Chern-Simons approach to three-manifold invariants

TL;DR

This paper introduces a new formal approach to three-manifold invariants using non-perturbative Chern-Simons theory, providing a direct surgery implementation and explicit results for SU(2) that align with known outcomes.

Contribution

It presents a novel non-combinatorial, quantum field theory-based method for computing three-manifold invariants, extending the framework to arbitrary compact simple Lie groups.

Findings

Explicit form of SU(2) invariant derived

Invariant agrees with Lickorish's results

Proposes a direct surgery implementation in quantum field theory

Abstract

% A new, formal, non-combinatorial approach to invariants of % three-dimensional manifolds of Reshetikhin, Turaev and % Witten in the framework of non-perturbative topological % quantum Chern-Simons theory, corresponding to an arbitrary % compact simple Lie group, is presented. A direct % implementation of surgery instructions in the context of % quantum field theory is proposed. An explicit form of the % specialization of the invariant to the group SU(2) is % derived, and shown to agree with the result of Lickorish. % % AMS subject classifications (1991): 57M25, 57N10, 57R65, 81T13.