Derived Representation Schemes and Nakajima Quiver Varieties

TL;DR

This paper develops a derived version of Nakajima quiver varieties using Koszul complexes, establishing conditions for vanishing higher homology and linking representation schemes to K-theoretic classes, with applications to instanton moduli spaces.

Contribution

It introduces a derived representation scheme for quivers, models it explicitly as a Koszul complex, and relates it to Nakajima varieties and integral formulas in physics.

Findings

Derived representation schemes have vanishing higher homology under flatness conditions.

Established a comparison theorem linking representation schemes to K-theory classes.

Extended the theory to include derived partial character schemes and equivariant derived functors.

Abstract

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on $S^4$. On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.