Attractive Strings and Five-Branes, Skew-Holomorphic Jacobi Forms and Moonshine

TL;DR

This paper demonstrates how BPS counting functions from strings and fivebranes in Calabi-Yau compactifications produce skew-holomorphic Jacobi forms, revealing connections to moonshine and Mathieu groups.

Contribution

It uncovers new instances of skew-holomorphic Jacobi forms arising from M5-branes and links them to moonshine phenomena involving Mathieu groups.

Findings

BPS functions yield skew-holomorphic Jacobi forms at rational points

M5-branes produce forms of negative weight and mock variants

Connections established between these forms and Mathieu group moonshine

Abstract

We show that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi--Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This suggests a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.