Matrix Models, Geometric Engineering and Elliptic Genera

TL;DR

This paper computes the prepotential of 4D N=2 supersymmetric gauge theories from 6D theories compactified on T^2 using three methods: matrix models, geometric engineering, and instanton calculus, establishing their equivalence.

Contribution

It demonstrates the equivalence of matrix models, geometric engineering, and instanton calculus for computing prepotentials in these theories and extends geometric engineering to include massive adjoint fields.

Findings

Established equivalence of three computational approaches.

Engineered theories with massive adjoint fields.

Connected moduli space of M5-branes to genus 2 curve periods.

Abstract

We compute the prepotential of N=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 T^2. 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 T^2. 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 R^4. We study the compactifications of N=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 T^2 combines the Kahler and complex moduli of T^2 and the mass parameter into the period matrix of a genus 2 curve.