2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theorythe two-dimensional Yang–Mills measure

Solvability in two dimensions and the Yang–Mills measure — on the plane the plaquettes become independent, and the measure can be made

Two dimensions is, so far, the only case in which a continuum Yang–Mills measure has been constructed for a non-abelian gauge group. Chatterjee (2018) writes that no continuum limit in which Wilson loops behave non-trivially has been constructed in any dimension higher than two. Known In two dimensions the field strength has one component.

$A$ is a Lie-algebra-valued connection (3-02), and $|\cdot|$ a conjugation-invariant length. $F$ has the dimension of an inverse length squared, so for the action to be dimensionless $g^2$ must have the dimension of an inverse area, and $g$ itself provides the scale. In four dimensions $g$ is dimensionless, and the scale is thought to be generated by quantum effects Physics (4-07). The protagonist of this chapter is a property found only in two dimensions: fixing the gauge suitably makes the degrees of freedom of the faces independent. In terms of the six stages (6-01), stages 1–4 close, stage 5 (the axioms) is partial (Euclidean invariance; §05), and the gap of stage 6 is not a question with the same meaning as in four dimensions (§07).

The set-up is as in 5-01: $U_\ell\in G$ on an edge $\ell$, $U_p$ for the product around a face $p$, and the weight $\prod_pe^{\beta\,\mathrm{Re}\operatorname{tr}U_p}$ ($dU$ is the Haar measure). The β of this chapter is half of the $\beta_W$ customary for $SU(2)$ (Wilson action $\beta_W\sum_p\bigl(1-\tfrac12\mathrm{Re}\operatorname{tr}U_p\bigr)$; in four dimensions $\beta_W=4/g^2$): $\beta=\beta_W/2$. β = 1 in §03 corresponds to $\beta_W=2$. A gauge transformation $U_{xy}\mapsto g_xU_{xy}g_y^{-1}$ does not change $\operatorname{tr}U_p$. The variables on a tree (a set of edges with no cycles) can be set to 1 by choosing $g_x$ in order from the root, without changing expectation values of gauge-invariant quantities. Known Counting on an $L\times L$ square lattice (free boundary),

the tree has $V-1$ edges, so the remaining edges are as many as the faces (Euler's formula for the disc). Taking the tree to be the vertical edges and the bottom edges (as in Driver 1989), what remains are the horizontal edges $h_{i,j}$ ($h_{i,0}=1$), and

can be solved from the bottom up in one-to-one correspondence, and since the Haar measure is invariant under multiplication the measure carries over. The face variables are mutually independent, each with density $\rho_\beta(U)=e^{\beta\,\mathrm{Re}\operatorname{tr}U}/z(\beta)$.Known Extending to the whole plane, the infinite-volume limit exists independently of the boundary condition and, on a finite region, agrees with the measure with free boundary (Theorem 7.2 of Driver, shown by integrating out the outer edges using the invariance of the Haar measure; Driver writes that this theorem is essentially contained in Dosch–Müller 1979). Known

ComputedThe numbers in this figure (numbers of vertices, edges and faces) were computed on this machine. The numbers are formula (2.1).

The tool is the character $\chi_\lambda(U)=\operatorname{tr}\lambda(U)$ ($\lambda$ an irreducible unitary representation, of dimension $d_\lambda$). It is conjugation-invariant, and $\int_G\chi_\lambda\overline{\chi_\mu}\,dU=\delta_{\lambda\mu}$. For $SU(2)$ the representation of spin $j$ has dimension $2j+1$, and for an element with eigenvalues $e^{\pm i\theta}$, $\chi_j=\sin((2j+1)\theta)/\sin\theta$ (1-04). The key is a lemma: if the density of $X$ is conjugation-invariant and $X$ is independent of $A$, then

This is because $\mathbb{E}[\lambda(X)]$ commutes with every $\lambda(V)$, so by Schur's lemma it is $w_\lambda I$. When a simple closed curve $C$ encloses $n$ faces, the variables of the enclosed faces appear once each in the tree-gauge $U_C$, accompanied by conjugations by other face variables. The densities are conjugation-invariant, so (3.1) applies as it is, and moving faces one at a time to the front by cyclicity of the trace and peeling them off gives

Here $W_C=\chi_{\mathrm{fund}}(U_C)/d_{\mathrm{fund}}$ (the value in the fundamental representation; $\tfrac12\operatorname{tr}U_C$ for $SU(2)$). Migdal (1975) showed that in two dimensions the transformation rule (recursion) for the weights under coarsening the lattice becomes exact, wrote the formula in which the expectation of a character is exponential in the enclosed area $A$, $\langle\chi_q\rangle=d_qe^{-g_q^2A}$ ($q$ a representation, $g_q^2$ a constant for each representation), and stated that it is "nothing but Wilson's area law". Known The value is fixed by the area alone, and holds at every β. This contrasts with four dimensions, where the area law has been shown only on the strong-coupling side (5-02). The coefficient can be written by the following formula in modified Bessel functions, and the difference from numerical integration was at most $2\times10^{-16}$. Computed

$I_n$ is the modified Bessel function, and for integer $n$, $I_n(x)=\frac1\pi\int_0^\pi e^{x\cos\theta}\cos n\theta\,d\theta$. When $x$ is large the ratio $I_n(x)/I_1(x)$ approaches 1, and §04 uses how it approaches.

ComputedThe numbers in this figure were computed on this machine. The $w$ in the figure is the Bessel series (3.3) summed on the spot, agreeing with values computed with scipy to the fifth decimal place. If JavaScript is disabled, see Table 1.

Table 1 $SU(2)$. "Random" is heat-bath Monte Carlo on a 6×6 lattice (free boundary) without gauge fixing. Brackets give the error in the last digit. Computed

The calculation without gauge fixing agrees with the tree-gauge formula within errors.

At lattice spacing $a$, a loop of area $A$ encloses $n=A/a^2$ faces. Taking the length in (1.1) as $|X|^2=-2\operatorname{tr}X^2$, the expansion $U_p\approx e^{a^2F_{12}}$ gives, for $SU(2)$, $\beta=2/(g^2a^2)$. From $I_n(x)/I_1(x)=1-\frac{n^2-1}{2x}+O(x^{-2})$, $w_j=1-j(j+1)/\beta+O(\beta^{-2})$, so

The right-hand side is a coefficient of the heat kernel $Q_t$ of the group. $\Delta$ is the Laplacian on the group (different from the $\Delta$ of the mass gap), the operator that has the characters as eigenfunctions, $\Delta\chi_\lambda=-c_\lambda\chi_\lambda$. $c_\lambda\ge0$ is the Casimir value determined by each representation, $c_j=j(j+1)$ for $SU(2)$. The heat kernel is the solution of the heat equation $\partial_tQ_t=\tfrac12\Delta Q_t$ concentrated at the identity at time 0, and expanded in characters

The right-hand side of (4.1) is the coefficient $e^{-c_jt/2}$ at $t=g^2A$. That is, the holonomy of a simple closed curve of area $A$ follows the heat kernel $Q_{g^2A}$. Known The semigroup property is "adding areas adds times". In Table 2 each of $j=\tfrac12,1,\tfrac32$ approaches the heat kernel, and the error becomes 1/4 each time $a$ is halved. Computed

Table 2 $SU(2)$, $g=1$, area 1, $\beta=2/a^2$. Values are $w_j^{\,n}$. Computed

Taking the face weights to be heat kernels from the start gives, by the semigroup property, a measure independent of how the faces are divided (Migdal's observation). Driver showed that lattices with heat-kernel weights (the Villain action) and with the Wilson action converge to the same continuum measure, using for the latter a group-valued central limit theorem (Borgs–Seiler). Known

Detour: a theory that sees only area

The continuum version of the tree gauge is the complete axial gauge ($A_2=0$, and $A_1=0$ on the $x$ axis). Then $[A_1,A_2]=0$, so the field strength becomes linear, and the weight $e^{-S}$ becomes Gaussian in $F$.

$F$ is Lie-algebra-valued white noise ($F^a(R)$ is the integral over a region $R$, $|R|$ its area; 1-09), and its values on disjoint regions are independent. It is the continuum version of "independent for each face" on the lattice. Known $F$ is not defined pointwise, so parallel transport becomes a stochastic differential equation: an ordinary differential equation whose driving term contains white noise (increments of Brownian motion), solved after fixing one way of taking stochastic integrals. The driving term of parallel transport along a horizontal curve is the integral of $F$ between the curve and the $x$ axis, and each component is a Brownian motion whose time is that area (1-09) (Proposition 3.2 of Driver). This method goes back to Gross–King–Sengupta (1989).

Driver's results are (i) that expectation values of gauge-invariant functions of holonomies can be written as finite-dimensional integrals with products of heat kernels as density (times are the face areas, arguments the way the curves cross; Theorem 6.4), (ii) invariance under area-preserving maps (Corollary 6.7), and (iii) convergence of lattices. Known From Corollary 6.7 follows Euclidean invariance of gauge-invariant expectation values (part of the axioms). Driver does not claim verification of the whole of the Osterwalder–Schrader axioms (4-10), including reflection positivity. The observables are quantities along curves, and Driver writes that Wilson-loop variables are "quite singular for $d=3$, and especially for $d=4$".

On a closed surface of genus $\gamma$ Euler's formula becomes $V-E+P=2-2\gamma$, and after fixing a tree $P-1+2\gamma$ edges remain. The face variables are subject to one overall constraint, and $2\gamma$ variables going around the holes are added, so the faces are not independent. Even so, the measure has been constructed. Sengupta (1992–97) constructed it on compact surfaces (original papers: bibliographic data only), and Lévy (2003) treated the discrete case and the continuum limit in a unified way and showed a Markov property with respect to cutting and gluing surfaces (when cut along a curve, fixing the values on the cut makes the two sides not influence each other). Ways of gluing gauge fields on a surface come in topological kinds (1-07); the non-trivial kinds (non-trivial principal bundles) are Lévy (2006), and the general theory of holonomy fields is Lévy (2010). Known The discrete version puts a heat kernel on each face $F$.

$h(\partial F)$ is the product along the boundary of the face, $T$ the total area, and $Z$ is up to a constant factor. Known The $Z_1$ of the torus has the same form as the $\operatorname{Tr}e^{-\tau H}$ of the Hamiltonian of §07.

Detour: letting the area go to 0

The mass gap is the gap in the spectrum of the Hamiltonian (4-11). On the lattice we use the transfer matrix $\mathsf T$: the operator whose kernel is the weight for advancing time by one step, defining the Hamiltonian $H$ by $\mathsf T=e^{-aH}$ (5-03). Take space to be a circle of $L$ edges, and do not make time periodic (an open cylinder). Then edges in the time direction form no cycles and can be fixed to 1, and gauge-invariant states are functions of the conjugacy class of the holonomy once around, $V_1\cdots V_L$, that is, superpositions of characters. One step of time evolution multiplies each edge by one independent face variable, so using (3.1) $L$ times gives

The lowest excitation is $E=\sigma_{\mathrm{lat}}L$ of the fundamental representation, the energy of a "ring of lines of force" going once around the circle. The tension is the same number as in the area law of §03. For $SU(2)$ at β = 1, $E_{1/2}$ is 3.3469 at $L=4$ and 6.6938 at $L=8$. The ratio $E_1/E_{1/2}$ at β = 1, 32, 512 is 2.4044, 2.6664, 2.6667, approaching the Casimir ratio $8/3$. Lemma (3.1) was also checked with random numbers ($2\times10^6$ samples). Computed

The continuum limit can be taken in the same way as in §04. Holding the physical length $L_{\mathrm{phys}}=La$ fixed and putting $\beta=2/(g^2a^2)$ and $-\ln w_\lambda=c_\lambda/\beta+O(\beta^{-2})$ into (7.1) gives

Combined with $\Delta\chi_\lambda=-c_\lambda\chi_\lambda$ (§04), the continuum Hamiltonian is a constant multiple of the Laplacian on the group, $-\tfrac12g^2L_{\mathrm{phys}}\Delta$ (acting on class functions). Paper The derivation is this chapter's own; for $SU(2)$ with $g=1$ and $L_{\mathrm{phys}}=1$, the value for $j=\tfrac12$ at $a=1/64$ was 0.375011 (limit 0.375), and the difference became 1/4 each time $a$ was halved. Computed

Making the circle longer ($L\to\infty$) sends every eigenvalue other than the vacuum off to infinity, and there are no excitations (particles) propagating with finite energy. We also define the correlation length here. The correlation length is the length $\xi_{\mathrm{lat}}$ (in lattice units) when the connected correlation $\langle P_xP_y\rangle-\langle P_x\rangle\langle P_y\rangle$ of the quantity $P_x=\mathrm{Re}\operatorname{tr}U_{p_x}$ of a plaquette $p_x$ decreases like $e^{-|x-y|/\xi_{\mathrm{lat}}}$. On the plane different plaquettes are independent, so the connected correlation is exactly 0, and $\xi_{\mathrm{lat}}=0$ at every β (what to hold fixed when taking the limit is in 5-09). The four-dimensional question (whether the correlation length diverges with a finite mass remaining in physical units) does not arise in two dimensions.

Stochastic quantisation is the method of not writing the measure directly, but solving a random time evolution $\partial_tA=-\tfrac12\nabla S(A)+\xi$ ($\xi$ is spacetime white noise) that has it as invariant measure. This equation is called the Langevin equation. In finite dimensions the invariant measure of this equation is proportional to $e^{-S}$. Even in two dimensions $A$ is a distribution, so renormalisation and consistency with gauge are needed. The tools for this (regularity structures, the DeTurck method) are treated in 6-09; here we look only at the results. Known

Chevyrev (2019) realised the measure on $T^2$ as a distributional 1-form, and Chandra–Chevyrev–Hairer–Shen (2022) built a space of distributional connections and gauge orbits (a Polish space, a good metric space on which probability theory can be used as usual) and defined a Markov process (a random process whose future distribution is determined by the present state alone) consistent with gauge in the sense of laws. Chevyrev–Shen (arXiv 2023; Comm. Pure Appl. Math. in 2026) showed that the Yang–Mills measure of the trivial principal bundle on $T^2$ is the unique invariant probability measure of this process (Theorem 2.16). As corollaries come long-time existence of the process, that the limits of the Wilson, Villain and Manton lattices are the same measure (the paper writes that for the Villain action this was already known from Lévy 2006), and a decomposition into a Gaussian free field (1-09) and an almost Lipschitz remainder. Known

This road too is not separate from solvability. Chevyrev–Shen use gauge fixing on the lattice and obtain the moment estimates from two-dimensional solvability. The paper calls extending these estimates to higher dimensions or to the two-dimensional Yang–Mills–Higgs model open, and writes that the construction of non-abelian Higgs models is open even in two dimensions. Known In July 2026 two preprints appeared (not refereed; from the abstracts). Chevyrev–Klose–Mohamed give a gauge-fixed representation of optimal regularity for the existing measure, and Chhaibi–Dang–Guedes Bonthonneau–Rivière–Tô say that they used Morse theory (a method of decomposing a surface by the critical points of a function on it) to make the measure on surfaces without passing through lattices, reproducing the formulas of Migdal, Witten and Lévy. Known

The core of the solvability was the counting of §02. On a $d$-dimensional lattice, per lattice point, the edges left after removing gauge number $d-1$ and the faces $\binom d2$: for $d=2,3,4$, 1 and 1, 2 and 3, 3 and 6. In three or more dimensions the face variables are tied by constraints on each cube (the lattice Bianchi identity) and cannot be taken independently. Known

What can be carried out is on the side of methods: taking holonomies along curves as observables, expansion in characters (the tool of the strong-coupling expansion), the pattern of proofs of universality of lattice actions, and the framework of stochastic quantisation. What cannot be carried out are the independence of faces, the dependence on area alone, and the absence of propagating degrees of freedom. What the problem statement of Jaffe–Witten cites as "the only complete example" of an interacting gauge theory satisfying the axioms is likewise the two-dimensional abelian Higgs model (Brydges–Fröhlich–Seiler). Known

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