Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots

TL;DR

This paper explores the large N duality and mirror symmetry in topological string theory, introducing a Q-deformed A-polynomial for knots that encodes knot invariants and relates to the classical A-polynomial.

Contribution

It constructs a quantum A-polynomial for knots from topological string theory, linking knot invariants, mirror symmetry, and large N duality in a novel way.

Findings

Introduces a polynomial A_K(x,p;Q) encoding knot invariants.

Shows A_K's relation to the classical A-polynomial at Q=1.

Provides a physical explanation for the generalized volume conjecture.

Abstract

We reconsider topological string realization of SU(N) Chern-Simons theory on S^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) in two variables x,p depending on the t'Hooft coupling parameter Q=e^{Ng_s}. Its vanishing locus is the quantum corrected moduli space of a special Lagrangian brane L_K, associated to K, probing the large N dual geometry, the resolved conifold. Using a generalized SYZ conjecture this leads to the statement that for every such Lagrangian brane L_K we get a distinct mirror of the resolved conifold given by uv=A_K(x,p;Q). Perturbative corrections of the refined B-model for the open string sector on the mirror geometry capture BPS degeneracies and thus the knot homology invariants. Thus, in terms of its ability to distinguish knots, the classical function A_K(x,p;Q) contains at least as much information as knot homologies. In the special case when N=2, our observations lead to a physical explanation of the generalized (quantum) volume conjecture. Moreover, the specialization to Q=1 of A_K contains the classical A-polynomial of the knot as a factor.