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

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 Teichmüller 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.

47 pages, 28 figures

Cited by in corpus (10)

• Aspects of Defects in 3d-3d Correspondence
• Holography of 3d-3d correspondence at Large N
• The complex volume of SL(n,C)-representations of 3-manifolds
• Floer homology, group orderability, and taut foliations of hyperbolic 3-manifolds
• Taming Supersymmetric Defects in 3d-3d Correspondence
• Triangulation Independent Ptolemy Varieties
• Octahedral developing of knot complement II: Ptolemy coordinates and applications
• Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)
• Verified computations for closed hyperbolic 3-manifolds
• The (twisted/$L^2$)-Alexander polynomial of ideally triangulated 3-manifolds