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

Abstract

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. This form appears naturally as a specific class of one-loop scattering amplitudes in type II string theory on T^2, which we calculate explicitly.

Figures (16)

• Figure 1: Four-fold approach to $\mathcal{N}=2^*$ gauge theories: The red squares represent three dual settings from which the partition function of certain mass deformed $\mathcal{N}=2$ gauge theories can be computed using various techniques. Each of these approaches has interesting dualities which can be exploited. The blue circle denotes a specific one-loop string amplitude which reproduces the partition function for a particular configuration of branes.
• Figure 3: 5-brane web with $N$ D5-branes (horizontal) and $M$ NS5-branes (vertical)
• Figure 4: (a) Web giving rise to 5D maximally supersymmetric $U(N)$ theory on the D5-branes, (b) the web giving rise to $U(1)^{N}$ theory on the D5-branes.
• Figure 5: The mass to the adjoint hypermultiplet is given by a deformation of the brane web. The mass parameter $m$ corresponds to the length of the blue interval.
• Figure 6: The Newton polygon of the conifold with triangulations shown.