3d analogs of Argyres-Douglas theories and knot homologies

TL;DR

This paper explores the algebraic curves linked to 3d N=2 theories associated with knots, deriving super-A-polynomials and analyzing their singularities to connect knot homologies with physical theories.

Contribution

It introduces the super-A-polynomial for new 3d N=2 theories T_K and studies their singularity structures, linking knot invariants with algebraic geometry and physics.

Findings

Derived super-A-polynomials for multiple knot-related theories

Cataloged singularity types and proposed physical interpretations

Connected superpolynomials with differential structures in HOMFLY homologies

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.