Abelian Chern-Simons Gauge Theory on The Lattice

TL;DR

This paper constructs an Abelian Chern-Simons gauge theory on a three-dimensional lattice, introducing a dual lattice approach that simplifies calculations and aligns with continuous topological field theory results.

Contribution

It presents a novel lattice formulation of Abelian Chern-Simons theory that avoids common issues and facilitates straightforward computation of Wilson loop expectations.

Findings

Lattice action is simple and symmetric across spacetime dimensions.

Wilson loop expectation values match continuous topological field theory.

Dual lattice approach resolves forward/backward difference and duplication problems.

Abstract

The Abelian Chern-Simons gauge theory is constructed on the three-dimensional spacetime lattice. This proposal introduces both lattice and dual lattice, and the gauge field on the dual lattice is expressed in terms of the gauge field on the original lattice. This treatment circumvents the issue of forward/backward difference, which is the common problem that many previous proposals have, and also avoids the duplication problem, which prevents people from introducing the dual lattice. The form of the lattice action is very simple, and is symmetric with respect to the three spacetime dimensions. These features make it straightforward to calculate the expectation values of Wilson loops, and the results agree with the topological field theory in continuous spacetime. Generalizations to multiple types of lattices are also discussed.