Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 088, 16 pages      arXiv:1210.5181      http://dx.doi.org/10.3842/SIGMA.2012.088
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

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

Balázs Szendrői
Mathematical Institute, University of Oxford, UK

Received June 12, 2012, in final form November 05, 2012; Published online November 17, 2012

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.

Key words: geometric engineering; Donaldson-Thomas theory; resolved conifold.

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