Finite-matrix formulation of gauge theories on a non-commutative torus with twisted boundary conditions

TL;DR

This paper introduces a new finite-matrix approach to gauge theories on a non-commutative torus that incorporates twisted boundary conditions and topological classification, facilitating non-commutative geometry applications.

Contribution

It develops an algebraic finite-matrix formulation for non-commutative gauge theories with twisted boundary conditions, enabling Morita equivalence and topological sector classification.

Findings

Finite-matrix formulation successfully incorporates twisted boundary conditions.

Ginsparg-Wilson Dirac operator defines an index for topological sectors.

Monte Carlo calculations validate the index as a topological invariant.

Abstract

We present a novel finite-matrix formulation of gauge theories on a non-commutative torus. Unlike the previous formulation based on a map from a square matrix to a field on a discretized torus with periodic boundary conditions, our formulation is based on the algebraic characterization of the configuration space. This enables us to describe the twisted boundary conditions in terms of finite matrices and hence to realize the Morita equivalence at a fully regularized level. Matter fields in the fundamental representation turn out to be represented by rectangular matrices for twisted boundary conditions analogously to the matrix spherical harmonics on the fuzzy sphere with the monopole background. The corresponding Ginsparg-Wilson Dirac operator defines an index, which can be used to classify gauge field configurations into topological sectors. We also perform Monte Carlo calculations for the index as a consistency check. Our formulation is expected to be useful for applications of non-commutative geometry to various problems related to topological aspects of field theories and string theories.