Mini course: Triangulations, gluing equations, and simplicial Chern-Simons theory (3/3)

Speaker: Christian Krogager Zickert (University of Maryland)

2012.09.20 | Christine Dilling

Date Tue 21 Aug
Time 14:15 15:15
Location Aud. D3 (1531-215)

Abstract

The purpose of this mini course is to study the set of representations of a 3-manifold group into a Lie group G, i.e. the set of flat G-connections. We give a concrete parametrization for the groups SL(n,C) and PGL(n,C), and use it to compute invariants such as the volume and Chern-Simons invariant, as well as certain quantum invariants. The parametrizations are inspired by coordinates on higher Teichmuller spaces due to Fock and Goncharov. In the first lecture we define the Chern-Simons invariant of a flat connection and discuss simplicial formulas for computing it using group homology and the dilogarithm. In the second lecture we discuss shape coordinates and Thurston's gluing equations for PGL(2,C)-representations, and the Ptolemy coordinates of Garoufalidis-Thurston-Zickert for SL(2,C)-representations. We also discuss symplectic properties of the gluing equations and some applications in quantum topology. In the third lecture we generalize the gluing equations to PGL(n,C)-representations and the Ptolemy coordinates to SL(n,C)-representations, and discuss the relationship to the X- and A-coordinates on higher Teichmuller spaces due to Fock and Goncharov.

Seminar