Indefinite theta functions for counting attractor backgrounds

TL;DR

This paper uses indefinite theta functions to regularize and analyze partition functions counting dyonic black hole degeneracies in string theory, linking microscopic counts to supergravity results.

Contribution

It introduces a novel regularization method for black hole partition functions using indefinite theta functions, connecting microscopic degeneracies with semi-classical supergravity.

Findings

Regularization of dyonic degeneracies via indefinite theta functions

Background independence achieved through elliptic transformations

Connection established between microscopic counts and supergravity results

Abstract

In this note, we employ indefinite theta functions to regularize canonical partition functions for single-center dyonic BPS black holes. These partition functions count dyonic degeneracies in the Hilbert space of four-dimensional toroidally compactified heterotic string theory, graded by electric and magnetic charges. The regularization is achieved by viewing the weighted sums of degeneracies as sums over charge excitations in the near-horizon attractor geometry of an arbitrarily chosen black hole background, and eliminating the unstable modes. This enables us to rewrite these sums in terms of indefinite theta functions. Background independence is then implemented by using the transformation property of indefinite theta functions under elliptic transformations, while modular transformations are used to make contact with semi-classical results in supergravity.