On the asymptotics of the meromorphic 3D-index (2109.05355v2)

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 additionally on a quantisation parameter q. In this paper, we study the asymptotics of this invariant as q approaches 1, developing a conjectural asymptotic approximation in the form of a sum of contributions associated to certain flat PSL(2,C)-bundles on the manifold. 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 a certain connected component of the space of circle-valued angle structures. To explain the topological significance of this component, we develop a theory connecting circle-valued angle structures to the obstruction theory for lifting a (boundary-parabolic) PSL(2,C)-representation of the fundamental group to a (boundary-unipotent) SL(2,C)-representation. We present strong theoretical and numerical evidence for the validity of our asymptotic approximations.