Surgery calculus for classical $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons theory

Abstract

Classical $\operatorname{SL}_2(\mathbb{C})$-Chern-Simons theory assigns a $3$-manifold $M$ with representation $\rho : \pi_1(M) \to \operatorname{SL}_2(\mathbb{C})$ its complex volume $\operatorname{V}(M, \rho) \in \mathbb{C} / 2 \pi^2 i \mathbb{Z}$, with real part the volume and imaginary part the Chern-Simons invariant. The existing literature focuses on computing $\operatorname{V}$ using a triangulation. In this paper we show how to compute $\operatorname{V}(M, L, \rho)$ directly from a surgery diagram for $M$ a compact oriented $3$-manifold with torus boundary components, embedded cusps $L$, and representation $\rho : \pi_1(M \setminus L) \to \operatorname{SL}_2(\mathbb{C})$. When $M$ has nonempty boundary $\operatorname{V}(M, L, \rho)(\mathfrak{s})$ depends on some extra data $\mathfrak{s}$ we call a log-decoration. Our method describes $\rho$ in a coordinate system closely related to quantum groups, and we think of our construction as a classical, noncompact version of Witten-Reshetikhin-Turaev's quantum $\operatorname{SU}(2)$ Chern-Simons theory.