Perturbative and nonperturbative aspects of complex Chern-Simons Theory

TL;DR

This paper reviews complex Chern-Simons theory with gauge group SL(N,C), exploring its challenges, connections to 3d-3d correspondence, and applications to topology and number theory, emphasizing both perturbative and nonperturbative aspects.

Contribution

It provides an accessible overview of complex Chern-Simons theory, highlighting recent progress and applications related to the 3d-3d correspondence and supersymmetric localization.

Findings

Insights into defining complex Chern-Simons as a TQFT

Connections between 3d-3d correspondence and topology

Applications to number theory and 3-manifold invariants

Abstract

We present an elementary review of some aspects of Chern-Simons theory with complex gauge group SL(N,C). We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d-3d correspondence. The 3d-3d correspondence relates partition functions (and other aspects) of complex Chern-Simons theory on a 3-manifold M to supersymmetric partition functions (and other observables) in an associated 3d theory T[M]. Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d-3d correspondence.