Bootstrap for Lattice Yang-Mills theory

TL;DR

This paper applies a numerical bootstrap method to lattice Yang-Mills theory in various dimensions, providing rigorous bounds on observables and showing potential as an alternative to Monte Carlo simulations.

Contribution

It introduces an efficient bootstrap algorithm that incorporates symmetries and relaxation techniques to study lattice Yang-Mills theory, yielding bounds consistent with known results.

Findings

Bounds on plaquette averages match Monte Carlo data in strong coupling.

Upper bounds align with 3-loop perturbation theory in weak coupling.

Method demonstrates potential as an alternative to Monte Carlo simulations.

Abstract

We study the $SU(\infty)$ lattice Yang-Mills theory at the dimensions $D=2,3,4$ via the numerical bootstrap method. It combines the Makeenko-Migdal loop equations, with a cut-off $L_{\mathrm{max}}$ on the maximal length of loops, and positivity conditions on certain matrices of Wilson loops. Our algorithm is inspired by the pioneering paper of P. Anderson and M. Kruczenski but it is significantly more efficient, as it takes into account the symmetries of the lattice theory and uses the relaxation procedure in line with our previous work on matrix bootstrap. We thus obtain rigorous upper and lower bounds on the plaquette average at various couplings and dimensions. For $D=4$, the lower bound data appear to be close to the MC data in the strong coupling phase and the upper bound data in the weak coupling phase reproduce well the 3-loop perturbation theory. Our results suggest that this bootstrap approach can provide a tangible alternative to the, so far uncontested, Monte Carlo approach.