Perturbative and nonperturbative aspects of complex Chern–Simons theory**This is a contribution to the review issue ‘Localization techniques in quantum field theories’ (ed Pestun and Zabzine) which contains 17 chapters available at [

Abstract

We present an elementary review of some aspects of Chern–Simons theory with complex gauge group . 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 . 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.