$K$-theory of moduli spaces of sheaves and large Grassmannians

TL;DR

This paper classifies equivariant K-theoretic pushforwards on moduli spaces of sheaves on P^2, linking them to Grassmannian formulas, with implications for gauge theory and vertex operator calculus.

Contribution

It provides a new classification theorem for K-theoretic pushforwards on moduli spaces of sheaves, connecting them to Grassmannian coefficients and extending previous work in gauge theory.

Findings

Derived a formula for coefficients on Grassmannians

Classified K-theoretic pushforwards on sheaf moduli spaces

Linked moduli space K-theory to Grassmannian geometry

Abstract

We prove a theorem classifying the equivariant $K$-theoretic pushforwards of the product of arbitrary Schur functors applied to the tautological bundle on the moduli space of framed rank $r$ torsion-free sheaves on $\mathbb{P}^2$, and its dual. This is done by deriving a formula for similar coefficients on Grassmannian varieties, and by thinking of the moduli space as a class in the $K$-theory of the Grassmannian, in analogy with the construction of the Hilbert scheme when the rank is one. Our motivations stem from some vertex operator calculus studied recently by Nekrasov, Okounkov, and the author when the rank is one, with applications to four-dimensional gauge theory.