Modules over plane curve singularities in any ranks and DAHA

TL;DR

This paper extends the construction of geometric superpolynomials for unibranch plane curve singularities from rank one to arbitrary ranks, introducing higher rank Jacobian factors and connecting to DAHA superpolynomials of algebraic knots.

Contribution

It introduces a higher rank generalization of geometric superpolynomials and defines new Jacobian factor counterparts related to vector bundles over algebraic curves.

Findings

Defined higher rank Jacobian factors related to compactified Jacobians.

Connected geometric superpolynomials with DAHA superpolynomials for algebraic knots.

Numerical evidence supports the conjectured relationship.

Abstract

We generalize the construction of geometric superpolynomials for unibranch plane curve singularities from our prior paper from rank one to any ranks. The new feature is the definition of counterparts of Jacobian factors (directly related to compactified Jacobians) for higher ranks, which is parallel to the classical passage from invertible bundles to vector bundles over algebraic curves. This is an entirely local theory, connected with affine Springer fibers for non-reduced (germs of) spectral curves. We conjecture and justify numerically the connection of our geometric polynomials in arbitrary ranks with the corresponding DAHA superpolynomials of algebraic knots colored by columns.