Super-A-polynomial

TL;DR

The paper introduces super-A-polynomials, a new two-parameter family of algebraic curves that generalize the A-polynomial, linking knot invariants, quantum invariants, and supersymmetric gauge theories.

Contribution

It constructs super-A-polynomials and their quantum versions, connecting knot theory, quantum invariants, and supersymmetric gauge theories with new algebraic structures.

Findings

Super-A-polynomials encode asymptotics of colored HOMFLY homology.

Quantum super-A-polynomials provide recursion relations for homology.

Super-A-polynomials have a physical interpretation as SUSY vacua equations.

Abstract

We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encode, respectively, the t-deformation which leads to the "refined A-polynomial", and the Q-deformation which leads to the augmentation polynomial of knot contact homology. For a given knot, the super-A-polynomial encodes the asymptotics of the corresponding S^r-colored HOMFLY homology for large r, while the quantum super-A-polynomial provides recursion relations for such homology theories for each r. The super-A-polynomial also admits a simple physical interpretation as the defining equation for the space of SUSY vacua in a circle compactification of the effective 3d N=2 theory associated to a given knot (complement). We discuss properties of super-A-polynomials and illustrate them in many examples.