Localization in Abelian Chern-Simons Theory

TL;DR

This paper rigorously defines and relates abelian Chern-Simons partition functions using gauge fixing and localization, connecting them to topological invariants like Reidemeister torsion on Sasakian three-manifolds.

Contribution

It provides a rigorous formulation of abelian Chern-Simons partition functions and links them to topological invariants via localization techniques.

Findings

Derived a rigorous abelian Chern-Simons partition function.

Connected the partition function to Reidemeister-Ray-Singer torsion.

Explicitly computed torsion using Seifert data.

Abstract

Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition function is also derived using the technique of non-abelian localization. This physically identifies the symplectic abelian partition function with the abelian Chern-Simons partition function as rigorous topological three-manifold invariants. This study leads to a natural identification of the abelian Reidemeister-Ray-Singer torsion as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections for the class of Sasakian three-manifolds. The torsion part of the abelian Chern-Simons partition function is computed explicitly in terms of Seifert data for a given Sasakian three-manifold.