2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsthe critical dimension, topology, the four hurdles · Kiichi, Shiori and Rin

Because it is four dimensions — the dimensionless coupling and topology

Writing the action of $d$-dimensional Yang–Mills as

the coupling $g^2$ is exactly dimensionless in four dimensions Cited (dimension counting. On the lattice, by Wilson's heuristic, $\beta\propto\varepsilon^{4-d}$, and in four dimensions $\beta$ does not depend on the lattice spacing $\varepsilon$; §3 of Chatterjee's review). In three dimensions or fewer, $g^2$ has the dimension of a negative power of length, and the interaction weakens like a power at short distances (super-renormalisable). In five dimensions or more it strengthens instead (non-renormalisable). In exactly four dimensions, the correction from each change of scale does not shrink like a power, and remains at the size of $g^2$. In the language of power counting, a quartic monomial in four dimensions is neither relevant nor irrelevant but marginal (Definition 5.2.1 of the book by Bauerschmidt–Brydges–Slade Cited).

Dimensional transmutation. In the non-abelian case, the sign of that correction points to the side "weaker at shorter distances" (asymptotic freedom, 4-06). The action has no ruler of length, yet on quantisation, at one loop,

an inverse length is born Cited (the standard one-loop form, 4-07). The mass gap and the confinement scale are expected to be constant multiples of $\Lambda$. All derivatives of $\exp(-1/g^2)$ vanish at $g=0$, so $\Lambda$ appears at no order of the perturbative expansion in $g$ ("invisible to perturbation theory" in 4-07).

How it shows in H. (a) The correlation length grows exponentially in $\beta$ ($\xi\propto e^{3\pi^2\beta/11}$ by the one-loop bookkeeping for $SU(2)$, 10-03 Table 2), and quantities measured in $\xi$ carry a factor $\ln\xi\sim\beta$. This is why Part 10 placed H as a power statement rather than a band statement (10-01 §07). (b) In the Gaussian approximation the coupling does not flow. In the Gaussian approximation on the hierarchical lattice, the static correlation of plaquettes decayed like the massless $R^{-4}$, and the correlation length was only cut off by the volume (11-02 §02). A scale is born only from the non-abelian correction. (c) The solution theory of the equations of stochastic quantisation has subcriticality ($d\lt4$) as its key condition, and four dimensions is exactly critical (§1.2 of Chevyrev's review Cited, 6-09).

The same marginal, but in the opposite direction. Four-dimensional $\phi^4$ also has a dimensionless (marginal) coupling, but under the renormalisation flow the coupling approaches 0 on the side of long distances ("infrared asymptotic freedom", Proposition 6.1.7 of Bauerschmidt–Brydges–Slade Cited). In the direction one wants to control (long distances) the theory approaches a Gaussian, so the tools reach, and the limit they reached was trivial (the scaling limit is Gaussian) (Aizenman–Duminil-Copin 2021 Cited (abstract), 6-08). In Yang–Mills the coupling approaches 0 on the side of short distances, and grows towards long distances.

Figure 1  Two arrows (the flow of the coupling g;
          left is short distance, right is long distance)

            short distance (UV)                long distance (IR)
  φ⁴₄ :  g toward the large side   ←――――――――――   g → 0 (approaches a Gaussian)
         → near a Gaussian in the direction to be controlled.
           The tools reach; the limit is trivial

  YM₄ :  g → 0 (approaches a Gaussian)  ――――→  g grows (g = O(1) at the scale Λ)
         → near a Gaussian only in the UV.
           No reference for an expansion on the IR side

We do not write "only in four dimensions". The two-dimensional $O(N)$ ($N\ge3$) model and the $CP^{N-1}$ model also have the same pair, a dimensionless coupling and asymptotic freedom, and $CP^{N-1}$ also has instantons with no scale. There too, the construction of the ultraviolet limit is not complete (Rivasseau 2000, Douglas 2004 Cited). Conversely, even at the critical dimension, an abelian ($U(1)$) or non-gauge ($\phi^4_4$) theory is equivalent to a free field or trivial. The "because it is four dimensions" of this chapter refers to being at the critical dimension (a dimensionless coupling) and being non-abelian (the direction of asymptotic freedom, topology, the geometry of gauge orbits) happening at the same time Estimate.

By $\pi_3(SU(N))=\mathbb Z$, four-dimensional gauge fields have an integer topological charge $Q$, and the action of a self-dual solution (an instanton) is $8\pi^2/g^2$, independent of its size $\rho$ Cited (the abstract of Belavin–Polyakov–Schwartz–Tyupkin 1975; the value of the action from the texts of Lucini–Teper 2001 and Athenodorou–Teper 2021). That the action does not depend on $\rho$ is a manifestation of classical Yang–Mills having no scale in four dimensions.

On the lattice, the smoothed topological charge changes only when a small instanton the size of the lattice spacing passes through the lattice ("disappears into a hypercube") (§6.1 of Lucini–Teper 2001 Cited). Its weight is estimated from the one-loop density of small instantons $D(\rho)\propto\rho^{11N/3-5}$, and for the time for the topological charge to move by one unit,

a primary source gives this count ((32)(34) of Athenodorou–Teper 2021 Cited). For $SU(2)$ the coefficient is $7/3$, and the original itself writes that it is "rather weak". The comparison with fitted slopes is Table 1.

The label of Table 1 is Cited (numbers from the original). There is one independent route. The figure of the original suggests that the probability that the field smoothed by the Wilson flow is between topological sectors (the probability of exceeding the smoothness threshold $h\ge0.067$) vanishes in proportion to $a^6$ on lattices at fixed physical size (§4.3 of Lüscher 2010, $SU(3)$, the word "suggests" Cited), and this is the same number as the coefficient 6 of (3) for $SU(3)$. For $SU(2)$, no measurement of the same type was found within the range searched.

Two ways of counting. In the crude count that sets the action of a dislocation to $c$ times the one-loop instanton action, the dynamical exponent is of the type $z=(11N/3)c-4$ Assumption. It differs by 1 from the counting of (3), in how the volume factor at $\rho\sim a$ and the width $d\rho$ are counted. Fitting $c$ to the numbers for $SU(3)$ (the fitted $6.82$, or $z\approx5$ from another original), in either counting $SU(2)$ falls near the boundary $z=2$ Estimate. This book does not judge which of the two counts is right. Nor has it checked that $c$ does not depend on the group.

Setting the two points side by side gives the following estimate Estimate. The time of the slow quantity (the topological charge) and the correlation length $\xi$ are both born from the same dimensionless coupling, in the form $e^{(\text{constant})\cdot\beta}$. But the exponents are different numbers. The correlation length is $\xi\propto\Lambda^{-1}$ from (2), and the time of the topological charge is $\tau_Q\propto(a\Lambda)^{-(11N/3-5)}$ from (3); in lattice units $\tau_Q$ is of the type $\xi^{11N/3-5}$, and its exponent does not match that of $\xi^2$, the thing it is compared with.

• For the 0+1-dimensional double well, the slow quantity (the number of kinks) and $\xi$ are born from the same kinks, the exponents match, and only a logarithmic difference remains (11-01 Table 3).
• On the two-dimensional $U(1)$ strip, the static length is linear in $\beta$, the length at which conservation breaks is $e^{0.92\beta}$, and the two separate exponentially (11-02 §03, a reading of a two-dimensional toy).
• For two-dimensional $CP^{N-1}$, varying $N$ split the strength of freezing and the length at which conservation breaks, while the local ratio did not split (11-02 §07, a conditional one-point reading).

The reading is that the candidate for the true nature of "even if the mass gap is born, it is not caught in the form of H (the form that ties $C_P$ and $\xi^2$ with the same power)" is the slow quantity and the correlation length splitting into different origins, and that the split comes from two things: the coupling being dimensionless, and topology. On this reading, the form that catches the mass gap is not the whole of H but lies on the side of the part local in time (11-04) and its reverse side, the Poincaré constant of the vacuum measure (11-05).

What this section can say: it is a reading, not a theorem. The numbers for the exponents are fits from the originals and semiclassical counts, and for $SU(2)$ they fall right near the boundary, so whether they "split" for $SU(2)$ is undecided (§07). The autocorrelations of the originals are numbers for heat-bath, overrelaxation and HMC dynamics, and rereading them as $C_P$ for the link Langevin dynamics is an assumption (11-01 §06).

Part 10 split Assumption H into four steps (10-03 §02): first step = the path measure of the 0+1-dimensional double well; second step = non-abelian gauge theory on the four-dimensional lattice (with the lattice spacing held fixed); third step = the continuum limit and the OS axioms; fourth step = the mass gap in the limit. Table 2 puts on one sheet, for each step, the hurdle, the known reach, whether the tools of this book reach it, and the kind of cost. There are three kinds of cost. "Weeks" = expected to be reached by accumulating work combining existing tools; "years" = a large extension of existing tools, or a reconstruction of a large body of work, is needed; "new mathematics" = no description of a route in the sources read (this phrase borrows as it is the wording of the question "why is there no mathematics for the third step?").

The "years" in Table 2 are guesses for the first and third roots, depending on descriptions in the sources Estimate: Rivasseau 2000's "simplify and rewrite Balaban's results; since they run to hundreds of pages, this is not a small task", and that Dimock's reconstruction, starting from three-dimensional $\phi^4$, reached the stability of the partition function of three-dimensional QED in a little over ten years (abstract). For the second root there are no sources to support a guess of the cost. The costs of the rows of the first step are as in the guesses of 10-15 §03, and not a single guess has been run even for the rows marked "weeks".

What this section can say: the rows of known reach are as their source labels say. "Does it reach" and the kind of cost are estimates, and for the steps marked as not reached we do not say "impossible".

First a summary in thirteen lines [Y3].

1. The third step is that "there is a single probability measure in the limit of finer and finer lattices, and it satisfies the OS axioms". The theorems at hand go only as far as a uniform estimate that, inside a finite box, however fine the lattice is made, the coarse-grained theory does not run wild (Balaban's ultraviolet stability).
2. Beyond that, three things are needed separately — estimates of correlation functions, uniqueness of the limit, and enlarging the box to infinity — and none of them has been shown. Even for three-dimensional non-abelian gauge theory, the measure inside a finite box has not been constructed.
3. The first difficulty is that in four dimensions the coupling is dimensionless. In three dimensions or fewer, the interaction weakens like a power as the scale is made finer, and only finitely many divergences had to be dealt with. In four dimensions the correction from each step does not shrink like a power, and the corrections from infinitely many steps must all be controlled in a form independent of the number of steps.
4. The second is that asymptotic freedom works "only at short distances". At short distances the coupling is small and expansions around the free field can be used. At long distances the coupling becomes large, and the confinement scale $\Lambda$ appears.
5. This $\Lambda$ appears at no order of the expansion in the coupling, so extending the expansion from short distances does not reach it. The strong-coupling expansion, which converges on the side where the coupling is large, can be used only on the side of large lattice spacing, and does not reach the continuum limit. There is no tool that crosses the scales between the two expansions.
6. The third is non-abelian gauge invariance. No continuous gauge fixing that works for all configurations at once exists (Gribov, Singer), and Balaban chose the gauge afresh scale by scale and place by place.
7. Where the field is large, expansions cannot be used, and the only way to control it is by its being "unlikely". On a group, large fields are necessarily non-convex, so methods that rely on convexity cannot be used, and methods that pay with rarity are needed (11-06).
8. Magnen–Rivasseau–Sénéor (1993) constructed the Schwinger functions in axial gauge, keeping a cutoff at long distances. They wrote in their introduction that the reasons for not removing the cutoff were that the coupling becomes large and enters the non-perturbative effects of confinement, and that there is no easy solvable model around which to expand.
9. In lattice numerics, the mass of the lightest glueball of $SU(3)$ has been extrapolated to the continuum limit as about $3.41$ times the square root of the string tension.
10. But the numbers are extrapolations from finitely many lattice spacings and finite boxes, and do not show that the limit exists, that it is unique, or uniformity in the size of the box.
11. Field theories with non-trivial interaction in four dimensions are thought to be limited to asymptotically free ones. The $\phi^4$ theory has been shown to be trivial in four dimensions (its scaling limit is Gaussian).
12. As a result there is not a single "interacting theory already constructed" in four dimensions to compare with, and Yang–Mills is sought as its first example.
13. The walls this book met in the first step (the double well) and the second step (the hierarchical gauge model) — paying for the lack of convexity at the summit with rarity, the convexity hypothesis not holding on the group, global quantities not reflected in the gap — are small versions of the third difficulty of the third step. In the third step, that "rarity" ceases to be rare at long distances Estimate.

The three roots

The relation among the three roots Estimate: (R1) is "how many steps there are" (infinitely many, each with the same weight), (R2) is "at which step the expansion breaks off" (the confinement scale), and (R3) is "what is lacking within one step" (convexity and a global gauge). Balaban handled (R3) at the ultraviolet steps in finite volume, and handled (R1) in the range where $g_j$ is small. None of the sources read has touched (R2).

Where Balaban and MRS stopped

Balaban. The gauge-theory series is 13 papers in Commun. Math. Phys. (1983–1989, including one on estimates of lattice Green's functions), about 426 pages when the pages are added up Cited (counted from the bibliographic data; the texts were not obtained). Combining the descriptions in the texts of other sources, the reach is as follows. In finite volume on the lattice, the effective action on the unit lattice stays bounded after arbitrarily many repetitions of the block-spin transformation (ultraviolet stability). From this it should follow that limits of gauge-invariant quantities such as smoothed Wilson loops exist along subsequences by compactness, but the limit is not necessarily unique (the summary in the introduction of MRS 1993 Cited). No source includes estimates of correlation functions, uniqueness of the limit, infinite volume, the OS axioms or the mass gap in Balaban's reach.

MRS 1993. What they showed is the construction of the Schwinger functions of pure $SU(2)$ four-dimensional Yang–Mills with a fixed infrared cutoff and no ultraviolet cutoff, in a regularised axial gauge, in the trivial topological sector, and that the Slavnov identities hold non-perturbatively. The authors themselves write that they do not claim to have given the proof of convergence in every detail, and that it is a detailed sketch Cited (introduction checked in the text; the body of the proof not read). The reason they stopped is the following sentence of the introduction (this book's translation).

We do not investigate invariance under large gauge transformations, or non-trivial topological effects such as instantons. Nor do we try to remove the infrared cutoff, because that leads to large values of the coupling constant, to the so-called non-perturbative effects, which probably correspond to confinement. These problems are for the moment beyond the reach of our constructive methods, in particular because there is no easy solvable model of these phenomena around which to expand.

The "we" of the quotation refers to the authors of MRS. Since the infrared cutoff is not removed, the introduction goes on, the OS axioms as a whole cannot be examined either.

The object, tools, theory, history

The object is undefined. The problem statement is "prove that a non-trivial quantum Yang–Mills theory exists on $\mathbb R^4$ and has a mass gap"; for the existence part it requires axioms as strong as Wightman's or OS, and in a footnote it says that weak existence by compactness alone is not counted as a solution (Jaffe–Witten Cited). The construction itself is the content of the problem; it is not in the form of asking about properties of an object already constructed. Douglas's report of 2004 also writes that "there is no satisfactory mathematical definition of quantum Yang–Mills theory" Cited. Moreover, the space on which to place the measure is not fixed. The natural space is the quotient of everything by gauge orbits, which for the non-abelian case is a non-linear space, and even the construction of the state space is not trivial. Chevyrev's review writes that already in three dimensions the Gaussian free field is too singular to be restricted to lines, and the space of holonomies that could be used in two dimensions cannot be used Cited. The objects up to the second step (finite-dimensional probability measures on the lattice) are defined, and there is mathematics that asks about their properties. For the object of the third step, even on which space and with which topology to place it is undecided. One step before "there are no numbers" there is "there is no object" Estimate.

The tools are perturbation around a Gaussian reference. Most of the models constructed are in the weak-coupling region, made with cluster expansions, Mayer expansions and their multiscale versions (Rivasseau 2000 Cited). The analysis of one step of the renormalisation group relies on perturbation in powers around the field being 0 and on Taylor approximation, which is a good approximation only when the field is small (the book by Bauerschmidt–Brydges–Slade Cited). The solution theory of stochastic quantisation uses the fact that, under subcriticality ($d\lt4$), the solution is a perturbation of the stochastic heat equation, which has the Gaussian free field as its invariant measure (Chevyrev Cited). All of them take as input "close to a Gaussian measure at each scale". The asymptotic freedom of Yang–Mills gives this input only at short distances. At long distances, the grounds for starting from a Gaussian reference run out. For four-dimensional $\phi^4$, on the contrary, it approaches a Gaussian at long distances, so the same tools work, and the conclusion was triviality. In four dimensions, the theories the tools reach become trivial, and the theory that should be non-trivial gives the tools no input Estimate.

There is no theory that crosses infinitely many scales with neither a small parameter nor convexity. The multiscale theorems within the range read all place a small parameter or convexity at each step: a small coupling (hierarchical and Euclidean $\phi^4_4$), uniform convexity of the conditional measure at each step (Bauerschmidt–Bodineau), a convergent perturbative expansion (two-dimensional Gross–Neveu, bibliographic data only), a small $g_j$ and the rarity of large fields (Balaban, MRS) Cited. For group-valued fields, the convexity hypothesis cannot hold in principle at weak-coupling scales (the lemma of 11-06: on a compact manifold of diameter $D$, if $\operatorname{Hess}V\ge-K$ then $\operatorname{osc}V\le KD^2/2$). Gauge invariance forbids this control Estimate: a gauge-invariant function is constant along orbits, so it cannot have convexity ($\operatorname{Hess}V\gt0$) in the direction of the orbits. If one tries to remove the orbit directions by fixing the gauge, there is no continuous global fixing. Balaban's local fixing and the seam between the two gauges of MRS both work only in the region of small fields. The Riemannian geometry of the orbit space has been described (Singer 1981, bibliographic data only), but a general theory handling measures on it that are neither Gaussian nor perturbations of one is not in the sources read.

History. The introduction of MRS writes that Balaban's work is "very impressive but hard to access; the results are scattered over many publications, and it is very difficult to check the consistency of all the arguments", and Rivasseau 2000 lists rewriting it among the open problems Cited. Dimock's reconstruction writes that "others have been slow to adopt this method", starts from three-dimensional $\phi^4$ (text), and has reached an estimate of the partition function of three-dimensional QED uniform in the lattice spacing (abstract) Cited. A reconstruction for the four-dimensional non-abelian case is not in the sources read. Intermediate goals (construction on a finite box $T^4$, the three-dimensional measure) are also open. There are also moves to make different mathematics: the formulation in the language of Wilson loops and probability (Chatterjee), and the road that makes a Markov process on gauge orbits by stochastic quantisation and poses the construction of the invariant measure as an open problem (Chandra–Chevyrev–Hairer–Shen, Chevyrev's review) Cited. Both go in the direction not of extending the lattice renormalisation group but of rebuilding where the objects live (the state space) or the means of construction (stochastic dynamics). However, the key condition $d\lt4$ of the latter shows that this new tool too becomes exactly critical in four dimensions.

What this section can say: an arrangement of descriptions in primary sources, and the matching to roots (an estimate). Balaban's texts were not obtained, so the description of his reach goes through the texts of other sources. The contents of Gribov and Singer are from memory; only the bibliographic data were checked.

The list of candidates for what to place as rules if the "new mathematics" in the table of §04 were to be made is dealt with in 11-08. Here we write only one Estimate. Combining axiomatising scale (a semigroup of coarse-grainings and a flow on it) with the class of measures that pay with rarity (locally convex on the good set, the bad set rare, a drift returning from the bad set, rarity paid only through the entropy inequality), it seems possible on the ultraviolet side to write a form in which the class closes at each step of coarse-graining. On the strong-coupling side, tools of the type that pay with the curvature of the group work in a narrow range. Between the two, a "coupling window" remains where neither currency is enough. It is the range where the rate of rarity, proportional to the effective coupling, runs out at the confinement scale, and the currency of curvature is not yet enough. This window is a restatement in the language of rules of the second root (R2) (that there is no expansion connecting the ultraviolet and the infrared), and is not a new obstacle. The hierarchical gauge model is a place where the three — the ultraviolet side, the window, and the strong-coupling side — come together within one model, but the results there imply nothing for four-dimensional Yang–Mills.

The periodic box. For reversible dynamics $\tau_{\rm int}(Q)\le C_P$ ((1) of 11-01), so if the autocorrelation of the topological charge grows faster than $\xi^2$, H↑ breaks ((3) of 11-01. Lean LatticeGauge.FourDTrial.not_hup_of_fast_tau, assumptions: $\tau_n\le C_n$, $c\,\xi_n^p\le\tau_n$, $p\gt2$, $\xi_n\to\infty$. Negative control z_two_compatible_with_hup, no assumptions: $\tau\propto\xi^2$ is compatible with H↑). On the periodic box, originals for $SU(3)$ and above report this, so under the reading that extends the conjecture that HMC and Langevin are in the same dynamic universality class (in the original, a phrase attached to observations with open boundaries) to the topological slowness on the periodic box, H↑ reads as false on the periodic box Assumption (the note in 10-01 §05, 11-01 §06).

Open boundaries rescue only finite boxes. With open boundaries, inside a box with fixed physical side, the growth of the autocorrelation goes back to $a^{-2}$. But if the spatial side of the box is held fixed and only the length in the time direction is extended, the autocorrelation time of $Q^2$ grows nearly in proportion to the length: $42.1(2.5)$, $113(6)$, $148(10)$ (lattices with time-direction lengths 24, 48, 80) (§5.3 of Lüscher–Schaefer 2011 Cited). This is because the charge enters and leaves only by transport through the boundary, and in this book's order, which takes the volume to infinity first, it is expected not to be a rescue Assumption (11-03 §05 and Table 2).

For $SU(2)$ there is one primary source, on the $a^{-2}$ side. Within the range searched, there is one primary source that measured the dependence on the lattice spacing of the autocorrelation of the topological charge in pure $SU(2)$ gauge theory (Abe–Morikawa–Suzuki 2025: HMC, periodic, a small box of physical side about $2.2/\sqrt\sigma$, three lattice spacings, reporting that it fits $a^{-2}$ fairly well Cited). The heat-bath originals are all observations on the side where freezing in $SU(2)$ does not become strong enough to measure, and $SU(2)$ was excluded in Table 1 as well. Both counts of §02 place $SU(2)$ near the boundary $z=2$. Hence, for this book's group $SU(2)$, it is undecided whether the freezing of the topological charge breaks Assumption H. The primary source leans to the boundary or below it Reading.

Correspondence with the two-dimensional test bed of 11-02 Estimate. For two-dimensional $CP^{N-1}$ at $\xi\approx4$, freezing split by $N$ (strongly frozen at $N=21$, not frozen at $N=2$), while the ratio of local quantities did not split (11-02 §07, a conditional one-point reading). That the strength of freezing splits with $N$, which plays the role of the size of the group, can be read as a two-dimensional analogue of $SU(2)$ falling on the boundary in four dimensions while $SU(3)$ and above clearly cross it. The two-dimensional count in the primary source is also of the type "$z\sim N$", and it writes that the same argument applies to four-dimensional $SU(N)$ (Del Debbio–Manca–Vicari 2004 Cited). However, $N=2$ carries an ultraviolet flag (the two definitions of the topological susceptibility differ by 13–14%), and the reason $N=2$ does not freeze may be that it is on the side where the lattice scale determines the topological charge. The analogy carries this reservation, and the two-dimensional results imply nothing for four dimensions.

What this section can say: "H↑ reads as false on the periodic box" is a statement on top of the originals for $SU(3)$ and above and a reading that extends a conjecture; we do not claim that H↑ is false. For $SU(2)$ it is undecided.

There are many physical mechanisms that explain the mass gap and confinement. This book uses none of them as tools of proof. Table 4 lists, one line each, what each mechanism says and why it is not a proof. The last row is an estimate that brings the leading-order statement of 11-05 of this book close to the language of physics, and its character differs from the other rows.

What this section can say: "what it says" in each row is a copy from the abstract or text of the original, and we judge none of the mechanisms right or wrong.

(a) The candidate for the true nature. "The slow quantity and the correlation length split into different origins" of §03 is an estimate. Neither the value of the ratio $c$ for the action of a dislocation nor its independence of the group has been checked. $SU(2)$ falls near the boundary in either of the two counts.

(b) Mechanisms. This book uses none of the mechanisms of Table 4 as tools of proof.

(c) Two dimensions. We do not say what the results for two-dimensional models (the $U(1)$ strip, $CP^{N-1}$) imply in four dimensions.

(d) We do not say "only in four dimensions". The two points were stated as properties of being at the critical dimension.

(e) Table 2. "Does it reach" and the kind of cost are estimates and guesses; "new mathematics" refers to the absence of a description of a route in the sources read, and does not mean impossible.

(f) Sources. §05 is an arrangement within the range of primary sources read, and the texts of Balaban, Gribov, Singer and Osterwalder–Seiler were not obtained. A theory written as "there is none" means "there is none in the sources read".

(g) $SU(2)$. For this book's group $SU(2)$, it is undecided whether the freezing of the topological charge breaks Assumption H. Nor do we claim that H↑ is false.

What the machine-checking column guarantees is only the algebra of sequences of reals; that $\tau_n$ is the autocorrelation time of a reversible dynamics, and that $C_n$ is the Poincaré constant of the Wilson measure, are outside Lean (10-13).

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

[01] CIT DIM    in four dimensions g² is dimensionless, marginal
[02] CIT DT     dimensional transmutation Λ = μ e^(−1/(2b₀g²))           one loop
[03] CIT TRIV   the limit of φ⁴₄ is Gaussian
[04] CIT BAL    Balaban's UV stability (subsequences, no uniqueness)      finite volume
[05] CIT MRS    MRS 1993 (IR cutoff, axial gauge, SU(2), sketch)
[06] CIT CHV    3D measure also not yet constructed, d < 4 is the key
[07] CIT INST   8π²/g², D(ρ) ∝ ρ^(11N/3−5), (3)
[08] CIT FIT    SU(3) fit 6.82(7)                                        heat bath, periodic
[09] CIT SU2    one primary source for SU(2) (close to a⁻²)              HMC, small box
[10] CIT OB     open boundaries: τ(Q²) proportional to time length       box with fixed physical side
[11] PRV NH     τ ≥ cξ^p (p > 2) ⇒ H↑ breaks                             Lean not_hup_of_fast_tau
[12] ASM HL     reading that extends the conjecture to the periodic box
[13] ASM R4     H↑ reads as false on the periodic box ⟸ [08][11][12]    SU(3) and above
[14] ASM IV     infinite volume first: open boundaries no rescue ⟸ [10]
[15] EST SEMI   SU(2) is near the boundary z = 2 ⟸ [07][08]
[16] EST UND    for SU(2), whether freezing breaks H is undecided ⟸ [09][15]
[17] EST TWO    slow quantity and correlation length split ⟸ [01][02][07]
[18] EST CRIT   properties of the critical dimension ⟸ [01][03]
[19] EST ROOT   the three roots ⟸ [01][02][04][05][06]
[20] EST WIN    the coupling window restates (R2) ⟸ [19]
[21] EST HUR    "does it reach" and costs in the four-step table ⟸ [04][05][06][19]
[22] EST ANA    the split in N in 2D is an analogue of the SU(2) boundary ⟸ [15][16]

The way to check is to see by eye that on every "⟸" the numbers on the right are smaller than the number on the left. There are no circles. The conclusion nodes of this chapter, [13]–[22], are Assumption or Estimate, and there is neither a node for the mass gap nor a node for the continuum limit. [09] (the primary source for $SU(2)$) enters only [16], not the premises of [13]. This is how the undecided reading for $SU(2)$ looks on the graph (the same form as the dependency graph of 11-01).

What this chapter can say

Sources and reproduction

Terms used in this chapter — critical dimension, marginal, dimensional transmutation, infrared asymptotic freedom, instanton, dislocation, the four hurdles, the three roots (R1)–(R3), ultraviolet stability, the coupling window, dual superconductivity, centre vortex. Their meanings are in A-02 Glossary and 11-A1.

Next chapter to read: 11-08 Questions, and what is not claimed — questions that can be settled at the size of an undergraduate thesis, the list of candidate rules, and what Part 11 does not claim.

← 11-06 Paying with rarity instead of convexity — renormalisation on groups and log-SobolevContents11-08 Questions, and what is not claimed →