M-strings, Elliptic Genera and N=4 String Amplitudes

Understand

We study mass-deformed N=2 gauge theories from various points of view.

• Their partition functions can be computed via three dual approaches: firstly, (p,q)-brane webs in type II string theory using Nekrasov's instanton calculus, secondly, the (refined) topological string using the topological vertex formalism and thirdly, M theory via the elliptic genus of certain M-strings configurations.
• We argue for a large class of theories that these approaches yield the same gauge theory partition function which we study in detail.
• To make their modular properties more tangible, we consider a fourth approach by connecting the partition function to the equivariant elliptic genus of R^4 through a (singular) theta-transform.