# Topology and index theorem with a generalized Villain lattice action -- a test in 2d

## TL;DR

This paper investigates two definitions of topological charge in 2D U(1) lattice gauge theory using a generalized Villain action, demonstrating the index theorem's rapid convergence to exactness in the continuum limit.

## Contribution

It introduces and compares two topological charge definitions, showing their effectiveness and the rapid approach to the index theorem's validity in the continuum limit.

## Key findings

- Both topological charge definitions quickly satisfy the index theorem near the continuum limit.
- One definition preserves charge conjugation symmetry exactly, aiding physics studies.
- Numerical results confirm the theoretical expectations for the generalized Villain action.

## Abstract

Using 2-d U(1) lattice gauge theory we study two definitions of the topological charge constructed from a generalized Villain action and analyze the implementation of the index theorem based on the overlap Dirac operator. One of the two definitions expresses the topological charge as a sum of the Villain variables and treats charge conjugation symmetry exactly, making it particularly useful for studying related physics. Our numerical analysis establishes that for both topological charge definitions the index theorem becomes exact quickly towards the continuum limit.