3d analogs of Argyres-Douglas theories and knot homologies

TL;DR

The paper studies algebraic curves attached to 3d $\mathcal{N}=2$ theories with flavor symmetry, focusing on knot-labeled theories $T_K$ whose partition functions encode $S^r$-colored HOMFLY homologies through super-$A$-polynomials $A^{\text{super}}(x,y;a,t)$. It develops a framework to derive these curves from the twisted superpotential, computes explicit super-$A$-polynomials for a variety of knots (including $5_1$, $5_2$, $6_1$, $(2,2p+1)$ torus knots, and twist knots), and analyzes their singularity structure, including universal singularities and loci associated with reducible flat connections. The work also reveals a 3d analog of Argyres-Douglas phenomena, proposes connections to augmentation polynomials via $t=-1$ limits, and uncovers dual 3d $\mathcal{N}=2$ descriptions (theory A and theory B) for certain torus knots, enriching the interplay between knot homology, gauge dynamics, and string/branes realizations. Overall, the results provide a rich landscape of spectral curves, their quantization, and a framework for understanding singularities and dualities in 3d $\mathcal{N}=2$ theories through knot theory data.

Abstract

We study singularities of algebraic curves associated with 3d N=2 theories that have at least one global flavor symmetry. Of particular interest is a class of theories T_K labeled by knots, whose partition functions package Poincare polynomials of the S^r-colored HOMFLY homologies. We derive the defining equation, called the super-A-polynomial, for algebraic curves associated with many new examples of 3d N=2 theories T_K and study its singularity structure. In particular, we catalog general types of singularities that presumably exist for all knots and propose their physical interpretation. A computation of super-A-polynomials is based on a derivation of corresponding superpolynomials, which is interesting in its own right and relies solely on a structure of differentials in S^r-colored HOMFLY homologies.

Figures (14)

• Figure 1: The Newton polygon of the super-$A$-polynomial for the knot $5_1$. Red circles denote monomials of the super-$A$-polynomial, and smaller yellow crosses denote monomials of its $a=-t=1$ specialization. These conventions are the same as in FGS.
• Figure 2: Matrix form of the super-$A$-polynomial for the $5_1$ knot.
• Figure 3: The Newton polygon of the super-$A$-polynomial for the knot $5_2$. Red circles denote monomials of the super-$A$-polynomial, and smaller yellow crosses denote monomials of its $a=-t=1$ specialization. These conventions are the same as in FGS.
• Figure 4: Matrix form of the $Q$-deformed $A$-polynomial for the $5_2$ knot.
• Figure 5: The Newton polygon of the super-$A$-polynomial for the knot $6_1$. Red circles denote monomials of the super-$A$-polynomial, and smaller yellow crosses denote monomials of its $a=-t=1$ specialization. These conventions are the same as in FGS.