Gluing equations for PGL(n,C)-representations of 3-manifolds

TL;DR

This paper introduces a new parametrization of PGL(n,C) representations of 3-manifolds using shape coordinates that generalize Thurston's gluing equations, with applications in quantum topology.

Contribution

It develops shape coordinates for PGL(n,C) representations that satisfy generalized gluing equations, extending previous Ptolemy coordinate parametrizations.

Findings

Shape coordinates satisfy Neumann-Zagier type equations.

Coordinates exhibit symplectic relations relevant to quantum topology.

Duality between Ptolemy and shape coordinates is established.

Abstract

In a previous paper, we parametrized boundary-unipotent representations of a 3-manifold group into SL(n,C) using Ptolemy coordinates, which were inspired by A-coordinates on higher Teichm\"uller space due to Fock and Goncharov. In this paper, we parametrize representations into PGL(n,C) using shape coordinates which are a 3-dimensional analogue of Fock and Goncharov's X-coordinates. These coordinates satisfy equations generalizing Thurston's gluing equations. These equations are of Neumann-Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.