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.
Highlights
The rational Gaussian modelThe c = 1 Gaussian model, corresponding to string compactification on a S1 of radius R, is defined by an embedding of the unique unimodular even lattice of signature (1, 1) into R1,1
Elliptic genera counting supersymmetric states in these theories are non-holomorphic modular forms of a certain kind
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
Summary
The c = 1 Gaussian model, corresponding to string compactification on a S1 of radius R, is defined by an embedding of the unique unimodular even lattice of signature (1, 1) into R1,1. In order to facilitate the comparison to skew-holomorphic Jacobi forms using the conventions of the previous section we specialize to the case ζ(z) = zp0 (this corresponds to choosing the normalization of Jsuch that the associated charge has integer eigenvalues). A short computation shows that nκ − wκ = ω0r mod 2κκ which allows us to write n2 q4κκ yn m2 q 4κκ r mod 2κκ n=ω0r mod 2κκ m=r mod 2κκ θκκ ,ω0r(τ, z)θκ0κ ,r(τ ) This is almost of the desired form except for the factor of ω0. Where we have suppressed the vector indices in the above equation
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