The degeneracies of $1/4$ BPS states in four-dimensional $mathcal{N}=4$ heterotic string theory are given in terms of the Fourier coefficients of a meromorphic Siegel modular form. In this talk I will explain how the symplectic symmetries of this form enable us to construct a fine-grained Rademacher type expansion which expresses these BPS degeneracies as a sum over residues of the poles of the Siegel modular form, revealing two distinct $SL(2,mathbb{Z})$ groups parametrizing the expansion. This construction can be used to improve on the existing formulation of the corresponding quantum entropy function obtained using supersymmetric localization with the result taking the form of a sum over Euclidean backgrounds which appear as orbifolds of the Euclidean $AdS_2 times S^2$ attractor geometry.

King’s College London