Super-A-polynomial for knots and BPS states

TL;DR

This work defines the two-parameter super-A-polynomial $A_{super}(x,y;a,t)$ and its quantum partner as a unifying refinement of the A-polynomial, encompassing both $t$- and $Q$-deformations linked to knot homologies and augmentation polynomials. It puts forward conjectures relating large-color asymptotics of colored superpolynomials and their recursions to a single quantum curve, with a physical picture as SUSY vacua in a circle-compactified 3d $N=2$ theory and as open-string moduli in brane constructions. Through explicit case studies of the unknot, figure-eight, trefoil, and $(2,2p+1)$ torus knots, it derives colored superpolynomials, classical and quantum super-A-polynomials, and analyzes their specialization limits to refine and augmentation polynomials, while discussing quantizability conditions. The results connect knot homologies, refined open BPS invariants, and knot contact homology within a single framework, offering new computational tools and guiding principles for higher-rank generalizations and physical interpretations.

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.

Figures (8)

• Figure 1: Various specializations of the super-$A$-polynomial.
• Figure 2: Newton polygon for the super-$A$-polynomial of the unknot (left). Red circles denote monomials of the super-$A$-polynomial, and smaller yellow crosses denote monomials of its $a=-t=1$ specialization. In this example both Newton polygons look the same, so that positions of all circles and crosses overlap. The coefficients of the super-$A$-polynomial are also shown in the matrix on the right. The role of rows and columns is exchanged in these two presentations: a monomial $a_{i,j}x^i y^j$ corresponds to a circle (resp. cross) at position $(i,j)$ in the Newton polygon, while in the matrix on the right it is shown as the entry $a_{i,j}$ in the $(i+1)^{\text{th}}$ row and in the $(j+1)^{\text{th}}$ column. These conventions are the same as in FGS.
• Figure 3: Matrix form of the super-$A$-polynomial for the figure-eight knot. The conventions are the same as in the unknot example in figure \ref{['fig-unknotNewtonMatrix']}.
• Figure 4: Newton polygon of the super-$A$-polynomial for the figure-eight knot and its $a=-t=1$ limit. The conventions are the same as in figure \ref{['fig-unknotNewtonMatrix']}.
• Figure 5: Matrix form of the super-$A$-polynomial for the trefoil knot. The conventions are the same as in figure \ref{['fig-unknotNewtonMatrix']}.