Refined Chern-Simons theory and Hilbert schemes of points on the plane

TL;DR

This paper refines Chern-Simons theory for symmetric links, connecting its large N limit to Hilbert schemes of points on the plane, and provides explicit formulas for Euler characteristics of universal sheaves.

Contribution

It introduces a refined version of Chern-Simons theory and links its large N limit to geometric structures like Hilbert schemes, offering new explicit formulas.

Findings

Large N limit of the S matrix relates to Hilbert schemes of points on the plane.

Derived explicit formulas for Euler characteristics of universal sheaves.

Connected deformation of S and T matrices to geometric invariants.

Abstract

Aganagic and Shakirov propose a refinement of the SU(N) Chern-Simons theory for links in three manifolds with S^1-symmetry, such as torus knots in S^3, based on deformation of the S and T matrices, originally found by Kirillov and Cherednik. We relate the large N limit of the S matrix to the Hilbert schemes of points on the affine plane. As an application, we find an explicit formula for the Euler characteristics of the universal sheaf, applied arbitrary Schur functor.