Siegel Modular Forms and Black Hole Entropy

TL;DR

This paper explores how Siegel Modular Forms can be used to count black hole microstates, extending known methods from the Igusa cusp form to other forms and analyzing their potential physical significance.

Contribution

It introduces a method to extract Fourier coefficients of various Siegel modular and paramodular forms for black hole entropy calculations, expanding the toolkit beyond the Igusa cusp form.

Findings

Fourier coefficients growth analyzed in different regimes

Identification of dominant contributions and logarithmic corrections

Comparison of new forms with the known Igusa cusp form behavior

Abstract

We discuss the application of Siegel Modular Forms to Black Hole entropy counting. The role of the Igusa cusp form $\chi_{10}$ in the D1D5P system is well-known, and its transformation properties are what allows precision microstate counting in this case. We apply a similar method to extract the Fourier coefficients of other Siegel modular and paramodular forms, and we show that they could serve as candidates for other types of black holes. We investigate the growth of their coefficients, identifying the dominant contributions and the leading logarithmic corrections in various regimes. We also discuss similarities and differences to the behavior of $\chi_{10}$, and possible physical interpretations of such forms both from a microscopic and gravitational point of view.