Perturbative and nonperturbative aspects of complex Chern–Simons theory

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.