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

Abstract

Classical ${\mathrm{SL}}_2(\mathbb{C})$-Chern-Simons theory assigns a $3$-manifold $M$ with representation $\rho :{\pi }_1(M)\to {\mathrm{SL}}_2(\mathbb{C})$ its complex volume $V(M,\rho )\in \mathbb{C}/2{\pi }^{2}i\mathbb{Z}$, with real part the volume and imaginary part the Chern-Simons invariant. The existing literature focuses on computing $V$ using a triangulation. In this paper we show how to compute $V(M,L,\rho )$ directly from a surgery diagram for $M$ a compact oriented $3$-manifold with torus boundary components, embedded cusps $L$, and representation $\rho :{\pi }_1(M\setminus L)\to {\mathrm{SL}}_2(\mathbb{C})$. When $M$ has nonempty boundary $V(M,L,\rho )(\mathfrak{s})$ 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 $\mathrm{SU}(2)$ Chern-Simons theory.

34 pages