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

Javad Komijani

arXiv:2610.09147v1hep-latcs.LGhep-th

TL;DR

Generative sampling can model individual plaquettes, but mapping them to links remains an obstruction because holonomies must satisfy a corner consistency condition. The paper replaces that constraint with conditional sampling and reweighting, deriving closed-form group-commutator densities for SU(2) and SU(3). The SU(2) corner sampler is exact, and its effective sample size approaches 6/π2 ≈ 0.61 for asymptotically large lattices.

  • Problem

    Generative methods can model a single plaquette, but mapping sampled plaquettes to links is obstructed by a four-corner consistency condition on holonomies.

    Source: §I · pp. 1–2

  • Method

    The paper uses a twisted prelink boundary condition to replace the corner constraint with conditional sampling and reweighting, using a closed-form SU(2) sampler and a trained flow for SU(3).

    Source: §I · p. 2, §5 · p. 12

  • Results

    SU(2) has an exact corner sampler whose effective sample size approaches 6/π2 ≈ 0.61 as the lattice becomes asymptotically large.

    Source: §V · p. 19

  • Takeaways & Limitations

    For SU(2), the corner correction's effective sample size is determined by the density of Z alone and remains moderate in the large-lattice limit.

    Source: §V · p. 19

    For SU(3), the conditional law of (D, Ω) given Z is not available in closed form, so the sampler is a trained approximation.

    Source: §V · p. 19

Abstract

from arXiv · show

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 $\mathrm{SU}(N)$ 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 $X,Y\in\mathrm{SU}(N)$ given the group commutator $Z=XYX^\dagger Y^\dagger$, where $Z\in\mathrm{SU}(N)$ 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 $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$. A normalizing flow models the single-plaquette distribution for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$; for the group-commutator problem we use a closed-form sampler for $\mathrm{SU}(2)$ and a trained normalizing flow for $\mathrm{SU}(3)$. 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.

I. INTRODUCTION

In two-dimensional lattice gauge theory, modeling plaquettes is straightforward, but recovering links from sampled plaquettes is obstructed by a corner consistency condition. The paper replaces that hard constraint with conditional sampling and reweighting in a statistical construction.

I. INTRODUCTION

  • The link-level map is the obstacle for generative sampling, and excluding one plaquette yields an effective sample size that depends drastically on gauge coupling.The single-plaquette distribution is easier to model than the mapping from plaquettes to links.Source: §I · p. 1
  • A statistical construction provides a weighted map between plaquettes and links, replacing direct recovery with conditional sampling and reweighting.The plaquette variables determine the holonomies, but the corner consistency condition must still be addressed.Source: §I · p. 2
  • A twisted prelink boundary condition trades the hard four-corner constraint for conditional sampling of two SU(N) variables given the corner-dependent group commutator Z.The corner condition is the only constraint on holonomy variables in two dimensions.Source: §I · p. 2
  • The conditional sampler is closed form for SU(2) and flow-based for SU(3), while the corner-correction effective sample size is typically above one half on tested 2 × 2 and 32 × 32 lattices.The tests use two couplings for each lattice size.Source: §I · p. 2
  • Unlike the single-plaquette-exclusion baseline, corner-correction effective sample size depends on coupling only through Z and remains moderate as the lattice becomes asymptotically large.For an exact SU(2) corner sampler, the effective sample size is determined by the density of Z alone.Source: §I · p. 2

B. Minimally extended lattice and prelinks

Prelinks replace link variables through a line-by-line recursion on a minimally extended lattice. Their boundary freedom and symmetries permit the extended-lattice construction used to address the corner constraint.

B. Minimally extended lattice and prelinks

  • The minimal extension adds one site beyond the physical lattice in each prelink's own direction, while the full extension adds boundary copies specified by a twisted-like condition.The boundary factor Q is shared along each prelink line and uses the initial-value symmetry.Source: B. Minimally extended lattice and… · pp. 3, 5
  • The recursion Vµ(x + ˆµ) = Vµ(x) Uµ(x) replaces link variables with prelinks on the minimally extended lattice without an additional Jacobian.Each recursion step is a left translation of a Haar-distributed link variable.Source: B. Minimally extended lattice and… · p. 4
  • Prelinks have a local right symmetry equivalent to link gauge transformations and a semi-global initial-value symmetry shared along each line.The latter is a redundancy of the prelink parametrization, invisible to the link formulation.Source: B. Minimally extended lattice and… · p. 5
  • Fixing the initial-value symmetry with the described scheme yields an L-shaped prelink pattern and reduces the semi-global freedom to a global one.The scheme also sets V1(0, 0) = V0(0, 0).Source: Figure 1 · p. 4, B. Minimally extended lattice and… · p. 5
  • Temporal gauge fixes V0 to the identity on the physical lattice, while the twisted boundary condition fixes its values on the added boundary sites.V0 outside the physical lattice need not be the identity.Source: B. Minimally extended lattice and… · p. 6

C. Extended lattice and prelink holonomies

Holonomies combine prelinks and allow the Wilson action to be expressed through an operator with the same plaquette trace. In two dimensions, the holonomy variables obey a constraint relating the four extended-lattice corners.

C. Extended lattice and prelink holonomies

  • The holonomy combines two prelinks at the same site, is invariant under the local right symmetry, and transforms covariantly under the initial-value symmetry.After fixing that symmetry to a global one, it transforms by global conjugation.Source: C. Extended lattice and prelink… · p. 6
  • The holonomy formulation defines a new operator with the same trace as the original plaquette, allowing the Wilson action to be written in holonomy variables.The holonomy variables nevertheless require constraints.Source: C. Extended lattice and prelink… · p. 7
  • With periodic prelink boundary conditions, the four corner holonomies must satisfy a group-valued constraint, so they are not independent.The two-dimensional holonomy parametrization is an open-boundary parametrization up to this corner constraint.Source: C. Extended lattice and prelink… · p. 7
  • Batrouni and Halpern imposed the finite-periodic-lattice corner constraint as a hard delta function and did not provide an inverse map to link or prelink data [18].The paper takes up the constraint for two-dimensional SU(N) gauge theory.Source: C. Extended lattice and prelink… · p. 7

III. OPEN-BOUNDARY HOLONOMY PARAMETRIZATION

The section replaces the four-corner consistency constraint of periodic prelinks with a conditional sampling problem using twisted prelink boundary conditions. This separates independently sampled open-boundary plaquettes from a corner-dependent conditional term.

III. OPEN-BOUNDARY HOLONOMY PARAMETRIZATION

  • Figure 4 labels the seed prelinks V01 and V11 and the semi-global freedoms Q0 and Q1 on the right column and top row, respectively.These labels are used to derive the corner relations.Source: Figure 4 · p. 8

A. The corner constraint as a group-commutator problem

  • The holonomy formulation trades the hard four-corner constraint for conditional sampling by allowing twisted prelink boundary conditions.The action retains its original link boundary condition; the twist affects the prelinks.Source: A. The corner constraint as a… · pp. 8–9
  • The section derives how to map holonomies back to prelinks while preserving the action, then restores the full extended lattice through a volume-preserving change of corner variable.The construction uses temporal gauge to express the prelinks in terms of corner holonomies and the twisted boundary terms.Source: A. The corner constraint as a… · pp. 9–10
  • The four corner holonomies determine Z, and solving the remaining consistency condition reduces to finding X, Y ∈ SU(N) with Z = XYX†Y†.Once X and Y are known, the twisted boundary terms and corner prelinks are fixed.Source: A. The corner constraint as a… · pp. 8–9
  • The resulting measure factorizes into an independently sampled open-boundary holonomy theory and a conditional term for sampling (D, Ω) given Z(h).The open-boundary theory factorizes plaquette by plaquette.Source: A. The corner constraint as a… · p. 10

B. Group-commutator encoding and sampling (X, Y )

Given the group commutator Z, the task is to sample X and Y from the conditional law induced by the SU(N) × SU(N) Haar measure. The section uses an encoder to make Z a coordinate, samples the remaining coordinates conditionally, and decodes X and Y.

B. Group-commutator encoding and sampling (X, Y )

1. Encoder and decoder

For SU(2) and SU(3), the section defines an encoder and decoder for (X, Y) using the spectrum and eigenvectors of Y and ZY. It then relates the encoder density to a Jacobian evaluated through a spectral twist.

1. Encoder and decoder

  • The encoder maps (X, Y) to (D, Ω, Z), with Ω the eigenvectors of Y and D = P†XΩ, where P contains ordered eigenvectors of ZY.The decoder returns the original pair, although re-encoding may change phases when eigenvectors are recomputed.Source: §1 · pp. 10–11
  • The decoder is restricted to SU(2) and SU(3), because equal traces are only a necessary condition for determining Λ when N ≥ 4.It also requires Z ≠ I, since then the condition does not determine Λ.Source: §1 · p. 10
  • For SU(3), the trace condition becomes triangle closure on the three |m_k| and fixes Λ up to six choices; for SU(2), it fixes λ1 up to the center ±Λ.The SU(3) choices comprise three center rotations and two chiralities.Source: §1 · p. 10

2. Jacobian: the spectral twist

  • The encoder density is the inverse Jacobian of the spectral twist Ẑ = GΛ†, whose Jacobian depends on the eigenvector matrix W through B_ij = |W_ij|^2.The Jacobian is given in closed form for both groups.Source: §2 · p. 11

3. The conditional law and sampling with Euler angles

The conditional density factorizes with a uniform diagonal variable, while dependence on Z enters through the relative frame. The sampler transforms Haar-distributed Euler-angle coordinates according to Z and importance-weights the resulting samples.

3. The conditional law and sampling with Euler angles

  • The encoded density factors as p_encode(D, Ω, Z) = ρ(Z)P(D, Ω|Z), with D uniform and independent of Z.Source: §3 · p. 11
  • Conditioning Ω on Z reduces to sampling the relative frame R given the eigen-angles of Z.The relative frame is invariant under simultaneous conjugation of X and Y.Source: §3 · p. 12
  • Haar-uniform coordinates make the measure flat on [0, 1]^3 for SU(2) and [0, 1]^8 for SU(3), while conditioning on Z deforms only the central coordinates.The outer phases remain uniform.Source: §3 · p. 12
  • The sampler transforms Haar-distributed R0 with a Z-dependent map T_Z, decodes (X, Y), and assigns importance weight w = p/q.The conditional density p is known in closed form; q is determined by the Jacobian of T_Z.Source: §3 · p. 12
  • Finding T_Z is the remaining sampling problem, and its solution differs between SU(2) and SU(3).Source: §3 · p. 12

4. Closed-form sampler for SU(2)

The SU(2) corner conditional is sampled exactly by deforming one relative-frame coordinate. The appendix also gives commutator-density results for SU(2) and SU(3), while the SU(3) conditional sampler remains flow-based.

4. Closed-form sampler for SU(2)

  • The SU(2) conditional sampler deforms only the relative-frame coordinate b; its closed-form conditional CDF yields an exact draw with weight w = 1.The other two relative-frame coordinates remain uniform.Source: §4 · p. 12

5. Trained sampler for SU(3)

  • For SU(3), a closed-form conditional sampler is unavailable, so a trained normalizing flow transforms four central coordinates conditioned on Z’s eigenvalues.The four outer phases and D remain uniform.Source: §5 · p. 12
  • At fixed Z, the SU(3) conditional-weight correction has empirical effective sample size 0.98.The weight is computed from the known density and spectral-twist Jacobian together with the flow log-Jacobian.Source: §5 · p. 13

6. Marginal distribution of the group commutator Z

  • The commutator density for SU(3) is obtained by summing a character series over irreducible representations.The same series-based derivation is presented for SU(2).Source: §6 · p. 13
  • For SU(2) and SU(3), the commutator density is, up to 1/N, the ratio of Vandermonde determinants of eigen-angles and eigenvalues.The density is finite and positive away from the identity, where it diverges as Z approaches I.Source: §6 · p. 14
  • For SU(2), the commutator density can also be derived by integrating the encoded density over D and Ω.This direct derivation uses the relative-frame coordinate b.Source: §6 · p. 14

C. Sampling workflow

The workflow samples independent plaquettes from the open-boundary holonomy theory, then corrects for the corner-dependent weight. Holonomies are converted deterministically to prelinks and links.

  • The open-boundary holonomy theory factorizes into independent plaquettes, while its difference from the exact holonomy marginal is a corner-coupling weight depending on Z(h).The action is summed over the physical lattice while holonomies live on an extended lattice.Source: C. Sampling workflow · p. 15
  • Plaquettes are proposed independently from trained single-plaquette flows and corrected by Metropolis–Hastings, yielding exact open-boundary samples in the stationary limit.The reported acceptance rate exceeds 99% in the tested cases.Source: C. Sampling workflow · p. 15
  • Cumulative plaquette products construct holonomies with identity values on the first row and column; each construction step is volume preserving.The action’s symmetry leaves ρ(Z(h)) invariant under this choice.Source: C. Sampling workflow · p. 15
  • Given Z(h), the workflow samples (D, Ω) conditionally, decodes (X, Y), and forms Q0 and Q1; the configuration carries ρ(Z(h)) and, for SU(3), the conditional weight.The SU(2) conditional sampler is exact, while SU(3) uses a trained model.Source: C. Sampling workflow · p. 15
  • Holonomies map deterministically to prelinks and links, but the configuration weight must be retained when computing weighted observables.Source: C. Sampling workflow · p. 15

IV. SIMULATIONS

Across 64 Wilson loops, weighting brings the 2×2 flow results into close agreement with HMC, while raw results on 32×32 lattices are already largely compatible at the tested statistics. Acceptance rates and final-weight ESS are also reported.

  • Final acceptance rates range from 0.998 (SU(2), 2×2, β = 2) to 0.72 (SU(3), 32×32, β = 12).The HMC step size is the same across settings and is not tuned.Source: §IV · p. 18
  • The simulations compare HMC, raw-flow, and weighted-flow Wilson-loop traces on 2×2 and 32×32 lattices for SU(2) and SU(3), at two couplings each.The study uses 16,384 configurations for 2×2 and 1,024 for 32×32 lattices.Source: §IV · pp. 17–18
  • For 2×2 lattices, weighted results have RMS z = 0.86 and |z| ≲ 2.1 across 64 loops, while raw RMS z reaches 22 and maximum |z| reaches 34.The loops are correlated, so these values are indicative rather than a formal test.Source: §IV · p. 18, Figure 7 · p. 18
  • On 32×32 lattices, raw flow results are already largely compatible with HMC at ncfg = 1024.Source: §IV · p. 18
  • Table I reports plaquette expectation values and final-weight ESS for all eight cases, with one run reported per case.ESS fluctuates between runs, especially on 32×32 lattices.Source: §IV · p. 18, Table I · p. 19

V. SUMMARY AND CONCLUDING REMARKS

The paper samples two-dimensional SU(N) lattice gauge theory from independent plaquettes using prelink and holonomy parametrizations, with corner corrections carried as weights. It reports moderate effective sample sizes, an exact SU(2) corner sampler, and an approximate SU(3) conditional sampler.

V. SUMMARY AND CONCLUDING REMARKS

  • 0.56–0.75 is the reported range for final-weight effective sample size, with no clear degradation from 2 × 2 to 32 × 32 lattices.The results are for SU(2) and SU(3) with the Wilson plaquette action.Source: §V · p. 19
  • Twisted prelink boundary conditions replace the four-corner consistency condition with conditional sampling given the group commutator, at the cost of a weight.The construction uses an encoder whose coordinates factor the measure into the marginal density of Z and the conditional law of (D, Ω); D is uniform and independent of Z.Source: §V · p. 19
  • For SU(2), the corner sampler is exact, with ρ(θ) = (π − θ)/(2 sin θ) and asymptotic effective sample size 6/π^2 ≈ 0.61.The conditional law’s only nontrivial part has an explicit cumulative distribution function that can be inverted.Source: §V · p. 19
  • For SU(3), the marginal density of Z is closed form, but the conditional law of (D, Ω) is sampled with a trained approximation whose fixed-Z effective sample size is about 0.98.The authors are still working on an exact conditional sampler.Source: §V · pp. 19–20
  • In higher dimensions, boundary holonomies face many additional constraints that are not straightforward to resolve.The paper leaves this problem for future work and notes gauge-equivariant networks as another use of the holonomy parametrization.Source: §V · p. 20

Note added

A related work explores direct plaquette-space sampling with multilevel normalizing flows. The paper also reports how AI tools contributed to text improvement, implementation, and a derivation that the author subsequently simplified.

Note added

  • Ref. [24] samples directly in plaquette space with multilevel normalizing flows satisfying Bianchi identities exactly, covering U(1) in two and four dimensions and SU(2) in two dimensions.It uses a periodic lattice and, to the authors’ knowledge, not twisted boundary conditions.Source: Note added · p. 20

Use of AI tools

  • Claude and ChatGPT helped improve the text; Claude also implemented the SU(3) Euler-angle parametrization and found a closed form for the N = 3 commutator density, which the author re-derived more simply.The author checked the parametrization and takes responsibility for the paper’s content.Source: Use of AI tools · p. 20
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