Topology in 2D non-Abelian Lattice Gauge Theories

TL;DR

This paper investigates the topological structure of 2D non-Abelian lattice gauge theories, focusing on $U(N_c)$ theories, their topological sectors, and the challenges of topological freezing, using gradient flow to find instanton-like solutions.

Contribution

It provides a detailed analysis of topological sectors in 2D $U(N_c)$ gauge theories and derives instanton-like solutions, especially for $N_c=2$, addressing topological freezing issues.

Findings

Identification of topological sectors in 2D $U(N_c)$ theories.

Derivation of instanton-like solutions using gradient flow.

Insights into topological freezing at small lattice spacings.

Abstract

In two dimensions, $U(N_c)$ gauge theories exhibit a non-trivial topological structure, while $SU(N_c)$ theories are topologically trivial. Hence, for $G = U(N_c)$ the phase space is divided into topological sectors, characterized by a topological index (a.k.a. ``topological charge''). These sectors are separated by action barriers, which diverge if the lattice spacing is taken small, resulting in an algorithmic problem known as ``topological freezing''. We study these theories in various box sizes and at various couplings. With the help of gradient flow we derive instanton-like solutions for 2D $U(N_c)$ theory with a specific focus on the case of $N_c = 2$.