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

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.

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The Refined Topological Vertex
• Non-commutative Donaldson-Thomas theory and the conifold
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
• The refined BPS index from stable pair invariants
• Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex

Cited by in corpus (1)

• Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds