A discretized Chern-Simons gauge theory on arbitrary graphs

TL;DR

This paper develops a method to discretize abelian Chern-Simons gauge theory on arbitrary planar graphs embedded in 2D manifolds, preserving key topological properties and identifying conditions for consistent discretization.

Contribution

It introduces a novel discretization scheme for abelian Chern-Simons theory on arbitrary graphs, establishing necessary and sufficient conditions for the discretized theory's consistency.

Findings

Discretization preserves essential properties of Chern-Simons theory.

A one-to-one vertex-face correspondence is necessary and sufficient for consistent discretization.

Application demonstrated on a tetrahedral graph.

Abstract

In this paper, we show how to discretize the abelian Chern-Simons gauge theory on generic planar lattices/graphs (with or without translational symmetries) embedded in arbitrary 2D closed orientable manifolds. We find that, as long as a one-to-one correspondence between vertices and faces can be defined on the graph such that each face is paired up with a neighboring vertex (and vice versa), a discretized Chern-Simons theory can be constructed consistently. We further verify that all the essential properties of the Chern-Simons gauge theory are preserved in the discretized setup. In addition, we find that the existence of such a one-to-one correspondence is not only a sufficient condition for discretizing a Chern-Simons gauge theory but, for the discretized theory to be nonsingular and to preserve some key properties of the topological field theory, this correspondence is also a necessary one. A specific example will then be provided, in which we discretize the Chern-Simons gauge theory on a tetrahedron.