Lattice Landau Gauge and Algebraic Geometry

TL;DR

This paper applies algebraic geometry techniques to efficiently find global minima of polynomial-like extremizing equations in physical systems, demonstrated through lattice Landau gauge fixing for compact U(1).

Contribution

It introduces algebraic geometry methods to solve gauge-fixing equations and finds all Gribov copies on small lattices, showcasing a novel approach in this context.

Findings

All Gribov copies for 1D and 2D lattices obtained.

Number of solutions for 3x3 systems can reach thousands.

Methods have potential applications beyond the studied examples.

Abstract

Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple 3x3 systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.