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

Understand

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.