Black holes and Bhargava’s invariant theory

Attractor black holes in type II string compactifications on K3 × T² are in correspondence with equivalence classes of binary quadratic forms. The discriminant of the quadratic form governs the black hole entropy, and the count of attractor black holes at a given entropy is given by a class number. Here, we show this tantalizing relationship between attractors and arithmetic can be generalized to a rich family, connecting black holes in supergravity and string models with analogous equivalence classes of more general forms under the action of arithmetic groups. Many of the physical theories involved have played an earlier role in the study of ‘magical’ supergravities, while their mathematical counterparts are directly related to geometry-of-numbers examples in the work of Bhargava et al. This paper is dedicated to the memory of Peter Freund. The last section is devoted to some of MG’s personal reminiscences of Peter Freund.