A topological state sum model for a scalar field on the circle

TL;DR

This paper develops a triangulation-independent state sum model for a scalar field on a circle, matching the continuum partition function and relating to the harmonic oscillator.

Contribution

It introduces a novel topological state sum model for scalar fields that reproduces continuum results and is independent of triangulation.

Findings

Model exactly reproduces the continuum partition function

Model is triangulation-independent

On the circle, equivalent to the harmonic oscillator path integral

Abstract

This paper is a follow-up to a previous paper on fermions. A simple state sum model for a scalar field on a triangulated 1-manifold is constructed. The model is independent of the triangulation and gives exactly the same partition function as the continuum functional integral with zeta function regularisation. For a certain choice of gauge group, the state sum model on the circle is equivalent to the path integral for the simple harmonic oscillator.