Topological gauge actions on the lattice as Overlap fermion determinants

TL;DR

This paper reviews how Overlap fermion determinants on the lattice can be used to construct and reproduce continuum topological gauge actions, like Chern-Simons, in a gauge-invariant way, verified through numerical tests.

Contribution

It introduces a method to derive topological gauge actions from Overlap fermion determinants on the lattice, bridging lattice regularization with continuum topological field theories.

Findings

Successfully reproduces continuum topological actions on the lattice.

Demonstrates gauge invariance and large gauge transformation properties.

Provides numerical evidence for the formalism's effectiveness.

Abstract

Overlap fermion on the lattice has been shown to properly reproduce topological aspects of gauge fields. In this paper, we review the derivation of Overlap fermion formalism in a torus of three space-time dimensions. Using the formalism, we show how to use the Overlap fermion determinants in the massless and infinite mass limits to construct different continuum topological gauge actions, such as the level-$k$ Chern-Simons action, ``half-CS" term and the mixed Chern-Simons (BF) coupling, in a gauge-invariant lattice UV regulated manner. Taking special Abelian and non-Abelian background fields, we demonstrate numerically how the lattice formalism beautifully reproduces the continuum expectations, such as the flow of action under large gauge transformations.