Machine learning automorphic forms for black holes

TL;DR

The paper tackles recovering the modular weight $k$ of automorphic counting functions for BPS black holes from a finite set of Fourier coefficients, reflecting the $SL(2,\,\mathbb{Z})$ (and congruence subgroups) symmetry. It trains feed-forward neural networks on coefficients from Dedekind \(\eta(\tau)\), Eisenstein \(E_2(\tau)\), and Jacobi theta functions to predict the weight $k$. Results show strong predictive performance for negative-weight modular and quasi-modular forms, with reduced accuracy for positive weights and more intricate Jacobi-theta combinations. Overall, the work provides a proof-of-concept that machine learning can reveal how modular data organize gravitational microstate information and suggests paths for automated symmetry detection in quantum gravity and related string-theoretic models.

Abstract

Modular, Jacobi, and mock-modular forms serve as generating functions for BPS black hole degeneracies. By training feed-forward neural networks on Fourier coefficients of automorphic forms derived from the Dedekind eta function, Eisenstein series, and Jacobi theta functions, we demonstrate that machine learning techniques can accurately predict modular weights from truncated expansions. Our results reveal strong performance for negative weight modular and quasi-modular forms, particularly those arising in exact black hole counting formulae, with lower accuracy for positive weights and more complicated combinations of Jacobi theta functions. This study establishes a proof of concept for using machine learning to identify how data is organized in terms of modular symmetries in gravitational systems and suggests a pathway toward automated detection and verification of symmetries in quantum gravity.