The volume conjecture, perturbative knot invariants, and recursion relations for topological strings

2010/10/21 by Robbert Dijkgraaf, Hiroyuki Fuji, Masahide Manabe · 4 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #hep-th

paper · pdf · doi:10.1016/j.nuclphysb.2011.03.014

published as Nucl.Phys.B849:166-211,2011 · 48 pages, 7 figures

arxiv created 2010/10/21 · openalex publication_date 2011/04/14 · arxiv updated 2011/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the relation between perturbative knot invariants and the free energies defined by topological string theory on the character variety of the knot. Such a correspondence between SL(2;C) Chern-Simons gauge theory and the topological open string theory was proposed earlier on the basis of the volume conjecture and AJ conjecture. In this paper we discuss this correspondence beyond the subleading order in the perturbative expansion on both sides. In the computation of the perturbative invariants for the hyperbolic 3-manifold, we adopt the state integral model for the hyperbolic knots, and the factorized AJ conjecture for the torus knots. On the other hand, we iteratively compute the free energies on the character variety using the Eynard-Orantin topological recursion relation. We check the correspondence for the figure eight knot complement and the once punctured torus bundle over S¹ with the holonomy L²R up to the fourth order. For the torus knots, we find trivial the recursion relations on both sides.

Citations

• Representations and the colored Jones polynomial of a torus knot
• Analytic Continuation Of Chern-Simons Theory
• Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
• Mahler's Measure and the Dilogarithm (II)
• Arithmeticity of Knot Complements
• 2 + 1 dimensional gravity as an exactly soluble system
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Proof of the volume conjecture for torus knots

Cited by

• Quantum Curve and the First Painlevé Equation
• On discrete surfaces: Enumerative geometry, matrix models and universality classes via topological recursion
• SL( 2,ℝ ) Chern-Simons, Liouville, and gauge theory on duality walls
• A short overview of the "Topological recursion"

Related

• GW/DT invariants and 5D BPS indices for strips from topological recursion
• On the volume and the Chern-Simons invariant for the hyperbolic alternating knot orbifolds
• Lecture Notes on Topological Field Theory
• Volume Conjecture, Regulator and SL₂(C)-Character Variety of a Knot
• All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory