  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.
