# BPS jumping loci are automorphic

## TL;DR

This paper demonstrates that BPS jumping loci in string compactification moduli spaces are naturally represented as Fourier coefficients of automorphic forms, linking physical phenomena with deep mathematical structures.

## Contribution

It establishes a direct connection between BPS jumping loci and automorphic forms, revealing their automorphic nature in various string compactification scenarios.

## Key findings

- BPS jumping loci correspond to Fourier coefficients of automorphic forms.
- In T^2 compactification, they relate to Hirzebruch-Zagier modular forms.
- In K3 compactification, they connect to arithmetic geometry via Kudla-Millson forms.

## Abstract

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.