Bootstrapping the Abelian Lattice Gauge Theories

Abstract

We study the $\mathbb{Z}_2$ and $U(1)$ Abelian lattice gauge theories using a bootstrap method, in which the loop equations and positivity conditions are employed for Wilson loops with lengths $L\leqslant L_{\textrm{max}}$ to derive two-sided bounds on the Wilson loop averages. We address a fundamental question that whether the constraints from loop equations and positivity are strong enough to solve lattice gauge theories. We answer this question by bootstrapping the 2D $U(1)$ lattice gauge theory. We show that with sufficiently large $L_{\textrm{max}}=60$, the two-sided bounds provide estimates for the plaquette averages with precision near $10^{-8}$ or even higher, suggesting the bootstrap constraints are sufficient to numerically pin down this theory. We compute the bootstrap bounds on the plaquette averages in the 3D $\mathbb{Z}_2$ and $U(1)$ lattice gauge theories with $L_{\textrm{max}}=16$. In the regions with weak or strong coupling, the two-sided bootstrap bounds converge quickly and coincide with the perturbative results to high precision. The bootstrap bounds are well consistent with the Monte Carlo results in the nonperturbative region. We observe interesting connections between the bounds generated by the bootstrap computations and the Griffiths' inequalities. We present results towards bootstrapping the string tension and glueball mass in Abelian lattice gauge theories.