Matrix models, geometric engineering and elliptic genera

TLDR: In this article, the prepotential of = 2 supersymmetric gauge theories in four dimensions was derived by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T2.

Abstract: We compute the prepotential of = 2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T2. We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T2. Geometric engineering involves considering F-theory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R4. We study the compactifications of = 2* theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5-branes wrapped on T2 combines the Kahler and complex moduli of T2 and the mass parameter into the period matrix of a genus 2 curve.