M String, Monopole String and Modular Forms

TL;DR

This paper explores the relationship between M-strings and m-strings in string theory, proposing a connection between their BPS state counting functions and elliptic genera, and analyzing their modular properties.

Contribution

It introduces a novel proposal linking M-string counting functions to m-string elliptic genera and investigates their modular transformation properties under specific conditions.

Findings

Complete agreement with known Taub-NUT and Atiyah-Hitchin spaces.

Existence of unique Jacobi form combinations for M-string free energies.

Necessary conditions for m-string moduli space factorization.

Abstract

We study relations between M-strings (one-dimensional intersections of M2-branes and M5-branes) in six dimensions and m-strings (magnetically charged monopole strings) in five dimensions. For specific configurations, we propose that the counting functions of BPS bound-states of M-strings capture the elliptic genus of the moduli space of m-strings. We check this proposal for the known cases, the Taub-NUT and Atiyah-Hitchin spaces for which we find complete agreement. Furthermore, we analyze the modular properties of the M-string free energies, which do not transform covariantly under SL(2,Z). However, for a given number of M-strings, we find that there exists a unique combination of unrefined genus-zero free energies that transforms as a Jacobi form under a congruence subgroup of SL(2,Z). These combinations correspond to summing over different numbers of M5-branes and make sense only if the distances between them are all equal. We explain that this is a necessary condition for the m-string moduli space to be factorizable into relative and center-of-mass parts.