Instantons and Donaldson-Thomas Invariants

TL;DR

This paper explores the connection between six-dimensional topological Yang-Mills theory and Donaldson-Thomas invariants, using localization techniques to evaluate partition functions and generalize instanton moduli spaces to Calabi-Yau threefolds.

Contribution

It introduces a higher-dimensional generalization of the ADHM formalism and applies equivariant localization to compute partition functions in this context.

Findings

Partition function evaluated for U(N) theory on flat space.

Establishment of a link between gauge theory and enumerative geometry.

Extension of formalism to generic toric Calabi-Yau manifolds.

Abstract

We review some recent progress in understanding the relation between a six dimensional topological Yang-Mills theory and the enumerative geometry of Calabi-Yau threefolds. The gauge theory localizes on generalized instanton solutions and is conjecturally equivalent to Donaldson-Thomas theory. We evaluate the partition function of the U(N) theory in its Coulomb branch on flat space by employing equivariant localization techniques on its noncommutative deformation. Geometrically this corresponds to a higher dimensional generalization of the ADHM formalism. This formalism can be extended to a generic toric Calabi-Yau.