2026-10-01 · chapter Yang–Mills and the mass gapPart 9 Where things stand, and the taskstimeline and where things stand

How far the world has come — reading where things stand from a timeline and tables by dimension

Every theorem about Yang–Mills comes with conditions on "where it holds". Drop the conditions, and a two-dimensional theorem looks like a statement about four dimensions, a lattice theorem like a statement about the continuum. This chapter gathers the conditions into six axes and calls them the address.

$d$ is the dimension of Euclideanised spacetime, and the coupling side is whether the lattice inverse coupling $\beta=2N/g^2$ (table of notation) is small (strong coupling) or large (weak coupling). The address of the problem statement is "4, continuum, $\mathbb R^4$, any compact simple group, none", and there it is required to pass through all six stages. Known The continuum limit of the four-dimensional lattice is believed to lie on the $\beta\to\infty$ side (Chatterjee 2018 §5). Physics

Example: Driver 1989 reaches stages 3 and 4 at (2, continuum, $\mathbb R^2$, —, connected compact, none), and Shen–Zhu–Zhu 2022 gives exponential decay on the lattice at (any, lattice with fixed spacing, infinite, strong, $SU(N)$, none). The "Scope" in the timeline abbreviates this address; the year is the year of first public appearance (the arXiv year if there is one), and the year in parentheses is the year of publication. Rows that combine several papers give a range of years.

The first half-century had three strands. In physics, the theory was written down (1954), and asymptotic freedom was found in perturbation theory (1973). In the framework, the axioms (1973, 1975) and the lattice (1974) were put in place. In construction came the strongly coupled lattice (1978), ultraviolet estimates in three and four dimensions (1985, 1987–89), the two-dimensional continuum measure (1989), and in four dimensions an outline of a construction that removes the ultraviolet cutoff while keeping the infrared one (1993). Known The formulas of 1974 and 1973 are the following two (5-01, 4-06).

Example: for $SU(2)$, $\beta=2.5$ is $g^2=2N/\beta=1.6$. The sign $b_0\gt0$ is asymptotic freedom. The "Comparison" rows are neighbouring theories, abelian groups and scalar fields, and do not move cells. The physics rows continue in Part 7 (7-04, 7-11, 7-09; lattice numbers in 5-08).

KnownTable 1 Timeline I (except rows with the Physics label). Pressing a row sets Figure 1 to that year.

On 24 May 2000 the Clay Mathematics Institute announced the Millennium Prize Problems in Paris. The problem statement (Jaffe–Witten) describes the state of four dimensions as follows. Known

'at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions.' (JW §6)

Since then, new constructions in two dimensions, the state space and dynamics in three dimensions, and new theorems on the lattice have been added, but a 2004 report writes 'The state of the art remains the works of Balaban and of Magnen, Rivasseau and Sénéor'. Known A 2018 review put the question on the lattice into a formula (Problem 5.1; 5-09). $f_\beta(x)$ is the correlation of the Wilson loops of the plaquettes nearest to the origin and to the point $x$.

Example: there are 16 rows from 2000 on (10 of them in the 2020s), and only 2 rows of 2022 (Shen–Zhu–Zhu and Adhikari–Cao) changed a colour on the board. Computed Problem 5.1 is a question on the lattice, and is a different thing from the requirements of the problem statement (continuum, $\mathbb R^4$, axioms, non-triviality) (0-03).

KnownTable 2 Timeline II.

Words in Table 2: stochastic quantisation = a way of building a measure as the invariant measure (a measure whose distribution does not change with time) of a stochastic time evolution. Langevin dynamics = a gradient flow with noise that leaves the measure invariant. Stochastic Yang–Mills heat flow = the flow in the direction that decreases the Yang–Mills action (the heat flow) with noise added. Regularity structures = a framework that gives meaning to singular stochastic PDEs and renormalises them. All in 6-09. Markov process = a stochastic process whose future distribution is fixed by the present state alone (6-02).

We transfer the timeline onto cells. The rows are continuum non-abelian Yang–Mills (no matter fields) in $d=2,3,4$ and the four-dimensional lattice, and the columns are the six stages (6-01). Stages 1 and 2 build the measure $\mu_{a,L}$ on the lattice, and close on the lattice (6-10 §03). In three and four dimensions the finite-volume measure has not yet been built while staying in the continuum, so stage 2 there is written "Closed (lattice)".

The names of the states are the same as in Table 2 of 6-01: Closed (there is a theorem), Partial (with assumptions, in outline, up to estimates of the partition function, or only part of the requirements; "with a hypothesis" in Table 2 of 6-10 also goes here), Not yet, and — (the question is hard to pose in that form; 6-02). The state in year $y$ is fixed by the record of the last change $(y_k,c_k,v_k)$ at or before $y$.

ComputedThe numbers in this figure were computed on this machine (formula (4)). How the cells are divided and how the states are decided is this chapter's choice.

ComputedTable 3 The board in 2026 (state, and the year it reached that state). The three cells of the lattice row do not correspond to the stage columns.

Example: the board moved in five years, 1974, 1978, 1985, 1989 and 2022, and the continuum cells have not moved since 1989. Of the 21 cells, 10 are Closed, 3 Partial, 7 Not yet and 1 —. Computed Table 2 of 6-10 makes stage 4 (infinite volume) of the lattice "Closed" at strong coupling by OS 1978. The lattice cells of this chapter count stage 4 and stage 6 (gap) as one, so 1978 is "Partial". Stage 5 in two dimensions goes as far as Euclidean invariance of gauge-invariant expectations (Corollary 6.7 of Driver 1989), with no claim in the form of the OS axioms, so it is "Partial". Stage 6 has no propagating degrees of freedom and the question is hard to pose, so it is "—" (both as in Table 2 of 6-01). Known

The strong-coupling lattice cell becomes "Partial" in 1978 and "Closed" in 2022. Osterwalder–Seiler 1978 gave the convergence of the cluster expansion in outline, and the gap (exponential clustering) follows from that convergence. The proof of uniqueness and a positive mass gap in an explicit range is Shen–Zhu–Zhu 2022 (both according to the introduction of SZZ; the abstract of OS 1978 states the existence and analyticity of the infinite-volume limit at strong coupling and Wilson's confinement estimate; 5-02 §08). Known That the continuum cells do not move is not because there has been no progress. Progress inside the cells, such as the characterisation of the two-dimensional measure by Langevin dynamics and the dynamics in three dimensions, does not show up on the coarse board (this chapter's reading; for the words of the literature stage by stage, see 6-10).

We rewrite the rows of the board in the language of addresses. The columns are the lattice (fixed spacing), the inside and the outside of the double limit (6-01), and the axioms and the gap. $\mu_{a,L}$ is the measure on a box with lattice spacing $a$ and side $L$.

KnownTable 4 Where things stand, by dimension (non-abelian, no matter fields).

That two dimensions is closed up to continuum and infinite volume is due to its exactly solvable structure (independence of plaquettes and the heat kernel of the group; 6-02). Three dimensions is said to be easier, but the continuum measure is 'open, even in finite volume' (6-06). In four dimensions, stage 3 is with a hypothesis and in outline in finite volume, and stages 4–6 have no theorems (6-07). Known

Example: the four-dimensional "Continuum, finite volume" of Table 4 corresponds to stage 3 (Partial) of Table 3, and the "Lattice" column to the lattice row of Table 3. The outline of MRS removes the ultraviolet cutoff but keeps the infrared cutoff.

Saying "it has been done in two and three dimensions" is correct for non-abelian Yang–Mills only in two dimensions. What it fits up to three dimensions is abelian groups and scalar fields (9-05). We line up the "Comparison" rows of the timeline. None of them moves a cell of the board. Known

KnownTable 5 How far neighbouring theories have got (the "Comparison" rows of the timeline).

Two-dimensional sigma models, unlike two-dimensional YM (6-02), have asymptotic freedom and dimensional transmutation, so they are the closer example of "a problem of the same shape" as four-dimensional YM. They have no gauge symmetry and no confinement. The four-dimensional cells of the board (Table 4) do not move (added 2026-10-09).

The two four-dimensional rows are both examples where the limit is a free (Gaussian) field. For asymptotically free (4-06) Yang–Mills a non-trivial limit is expected, but there is no proof (6-08). Physics Even in the same four dimensions, φ⁴ becomes strongly coupled at short distances, while Yang–Mills becomes weakly coupled. The directions are opposite, so the triviality theorems for φ⁴ do not apply to Yang–Mills. Example: comparing in three dimensions, φ⁴₃ has passed all six stages (Table 2 of 6-01). Non-abelian Yang–Mills is Partial at stage 3, and Not yet from stage 4 on (Table 3).

At strong coupling the high-temperature expansion (expansion in powers of small $\beta$, also called the strong-coupling expansion; 5-02) works routinely, and gives exponential decay even for continuous Lie groups (according to the introductions of Adhikari–Cao and Shen–Zhu–Zhu; Osterwalder–Seiler 1978 abstract only). At weak coupling, uniqueness of the infinite-volume limit of lattice gauge theory is in general not known for $d\gt2$ with large $\beta$ (Chatterjee 2018 §3). That is the side the continuum limit needs (5-05). Known

Converting the conditions of two theorems to one β

Converting the conditions of Shen–Zhu–Zhu, with weight $\exp(N\beta\,\mathrm{Re}\sum_P\mathrm{Tr}\,U_P)$, and of Adhikari–Cao, with weight $\exp(-\beta\sum_P\mathrm{Re}(\chi(1)-\chi(U_P)))$ ($\chi$ is the character of a representation), to the $\beta_W$ of formula (2) gives

($\Delta_G$ is the notation of Adhikari–Cao, not the mass gap $\Delta_G$ of 9-02 and 9-05). It is related to the $\beta$ of 6-10 §05 (Chatterjee's convention $e^{-\beta\sum_p\mathrm{Re}\,\mathrm{tr}(I-U_p)}$) by $\beta_W=N\beta_{\mathrm{Ch}}$, and the $|\beta|\lt N/48$ of 6-10 is the same condition as $\beta_W\lt N^2/48$ ($d=4$) in this chapter. Adhikari–Cao allow any unitary representation $\rho$ of $G$, with $\chi=\mathrm{tr}\,\rho$. Here we take $\rho$ to be the inclusion $G\subset SU(2)$, which gives Wilson's model with link variables restricted to $G$ ($\chi(1)=2$, so $\beta_W=2\beta_{\mathrm{AC}}$). $\log$ is the natural logarithm.

The former is $\beta_W\lt0.25,\ 0.125,\ 0.083$ for $d=2,3,4$ with $SU(2)$, and 9/4 times that for $SU(3)$. Computed Example: $\beta_W\lt1/12$ at $d=4$ is 1/27 of the $\beta_W=2.25$–$2.60$ ($SU(2)$, Lucini–Teper 2001) Physics that lattice numbers use to extrapolate to the continuum, and the condition $\beta_W\ge697$ for the binary icosahedral group is 268 times the upper end of that range. Computed

ComputedThe numbers in this figure were computed on this machine (the groups built by closure from generators with their orders checked, and formula (6) computed). The range of lattice numbers is from the literature.

ComputedTable 6 Finite subgroups of $SU(2)$ (defining representation) and the condition of formula (6). The defining representation is the matrices themselves, and the character is the trace (1-02). The smallest angle is the $\theta$ of the eigenvalues $e^{\pm i\theta}$ of the element nearest the identity.

How to read Table 6

(a) The thresholds are constants that appear in the proofs, not the positions of phase transitions. (b) The finite subgroups of $SU(2)$ are only the cyclic groups, the binary dihedral groups, and three exceptions of order 24, 48 and 120. Known The groups whose names carry "binary" are rotation groups in $SO(3)$ pulled back by the double cover $SU(2)\to SO(3)$ (1-02). The binary tetrahedral, octahedral and icosahedral groups are the inverse images of the rotation groups of the regular polyhedra, and the binary dihedral group $2D_n$ is the inverse image of the rotation group of the regular $n$-gon (the dihedral group of order $2n$); each has twice the order of the original. For $2D_n$ (order $4n$), $\Delta_G=2-2\cos(\pi/n)\approx\pi^2/n^2$, so the condition recedes roughly in proportion to $n^2$. Computed (c) With the one-loop $\xi\propto e^{3\pi^2\beta_W/11}$ ($SU(2)$), each reduction of the lattice spacing to 1/10 increases $\beta_W$ by about 0.86 (a one-loop rough guide; for two-loop corrections and the fact that asymptotic scaling does not yet hold in this range of $\beta$, see 5-05). Physics Why results for finite groups do not carry over to continuous groups is in 5-10, and the strong-coupling expansion in 5-02.

The β knob — two shores and a band on one axis

On the number line of Figure 2 we overlay the correlation length $\xi$ in lattice units. The left shore is the strong-coupling theorems (the condition (9) of SZZ, and the paper of this site Paper; 10-01 The line of construction and Assumption H) and the leading strong-coupling term $\xi=-1/(4\ln u)$; the right shore is the two-loop formula (9-02 (10), $C=185$) and Bałaban's hypothesis. The hypothesis is that the effective coupling is small enough, and no value is written (6-07 §04), so a rough guide reading $g^2\le\gamma$ as $\beta\ge4/\gamma$ is moved with the $\gamma$ knob. In the middle are the numerical window 2.25–2.60 (Lucini–Teper 2001) and the band 1.75–2.6 where neither leading term is within 5% for the plaquette (9-02 Figure 1). Physics

The lower panel is the range of β over which the statements of 10-01 The line of construction and Assumption H are needed (this book's arrangement). The role of crossing the band falls to (S1), (S4) and (S5), which are needed at each β. (S2) holds at every β, (S3)(S6)(S8) are statements about the $\beta\to\infty$ limit, and (S7) is a theorem independent of β. Known Theorems exist only at the left end, and the smaller $\gamma$ is made, the further the rough guide for the right shore recedes.

ComputedThe curves are (9)(10) of 9-02 computed on the spot (Bessel functions by series; Table 7 gives the same values with scipy). The points and window values are Lucini–Teper 2001 Physics.

ComputedTable 7 Markers of β in Figure 3 and ξ (in lattice units; the column "ξ numerical" is the $1/(am)$ of 9-02 Table 3).

What this chapter can say

Sources and reproduction

Next chapter to read: 9-02 Where the two ways of digging part. The seven places, in front of the Not-yet cells, where the ways physics and mathematics dig come apart.

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