3D Gravity and Gauge Theories

Abstract

I argue that the complete partition function of 3D quantum gravity is given by a path integral over gauge-inequivalent manifolds times the Chern-Simons partition function. In a discrete version, it gives a sum over simplicial complexes weighted with the Turaev-Viro invariant. Then, I discuss how this invariant can be included in the general framework of lattice gauge theory (qQCD₃). To make sense of it, one needs a quantum analog of the Peter-Weyl theorem and an invariant measure, which are introduced explicitly. The consideration here is limited to the simplest and most interesting case of SL _q (2), q = ${{e}^{{i\frac{{2\pi }}{{k + 2}}}}}$. At the end, I dwell on 3D generalizations of matrix models.