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.The hyperbolic three-manifold M is given bywhere Γ is a torsion-free and discrete subgroup of P SL(2; C) with the action (2.2).The volume of an ideal tetrahedron T αβγ is computed directly by using the metric (2.1) [33].The function Λ(θ) is called the Lobachevsky function. The volume formula can also be rewritten in terms of the Bloch-Wigner function D(z)(2.9)The gluing condition along each edge imposes constraints on the face angles of the ideal tetrahedra.(2.10)Furthermore, we should also consider the gluing condition for the boundary of the knot complement. In order to realize the torus as a boundary, the face angles must satisfy the completeness conditions along the meridian and the longitude.Mostow's rigidity theorem [34] implies that these conditions are solved uniquely with Im z i > 0 for all i. By summing all volumes for the ideal tetrahedra, we are able to determine the hyperbolic volume uniquely for each hyperbolic three-manifold.

A-polynomial, B-model, and quantization

“…Indeed, applying our general result (3.20) to this particular model we immediately obtain 11) which is also consistent with (7.5). As we pointed out earlier, however, this result is based only on the elementary computation of the annulus amplitude S 1 , and now we wish to verify that computing S n and A n to higher order does not lead to any modifications and merely confirms the result (7.11).…”

“…The Bergman kernel B(p, q) is defined as the unique meromorphic differential with exactly one pole, which is a double pole at p = q with no residue, and with vanishing integral over A I -cycles A I B(p, q) = 0 (in a canonical basis of cycles (A I , B I ) for C). Thus, for curves of genus zero the Bergman kernel takes a particularly simple form 11) and its form for curves of higher genus is presented in section 2.5. A closely related quantity is a 1-form, defined in a neighborhood of a branch point…”

Topological strings, D-model, and knot contact homology

“…The graph Γ K for the (3,3) torus link is shown in Figure 25, and the intersection dimensions are in accord with Conjecture 5.3.…”

“…As we vary x, we can probe all of the Riemann surface and go to a region where this is no longer the case. 3 We cannot expect to specify holonomies around both cycles of Λ K simultaneously, as the theory on the Lagrangian A-brane in a Calabi-Yau three-fold is Chern-Simons theory, and in Chern-Simons theory on a manifold with a T 2 boundary holonomies of the 1-cycles generating H 1 (T 2 ) are canonically conjugate.…”

Three-dimensional extensions of the Alday-Gaiotto-Tachikawa relation

“…dw w e 2xz sinh w sinh τ w (54) In the previously discussed context of CFT one can encounter a similar function though with rescaled parameters…”