On the asymptotics of the meromorphic 3D-index

TL;DR

This paper investigates the asymptotic behavior of the meromorphic 3D-index invariant of 3-manifolds as the quantum parameter approaches 1, revealing connections to hyperbolic geometry, angle structures, and introducing a new topological invariant.

Contribution

It develops a conjectural asymptotic formula for the 3D-index, introduces the beta invariant, and links the invariant's asymptotics to angle structures and volume optimization.

Findings

Asymptotic approximation involves conjugacy classes of boundary PSL(2,C) representations.

Introduction of the beta invariant as a new topological invariant.

Stationary phase analysis connects the invariant to angle structures and volume.

Abstract

In their recent work, Garoufalidis and Kashaev extended the 3D-index of an ideally triangulated 3-manifold with toroidal boundary to a well-defined topological invariant which takes the form of a meromorphic function of 2 complex variables per boundary component and which depends in addition on a quantisation parameter q. In this paper, we study asymptotics of this invariant as q approaches 1 and develop a conjectural asymptotic approximation in the form of a sum of contributions associated to conjugacy classes of certain boundary parabolic PSL(2,C) representations of the fundamental group. Furthermore, we study the coefficients appearing in these contributions, which include the hyperbolic volume, the '1-loop invariant' of Dimofte and Garoufalidis, as well as a new topological invariant of 3-manifolds with torus boundary, which we call the 'beta invariant'. The technical heart of our analysis is the expression of the state-integral of the Garoufalidis--Kashaev invariant as an integral over one connected component of the space of circle-valued angle structures introduced by Luo. Our stationary phase analysis of the asymptotics of this integral reveals many connections to the theory of angle structures and volume optimization. This investigation was motivated by extensive numerical experiments. In addition, we prove a variety of theorems about the quantities appearing in the analysis which support the overall conjectural picture.