Notes on holonomy matrices of hyperbolic 3-manifolds with cusps

TL;DR

This paper introduces a method to construct holonomy matrices for hyperbolic 3-manifolds with cusps, extending techniques from 2-manifolds, and applies it to ideal tetrahedra to analyze nontrivial holonomies and their partition functions.

Contribution

It presents a novel method for constructing holonomy matrices of hyperbolic 3-manifolds with nontrivial holonomies, expanding the analytical tools available for such geometries.

Findings

Successfully constructed nontrivial holonomies for ideal tetrahedra.

Derived the partition function for tetrahedra with nontrivial holonomies.

Extended the known methods from hyperbolic 2-manifolds to 3-manifolds.

Abstract

In this paper, we give a method to construct holonomy matrices of hyperbolic 3-manifolds by extending the known method of hyperbolic 2-manifolds. It enables us to consider hyperbolic 3-manifolds with nontrivial holonomies. We apply our method to an ideal tetrahedron and succeed in making the holonomies nontrivial. We also derive the partition function of the ideal tetrahedron with nontrivial holonomies by using the duality proposed by Dimofte, Gaiotto and Gukov.