Super-A-polynomial for knots and BPS states

TL;DR

This paper introduces a two-parameter deformation of the A-polynomial called the super-A-polynomial, which encodes knot invariants, unifies various deformations, and has applications in physics and knot theory, including new formulas for specific knots.

Contribution

The authors define and compute the super-A-polynomial and its quantum version, unifying several known deformations and providing new formulas for colored superpolynomials of knots.

Findings

Defined the super-A-polynomial as a two-parameter deformation of the A-polynomial.

Computed the quantum super-A-polynomial encoding recursion relations for HOMFLY homology.

Derived new formulas for colored superpolynomials of the trefoil and figure-eight knots.

Abstract

We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the "refined A-polynomial" introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for S^r-colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d N=2 theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot.