Sampling SU(N) gauge theory on a 2D lattice from independent plaquettes via holonomies and corner reweighting

The gist The authors explore a statistical way around mapping sampled plaquettes to links in two-dimensional SU(N) gauge theory by trading an exact consistency condition for a conditional sampling problem involving the group commutator, which allows for effective sampling even when using generative models.

How it works

1. The method employs an equivalent formulation of lattice gauge theory using string and corner variables and introduces prelinks Vµ(x) (Page 2).

2. The link field is recovered from the holonomy configuration by resolving a consistency condition at the four corners of the extended lattice (Page 2).

3. This hard, group-valued delta-function constraint is traded for sampling two SU(N) variables, X and Y, conditional on Z = XY X†Y† (Page 2).

4. The full prelink measure factorizes as p(V) dV dX dY = p(h) pencode(D, omega, Z(h)) dh dD domega (Page 3).

Holonomy and Corner Constraint

The holonomy variables hµν(x) are defined on the extended lattice Λ˜ (Page 6). The plaquette action can be written in terms of these holonomy variables through the combination P˜µν(x) = h†µν(x) hµν(x + ˆµ) h†µν(x + ˆµ + ˆν) hµν(x + ˆν) (Page 6). The constraint arising from periodic prelinks on the extended lattice is a consistency condition at the four corners, which is hard to sample directly (Page 8). This constraint leads to the group-commutator problem for (X, Y) given Z = XY X†Y† where Z depends on the four corners (Page 9).

Encoding and Conditional Sampling

The corner condition is relaxed by using a twisted boundary condition on the prelinks which introduces two SU(N) variables (X, Y) sampled conditional on Z = XY X†Y† (Page 2). The encoding of (X, Y) into three matrices (D, omega, Z) allows the corner condition to become an ordinary probability density pencode(D, omega, Z(h)) dh dD domega (Page 3). The full prelink measure then factorizes as p(V) dV dX dY = p(h) pencode(D, omega, Z(h)) dh dD domega (Page 3).

Sampling Workflow

The sampling workflow involves several steps:

1. Plaquettes are sampled independently using a normalizing flow model for SU(2) and SU(3) (Page 15).

2. A Metropolis–Hastings accept/reject step is used to correct the plaquette samples based on the ratio of target weight to q (Page 16).

3. Holonomies are constructed from plaquettes by cumulative matrix products along both directions, fixing the holonomies on the first column and first row to the identity (Page 17).

4. Given Z(h), (D, omega) are sampled from P(D, omega Z(h)) using a closed-form sampler for SU(2) and a trained normalizing flow for SU(3) (Page 17).

5. The resulting samples are decoded to obtain (X, Y), and the configuration carries the weight ρ(Z), multiplied by the conditional weight w of the corner sampler (Page 17).

Results and Effectiveness

For SU(2), an exact corner sampler is available, and the effective sample size is determined by the density of Z alone, which approaches a Haar distribution for asymptotically large lattices (Page 19). For SU(3), a trained normalizing flow is used for the conditional sampling of (D, omega) given Z, and its effective sample size at fixed Z is about 0.98 (Page 19). The tests on 2 × 2 and 32 × 32 lattices show that per-plaquette acceptance rates exceed 99% and the effective sample size of the final weights is moderate, typically above one half (Page 1).

Conclusion

The paper successfully demonstrates a statistical construction for sampling SU(N) gauge theory from independent plaquettes by resolving the corner constraint into a conditional sampling problem involving the group commutator Z, yielding moderate effective sample sizes in tests on various lattices (Page 1). The key finding is that this approach circumvents the obstruction of mapping sampled plaquettes to links by carrying a weight ρ(Z) that depends only on the lattice size and coupling (Page 19). The construction is exact for SU(2) and provides a trained approximation for SU(3), suggesting a viable path forward for more general gauge groups (Page 19).

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Sampling SU(N) gauge theory on a 2D lattice from independent plaquettes via holonomies and corner reweighting Javad Komijani (Dated: October 8, 2026) In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the SU(N) gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling X, Y ∈ SU(N) given the group commutator Z = XY XY†Y†, where Z ∈ SU(N) depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of Z, which determines these weights, in closed form for SU(2) and SU(3). A normalizing flow models the singleplaquette distribution for SU(2) and SU(3); for the group-commutator problem we use a closed-form sampler for SU(2) and a trained normalizing flow for SU(3). In our tests on 2 × 2 and 32 × 32 lattices at two couplings each, per-plaquette acceptance rates exceed 99% and the effective sample size of the final weights is moderate, typically above one half.

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We circumvent the problem by giving a statistical construction in two dimensions, a weighted map between plaquettes and links. To this end, we employ an equivalent formulation of lattice gauge theory using string and corner variables [17, 18]; building on those works, we introduce prelinks Vµ(x), defined on an extended lattice, and the holonomy hµ0(x) = Vµ(x)V†0 (x). (1.1) Unlike link variables, which are bilocal in the sense that their gauge transformation depends on two neighboring sites, prelinks are local up to an additional semi-global freedom, the same for all prelinks along a line. This semi-global freedom allows us to impose a boundary condition on the prelinks that differs from the original boundary condition of the links; this freedom is exploited in this paper.

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The plaquette variables can be written in terms of the holonomy variables in a unique way, but recovering the link field from a given holonomy configuration requires resolving a consistency condition at the four corners of an extended lattice; the same obstruction Batrouni and Halpern [18] identified on the periodic lattice but left as an imposed condition. This condition is a hard, group-valued delta-function constraint, so reweighting samples directly against it is impractical. In two dimensions it is the only constraint on the holonomy variables. Using a twisted boundary condition on the prelinks, without compromising the original boundary condition on the links, we trade it for two SU(N) variables (X, Y) sampled conditional on Z = XY X†Y†, where Z ∈ SU(N) built from the corner holonomies.

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We do this by encoding (X, Y) into three SU(N) matrices (D, omega, Z), with D diagonal, and decoding them back in closed form for SU(2) and SU(3); this turns the corner condition from a delta function into an ordinary probability density pencode(D, omega, Z(h)). The full prelink measure then factorizes as p(V) dV dX dY = p(h) pencode(D, omega, Z(h)) dh dD domega (Page 4).

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This is not a map but a statistical construction: computing (V, X, Y) from (h, D, omega) is deterministic, but (D, omega) must itself be drawn from P(D, omega Z(h)) and the resulting samples reweighted against it. For SU(2) we solve this conditional sampling problem in closed form. For SU(3) this work is in progress, and for now we set up the sampler with a normalizing flow. We also use normalizing flows for the single-plaquette distribution.

1. The system can perform generative sampling of SU(N) gauge theory on a 2D lattice from independent plaquettes by implementing a statistical construction that factors the measure into an open-boundary holonomy theory and a conditional sampling problem for (X, Y) given the group commutator Z = XY X†Y†.

2. For SU(2), the system can perform exact corner sampling using a closed-form sampler based on inverting the cumulative distribution function of the modulus coordinate b: b = 1/2 [1 + tan(θ/2) tan(π − θ)(s − 1/2)] where e±iθ are the eigenvalues of Z.

3. For SU(3), the system can perform conditional sampling for (D, Ω) given Z using a trained normalizing flow, achieving an effective sample size of about 0.98 at fixed Z, which is then used to compute weights in the final configuration samples.

4. The system can efficiently map sampled plaquettes to gauge links by first constructing holonomies and then prelinks deterministically: "V1(x) = h1,0(x), x0 < L0, x1 < L1, h1,0(x)Q0, x0 < L0, x1 = L1".

5. The system can incorporate the weight correction derived from the spectral twist Jacobian into its sampling process by computing log w = − log det(dZ/dG ˜) − log ρ(Z) + log jT for both SU(2) and SU(3).

6. The system can generate configuration samples that are statistically independent of one another, as shown in the comparison between NF:raw and NF:weighted results, with the effective sample size (ESS) fluctuating between 0.56 and 0.75 across various lattice sizes and couplings.

In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the SU(N) gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling X,Y in SU(N) given the group commutator Z=XYX Y, where Z in SU(N) depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of Z, which determines these weights, in closed form for SU(2) and SU(3). A normalizing flow models the single-plaquette distribution for SU(2) and SU(3); for the group-commutator problem we use a closed-form sampler for SU(2) and a trained normalizing flow for SU(3). In our tests on 2 times2 and 32 times32 lattices at two couplings each, per-plaquette acceptance rates exceed 99% and the effective sample size of the final weights is moderate, typically above one half.