The descendants of the 3d-index

TL;DR

This paper explores the algebraic structure and properties of the 3d-index in 3d-3d correspondence, proposing conjectures on its factorization, difference equations, and asymptotic behavior, supported by computations on hyperbolic knots.

Contribution

It introduces conjectural structural properties of the 3d-index related to $q$-Weyl algebra modules, including factorization, difference equations, and asymptotics.

Findings

Conjectured bilinear factorization in terms of holomorphic blocks.

Proposed pair of linear $q$-difference equations for the 3d-index.

Determination of the 3d-index via a finite matrix of rational functions.

Abstract

In the study of 3d-3d correspondence occurs a natural $q$-Weyl algebra associated to an ideal triangulation of a 3-manifold with torus boundary components, and a module of it. We study the action of this module on the (rotated) 3d-index of Dimofte-Gaiotto-Gukov and we conjecture some structural properties: bilinear factorization in terms of holomorphic blocks, pair of linear $q$-difference equations, the determination of the 3d-index in terms of a finite size matrix of rational functions and the asymptotic expansion of the $q$-series as $q$ tends to 1 to all orders. We illustrate our conjectures with computations for the case of the three simplest hyperbolic knots.