The 3d-index of the 3d-skein module via the quantum trace map

TL;DR

This paper introduces a new map from the skein module of cusped hyperbolic 3-manifolds to Laurent series, linking the 3d-index with quantum topology and physics predictions.

Contribution

It constructs a map that connects the 3d-index to the skein module via the quantum trace, fulfilling supersymmetry predictions and contributing to TQFT conjectures.

Findings

The map evaluates to the 3d-index at peripheral curves.

It factors through the 3d-quantum trace map.

Supports the conjectural 3+1D TQFT framework.

Abstract

We define a map from the skein module of a cusped hyperbolic 3-manifold to the ring of Laurent series in one variable with integer coefficients that satisfies two properties: its evaluation at peripheral curves coincides with the Dimofte--Gaiotto--Gukov 3d-index, and it factors through the 3d-quantum trace map associated to a suitable ideal triangulation of the manifold. The map fulfills a supersymmetry prediction of mathematical physics and is part of a conjectural 3+1 dimensional topological quantum field theory.