The Volume Conjecture and Topological Strings

TL;DR

This paper explores the deep connection between knot invariants, topological string theory, and the volume conjecture, revealing a shared Hamiltonian structure and interpreting the AJ conjecture through D-module theory, supported by explicit computations.

Contribution

It uncovers a Hamiltonian structure linking Jones-Witten theory and topological strings, and interprets the AJ conjecture as a D-module structure, verified through free energy calculations.

Findings

Hamiltonian structures are similar in both theories.

AJ conjecture can be viewed as a D-module structure.

Computed free energy matches Reidemeister torsion for specific knots.

Abstract

In this paper, we discuss a relation between Jones-Witten theory of knot invariants and topological open string theory on the basis of the volume conjecture. We find a similar Hamiltonian structure for both theories, and interpret the AJ conjecture as the D-module structure for a D-brane partition function. In order to verify our claim, we compute the free energy for the annulus contributions in the topological string using the Chern-Simons matrix model, and find that it coincides with the Reidemeister torsion in the case of the figure-eight knot complement and the SnapPea census manifold m009.