Attractor black holes in type II string compactifications on K 3 × 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 MG’s personal reminiscences of Peter Freund.

中文翻译：

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黑洞与巴尔加瓦不变性理论

在K 3×T 2上的II型弦压实中的吸引子黑洞与二元二次型的等价类相对应。二次型的判别式控制着黑洞的熵，并且在给定熵下的吸引子黑洞的数量由一个类数给出。在这里，我们显示出吸引子和算术之间的这种诱人关系可以推广到一个丰富的族，在算术组的作用下，将超重力和弦模型中的黑洞与更一般形式的类似等价类联系起来。涉及的许多物理理论在“魔术”超重力的研究中起着较早的作用，而它们的数学对应则直接与Bhargava等人的工作中的几何数示例相关。本文致力于纪念Peter Freund。