BPS jumping loci are automorphic

We show that BPS jumping loci – loci in the moduli space of string compactifications where the number of BPS states jumps in an upper semi-continuous manner – naturally appear as Fourier coefficients of (vector space-valued) automorphic forms. For the case of $T^2$ compactification, the jumping loci are governed by a modular form studied by Hirzebruch and Zagier, while the jumping loci in K3 compactification appear in a story developed by Oda and Kudla-Millson in arithmetic geometry. We also comment on some curious related automorphy in the physics of black hole attractors and flux vacua.