Instantons and Curve Counting
Seminar Room 1, Newton Institute
AbstractWe review how the counting of BPS states of D-branes in Type IIA string theory can be captured by enumerating the instanton solutions in a six-dimensional noncommutative N=2 gauge theory; mathematically, the instanton contributions are related to Gromov-Witten and Donaldson-Thomas curve counting invariants, which in this context have natural higher-rank generalisations in the Coulomb branch of the gauge theory. We discuss some of the integrability properties of the resulting partition functions, including a novel reformulation as an infinite rank unitary matrix model. We define a stacky version of the gauge theory whose instanton solutions compute noncommutative Donaldson-Thomas invariants for abelian orbifold singularities. We also discuss how N=4 supersymmetric Yang-Mills theory in four dimensions is analogously related to curve counting on toric surfaces
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