M‐strings, elliptic genera and N = 4 string amplitudes

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 ℂ2 through a (singular) theta‐transform. This form appears naturally as a specific class of one‐loop scattering amplitudes in type II string theory on T2, which we calculate explicitly. Mass‐deformed N= 2 gauge theories are studied 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. For a large class of theories these approaches yield the same gauge theory partition function. To make modular properties more tangible, a fourth approach is considered by connecting the partition function to the equivariant elliptic genus.