[Paper Review] Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots

Mina Aganagic, Cumrun Vafa|arXiv (Cornell University)|Apr 20, 2012

Black Holes and Theoretical Physics44 references65 citations

TL;DR

This paper introduces a Q-deformed A-polynomial $A_K(x,p;Q)$ that encodes the quantum-corrected moduli space of a Lagrangian brane $L_K$ associated with a knot $K$ in $S^3$ via large $N$ duality and mirror symmetry. Using a generalized SYZ conjecture, it shows that each knot defines a distinct mirror geometry $uv = A_K(e^x, e^p; Q)$, whose open topological string amplitudes compute HOMFLY and Khovanov-type knot invariants, implying the Q-deformed A-polynomial captures at least as much information as knot homologies.

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.

Motivation & Objective

• To resolve the difficulty in computing knot invariants for general knots using large $N$ duality by characterizing the Lagrangian brane $L_K$ after the large $N$ transition.
• To extend the SYZ conjecture to local non-compact Calabi-Yau geometries to construct a distinct mirror for each knot $K$.
• To show that the Q-deformed A-polynomial $A_K(x,p;Q)$ captures all HOMFLY and Khovanov-type invariants via open topological string amplitudes on the mirror geometry.
• To establish a physical realization of the generalized (quantum) volume conjecture for $N=2$ and relate the $Q \to 1$ limit to the classical A-polynomial.

Proposed method

• Use large $N$ duality to map $SU(N)$ Chern-Simons theory on $S^3$ to topological strings on the resolved conifold.
• Apply a generalized SYZ conjecture to associate a unique mirror Calabi-Yau $Y_K$ to each knot $K$, defined by $uv = A_K(e^x, e^p; Q)$.
• Construct the Q-deformed A-polynomial $A_K(x,p;Q)$ as the quantum-corrected moduli space of the Lagrangian brane $L_K$ probing the resolved conifold.
• Relate open topological string amplitudes on $Y_K$ to BPS degeneracies and knot homology invariants via Nekrasov deformation.
• Use perturbative computations in the refined B-model to recover HOMFLY and Khovanov invariants from the mirror geometry.
• Verify that in the $Q \to 1$ limit, $A_K(x,p;1)$ contains the classical A-polynomial of the knot as a factor.

Experimental results

Research questions

• RQ1 Can a unique mirror Calabi-Yau geometry be associated to each knot $K$ via large $N$ duality and generalized mirror symmetry?
• RQ2 Does the Q-deformed A-polynomial $A_K(x,p;Q)$ encode all HOMFLY and Khovanov-type knot invariants through open topological string amplitudes?
• RQ3 How does the $Q \to 1$ limit of $A_K(x,p;Q)$ relate to the classical A-polynomial of the knot?
• RQ4 Can the generalized (quantum) volume conjecture for $N=2$ be physically explained via this construction?
• RQ5 Is there a correspondence between knot contact homology and open Gromov-Witten theory via the Q-deformed A-polynomial?

Key findings

• The Q-deformed A-polynomial $A_K(x,p;Q)$ is derived as the quantum-corrected moduli space of the Lagrangian brane $L_K$ associated with knot $K$ in the large $N$ dual geometry.
• Each knot $K$ defines a distinct mirror Calabi-Yau $Y_K$ via $uv = A_K(e^x, e^p; Q)$, generalizing the SYZ conjecture to local Calabi-Yau geometries.
• Perturbative open topological string amplitudes on $Y_K$ compute BPS degeneracies and thus fully capture HOMFLY and Khovanov-type invariants.
• The $Q \to 1$ limit of $A_K(x,p;Q)$ contains the classical A-polynomial of the knot as a factor, establishing a direct link to classical knot invariants.
• For $N=2$, the construction provides a physical explanation of the generalized (quantum) volume conjecture via the Q-deformed A-polynomial.
• The Q-deformed A-polynomial is shown to be equivalent to the operator annihilating the knot partition function, consistent with open topological string theory and difference equations in knot theory.