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

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 S 1 with the holonomy L 2 R up to the fourth order. For the torus knots, we find trivial the recursion relations on both sides.

A-polynomial, B-model, and quantization

“…(2.7) differs from that in [16]. As will be explained below, these differences are important for overcoming the obstacles in [16] and reproducing the "quantum" q-corrections in the quantization of the A-polynomial (2.1).…”

“…We plan to elucidate the relation between these two properties in the future work. 16 At first, this may seem a little surprising, because the quantization problem is about symplectic geometry and not about complex geometry of C. (Figuratively speaking, quantization aims to replace all classical objects in symplectic geometry by the corresponding quantum analogs.) However, our "phase space," be it C × C or C * × C * , is very special in a sense that it comes equipped with a whole CP 1 worth of complex and symplectic structures, so that each aspect of the geometry can be looked at in several different ways, depending on which complex or symplectic structure we choose.…”

“…In most applications, including matrix models, one can set κ = 0, which leads to the ordinary Bergman kernel given above. 20 Taking the common denominator of the two square roots, the dependence on branch points in numerator can be expressed in terms of symmetric functions of ai, which leads to the formula presented in [16]. Note that this expression, contrary to (2.58), is manifestly holomorphic in the elliptic modulus τ .…”

Formal multidimensional integrals, stuffed maps, and topological recursion

“…. ), and has appeared provably or experimentally in many problems of 2D enumerative geometry: the two hermitian matrix model [EO08] and the chain of hermitian matrices [CEO06], topological string theory and Gromov-Witten invariants [BKMP09, BEMS10, EMS09, MP12, NS11, EO12], integrable systems [BE09,BE10,BE11], intersection numbers on the moduli space of curves [EO07b,Eyn11b,Eyn11a], asymptotic of knot invariants [DFM11,BE12,BEM12], . .…”

Checks of integrality properties in topological strings

“…This has been achieved for the torus knots [101,102]. There are still difficulties implementing AMM/EO topological recursion for twist knots [103,104] but there is some evidence of applicability of AMM/EO recursion to the non-torus knot 4 1 [105].…”