The Volume Conjecture and Topological Strings

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.

52 pages, 10 figures, references added, typos corrected, comments included

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Cited by in corpus (12)

• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Super-A-polynomial for knots and BPS states
• A-polynomial, B-model, and Quantization
• The matrix model version of AGT conjecture and CIV-DV prepotential
• Semiclassical Analysis of the 3d/3d Relation
• On 3d extensions of AGT relation
• Knots, BPS states, and algebraic curves
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• Quantum curves and conformal field theory
• Challenges of beta-deformation
• Loop equations and topological recursion for the arbitrary-$\beta$ two-matrix model
• Wronskians, dualities and FZZT-Cardy branes