Three-manifold invariant from functional integration

TL;DR

This paper defines and computes a path-integral for the abelian U(1) Chern-Simons theory on any closed 3-manifold, connecting it with the Reshetikhin-Turaev invariant.

Contribution

It provides a rigorous path-integral formulation of the abelian U(1) Chern-Simons invariant using Deligne-Beilinson formalism and compares it with known topological invariants.

Findings

Path-integral computation matches Reshetikhin-Turaev invariant

Uses Deligne-Beilinson formalism for gauge fields

Summation over inequivalent U(1) bundles

Abstract

We give a precise definition and produce a path-integral computation of the normalized partition function of the abelian U(1) Chern-Simons field theory defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson formalism, we sum over the inequivalent U(1) principal bundles over the manifold and, for each bundle, we integrate over the gauge orbits of the associated connection 1- forms. The result of the functional integration is compared with the abelian U(1) Reshetikhin-Turaev surgery invariant.