Three-Manifold Invariants from Chern-Simons Field Theory with Arbitrary Semi-Simple Gauge Groups

TL;DR

This paper extends the construction of three-manifold invariants from Chern-Simons theory to arbitrary semi-simple gauge groups, generalizing previous SU(2) results and providing computational methods.

Contribution

It introduces a new class of three-manifold invariants derived from Chern-Simons theory with arbitrary semi-simple gauge groups, generalizing earlier SU(2) based invariants.

Findings

Constructed invariants are unchanged under Kirby moves.

Provided explicit computations for specific three-manifolds.

Related invariants to the Chern-Simons partition function.

Abstract

Invariants for framed links in $S^3$ obtained from Chern-Simons gauge field theory based on an arbitrary gauge group (semi-simple) have been used to construct a three-manifold invariant. This is a generalization of a similar construction developed earlier for SU(2) Chern-Simons theory. The procedure exploits a theorem of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate three-manifolds to surgeries on framed unoriented links. The invariant is an appropriate linear combination of framed link invariants which does not change under Kirby calculus. This combination does not see the relative orientation of the component knots. The invariant is related to the partition function of Chern-Simons theory. This thus provides an efficient method of evaluating the partition function for these field theories. As some examples, explicit computations of these manifold invariants for a few three-manifolds have been done.