The Volume Conjecture and Topological Strings

TL;DR

This work links the volume conjecture and AJ conjecture for hyperbolic knot complements to open topological string theory on a mirror Calabi–Yau, positing a shared Hamiltonian structure and a D-module interpretation of the AJ constraint. It identifies the colored Jones polynomial with the open-string partition function and predicts that subleading terms correspond to Reidemeister torsion, realized explicitly as annulus free energies in a Chern–Simons matrix-model framework. The authors verify the proposal for the figure-eight knot complement and the SnapPea census manifold $m009$ by showing that the annulus free energy matches the Reidemeister torsion, thereby linking quantum knot invariants, hyperbolic geometry, and topological string theory through concrete computational evidence. The results illuminate a deeper geometric-engineering perspective on knot invariants and suggest a broader role for ${\cal D}$-modules and spectral-curve methods in quantum topology.

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.

Figures (10)

• Figure 1: Knot complement
• Figure 2: Ideal tetrahedron
• Figure 3: Gluing condition along each edge and completeness condition
• Figure 4: Figure-eight knot complement as two ideal tetrahedra.
• Figure 5: Profile of $\ell_+(e^{i\theta})$ and the integration path.