Chern-Simons theory on an arbitrary manifold via surgery

TL;DR

This paper derives a general formula for Chern-Simons theory observables on any closed 3-manifold using surgery techniques and Wilson loop expectations, with explicit calculations for SU(n).

Contribution

It introduces a universal method to compute Chern-Simons invariants on arbitrary manifolds via surgery and Wilson loops, expanding the applicability of the theory.

Findings

Derived a formula for observables on arbitrary manifolds

Implemented surgery and Kirby moves in the calculation process

Explicitly calculated for SU(n) gauge group

Abstract

A general formula for physical observables in Chern-Simons theory with an arbitrary compact Lie group $G$, on an arbitrary closed oriented three-dimensional manifold $\cM$ is derived in terms of vacuum expectation values of Wilson loops in ${\cal S}^3$. Surgery presentation of $\cM$ and the Kirby moves are implemented as the main ingredients of the approach. The case of $G={\rm SU}(n)$ is explicitly calculated.