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

Javad Komijani

hep-latOct 6, 2026

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

Authors: Javad Komijani

Abstract

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.

Figures & tables

Figure 1: Prelinks

V_{\mu}(x)

on the extended lattice

\tilde{\Lambda}

. The gray band shows the extension

\tilde{\Lambda}\setminus\Lambda

. Left: the straight-line recursion eq. ( 10 ), colored by direction (

\mu

blue,

\nu

red); the faded lines are the boundary copies of eq. ( 16 ), lying entirely inside the extension. Right: the L-shaped pattern arising from the scheme of eq. ( 17 ) for fixing the initial-value symmetry.

Figure 2: Holonomy

h_{\mu\nu}(x)=V_{\mu}(x)V_{\nu}^{\dagger}(x)

and the plaquette decomposition, eq. ( 22 ): the four corner rectangles combine so that shared interior Wilson-line segments cancel, leaving only the plaquette boundary.

Figure 3: Geometric picture of the corner constraint ( 24 ).

Figure 4: Corner labeling used to derive eqs. ( 25 )–( 29 ): seed prelinks

V_{01},V_{11}

and semi-global freedoms

Q_{0}

(right column,

x_{1}=L_{1}

) and

Q_{1}

(top row,

x_{0}=L_{0}

Panel of Figure 5 — Figure 5: $\mathrm{SU}(2)$ : $\operatorname{Tr}(W_{m,n})$ from HMC, raw flow samples (NF:raw), and reweighted flow samples (NF:weighted); ratio panels show NF:weighted against HMC.

Panel of Figure 6 — Figure 6: $\mathrm{SU}(3)$ : same comparison as fig. 5 .

Figure 7: Normalized deviation

z=(O_{\rm NF}-O_{\rm HMC})/\sqrt{\sigma_{\rm NF}^{2}+\sigma_{\rm HMC}^{2}}

of the Wilson loops

\operatorname{Tr}(W_{m,n})

of figs. 5 and 6 (eight loops per setting, in the order

(1,1)

(1,2)

(1,3)

(1,4)

(2,2)

(2,3)

(2,4)

(3,3)

), for NF:raw and NF:weighted. The vertical axis is linear for

|z|<2

(shaded) and logarithmic beyond. Right: histogram of the

64

weighted deviations against a unit normal.

Group	Size	$\beta$	HMC	NF:raw	NF:weighted	ESS
$\mathrm{SU}(2)$	$2\times 2$	2	0.4471(16)	0.4312(16)	0.4434(21)	0.627
$\mathrm{SU}(2)$	$2\times 2$	8	0.8340(6)	0.8186(6)	0.8339(7)	0.600
$\mathrm{SU}(2)$	$32\times 32$	4	0.6579(3)	0.6584(3)	0.6585(3)	0.684
$\mathrm{SU}(2)$	$32\times 32$	8	0.8191(1)	0.8195(1)	0.8195(2)	0.564
$\mathrm{SU}(3)$	$2\times 2$	3	0.2046(11)	0.2039(11)	0.2049(13)	0.724
$\mathrm{SU}(3)$	$2\times 2$	12	0.6921(6)	0.6779(6)	0.6930(8)	0.639

Table 1: Plaquette expectation value

\operatorname{Tr}(W_{1,1})

for all eight cases of figs. 5 – 6 : HMC, raw flow samples, and importance-reweighted flow samples. Digits in parentheses are the error on the last digits. The last column is the ESS of the final weights, i.e., the weight

\rho(Z(h))

times the conditional weight of the corner sampler, which is unity for

\mathrm{SU}(2)

(secs. III.2 , III.3 ). The ESS values fluctuate from one run to another, especially for the

32\times 32

lattices, and we report a single run for each case.

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