a reading that extends to four dimensions the facts about the twisted part of $T^2$ in 12-01; the four-dimensional classification has not been checked against the sources). At fine steps, on the other hand, the twist has no effect. The twist's plane can be moved by central gauge transformations and is not visible locally Reading (a physical estimate). Lemma R (15-07 §02) holds as it is for the twisted plaquette $z_pU_{\partial p}$: since $z_p$ is a central element, one only replaces $W_{\partial p}$ in the proof by $z_pW_{\partial p}$ Proved here.

What the statement says and does not say. If shown, what it says is an estimate of the effective action uniform in the lattice spacing, and non-degeneracy at the last step; this amounts to Bałaban's Theorem 1 + the unpublished Theorem 2 ((T-U3)) + an extension to observables ((T-Obs)) + non-degeneracy at the last step from the twist. Subsequential limits of the smoothed observables follow from it by compactness. What it does not say is uniqueness of the limit, the OS axioms, recovery of rotations, non-triviality, or a gap uniform in the volume. The difficulty of (U1)–(U3) at the fine steps is exactly the same as for the periodic $T^4$; what the twist removes is only the incidental difficulty of the degeneracy of the last few steps Reading (a physical estimate). The role of T⁴-UV is not to reduce the difficulty but to state ① in a pure form without zero modes.

A test bench with the gauge taken out. The two-dimensional $O(4)$ sigma model (the principal chiral model of $SU(2)$) has a field taking values in $SU(2)$, a dimensionless and asymptotically free coupling, and large fields, and no gauge. Writing (U1)–(U3) on the twisted cylinder $\mathbb R\times S^1_L$ (no zero modes because of the twist) gives the four-dimensional ① with the gauge half of root (R3) taken out Reading (a physical estimate). As in 15-01 §04, the ultraviolet limit has not been constructed for this model either, and transplantation has reached the variational problem and the background-field stage Cited. Writing stage 4 of Table 1 (one step of the small-field expansion) for this model is cheaper than rereading the second half of the four-dimensional series, and is a test that cuts out the "group-valued field × dimensionless coupling" part of ①. Two-dimensional results imply nothing about four dimensions.

Table 2 sets out, for each row of Statement T⁴-UV, where it falls in Bałaban's series, whether it changes with the twist, and whether it is absent from the sources. The last column gives the correspondence in three-dimensional QED, where the method has been rewritten for an easier model (Dimock's three papers in §03).

What to read from Table 2 Reading. What is absent from the sources are two rows, the content of (T-U3) (the one-step remainder (S) including large fields) and (T-Obs); these two are where new mathematics is needed. The normalisation of (T-Z) and the non-degeneracy of the last step are specific to the twist, but they are problems of finite-dimensional integrals, and the difficulty is of a different quality. Three-dimensional QED in 1512.04373 has, in the bounded-field approximation, a statement of the type that tunes the bare relevant quantities to fix the endpoint of the flow, and this is the form of the conclusion of Lemma F. Writing the exponent of (T-U2) with $p(\gamma_0)^2$ matches the form of the large-field factor that comes from the original threshold $\varepsilon=g\,p(g)$ (equation (7) of 6-05); writing it with a negative power of $\gamma_0$ would give a statement stronger than the original threshold. A one-line statement of what the two rows must show to close, and proposals for them, are in 15-08.

(a) The content of the series in Table 1, apart from the opening of the three-dimensional paper and the abstract and introduction of the last paper, is from abstracts for stages 4 and 5 and 119, and a reading from the titles for stages 1–3.

(b) We do not know the content of Theorem 2. The judgement that it is unpublished rests on the sentence in the introduction and on the bibliography within the range we searched.

(c) The chain of §04 is the text of a source on flat fields and says nothing about the group. "The measure of property (iii) does not depend on the background" is a reading of the chain of the proof.

(d) Statement T⁴-UV is a draft and has not been proved. Even if proved, it says nothing about uniqueness, the OS axioms, non-triviality, infinite volume, or a gap uniform in the volume. The classification of twisted flat connections in four dimensions has not been checked against the sources.

(e) The correspondence in Table 2 is at the level of abstracts and of introductions (via 6-07); it does not reach the section and theorem numbers in the text of the series.

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

[01] CIT ABS    abstracts of the series and of three-dimensional QED
[02] CIT THM    Theorem 1, Theorem 2 unpublished
[03] CIT MRS    what MRS reached
[04] CIT BBS    Lemma A.6.1 and the chain                         flat fields
[05] RDG FLAT   measure of (iii) independent of background ⟸ [04]
[06] ASM T4     Statement T⁴-UV ⟸ [02]                            draft
[07] PRV RTW    Lemma R unchanged by the twist                    z_p central
[08] RDG TW     the twist only at the last step ⟸ [06]
[09] RDG O4     the two-dimensional O(4) test bench ⟸ [06]
[10] RDG MAP    T⁴-UV and the sources ⟸ [01][02][03][06][07]       level of abstracts

The way to check it is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. [06] is an assumption row, in the form of a question that restates ①. Among the nodes of this chapter there is no node saying that ① has been shown.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the five stages of Bałaban's series, Theorem 1, Theorem 2, MRS, Lemma A.6.1, the interchange of expectation and seminorm, Statement T⁴-UV, 't Hooft's twist, the test bench, the two rows absent from the sources. Their meanings are in the A-02 glossary.

Next chapter: 15-03 The decomposition statement — splitting ① into a class that closes with finitely many parameters and three remainders.

← 15-01 Why the ultraviolet is hard — dimensionless coupling, group-valued fields, gaugeContents15-03 The decomposition statement — a class that closes with finitely many parameters, and three remainders →