# Black holes and Bhargava's invariant theory

## TL;DR

This paper explores the deep connection between black hole physics in string theory and advanced arithmetic invariant theory, extending known relationships to more general forms and arithmetic groups.

## Contribution

It generalizes the correspondence between black holes and quadratic forms to broader classes of forms and arithmetic groups, linking supergravity and string models with Bhargava's invariant theory.

## Key findings

- Black hole entropy relates to discriminants of generalized forms.
- Counting attractor black holes corresponds to class numbers of these forms.
- The work connects physical theories with geometric number theory.

## Abstract

Attractor black holes in type II string compactifications on $K3 \times T^2$ 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 M.G's personal reminiscences of Peter Freund.