Nekrasov's Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case

TL;DR

This paper explores the connection between Nekrasov's partition function and refined Donaldson-Thomas theory in the rank one case, revealing geometric and algebraic structures on the affine plane and conifold.

Contribution

It identifies the vector space underlying refined Donaldson-Thomas theory as an exterior space of polynomial functions, linking it to the cohomological Hall algebra of the conifold.

Findings

Identification of the vector space as an exterior space of polynomial functions

Duality between SL(2)-actions on the threefold and affine plane

Proposal that the exterior space is a module for the cohomological Hall algebra

Abstract

This paper studies geometric engineering, in the simplest possible case of rank one (Abelian) gauge theory on the affine plane and the resolved conifold. We recall the identification between Nekrasov's partition function and a version of refined Donaldson-Thomas theory, and study the relationship between the underlying vector spaces. Using a purity result, we identify the vector space underlying refined Donaldson-Thomas theory on the conifold geometry as the exterior space of the space of polynomial functions on the affine plane, with the (Lefschetz) SL(2)-action on the threefold side being dual to the geometric SL(2)-action on the affine plane. We suggest that the exterior space should be a module for the (explicitly not yet known) cohomological Hall algebra (algebra of BPS states) of the conifold.