Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional and Hilbert Modular Forms

TL;DR

This paper explores the mathematical structure of BPS black hole degeneracies in octonionic magical supergravity, linking them to Fourier coefficients of modular forms, Niemeier lattices, and exceptional Jordan algebra, revealing deep connections between physics and advanced number theory.

Contribution

It establishes a novel connection between black hole quantum degeneracies and Fourier coefficients of modular forms associated with exceptional groups and lattices, extending previous work to octonionic supergravity.

Findings

Quantum degeneracies are given by Fourier coefficients of modular forms of $E_{7(-25)}$.

Degeneracies for rank one and two black holes relate to singular modular forms $E_4$ and $E_8$.

Charge lattices correspond to Niemeier lattices derived from cubic ring embeddings.

Abstract

We study the quantum degeneracies of BPS black holes of octonionic magical supergravity in five dimensions that is defined by the exceptional Jordan algebra. We define the quantum degeneracy purely number theoretically as the number of distinct states in the charge space with a given set of invariant labels of the discrete U-duality group. We argue that the quantum degeneracies of spherically symmetric stationary BPS black holes of octonionic magical supergravity in five dimensions are given by the Fourier coefficients of the modular forms of the exceptional group $E_{7(-25)}$. The charges of the black holes take values in the lattice defined by the exceptional Jordan algebra $J_3^{\mathbb{O}}(\mathcal{R})$ over integral octonions $\mathcal{R}$. The quantum degeneracies of charge states of rank one and rank two BPS black holes (zero area) are given by the Fourier coefficients of singular modular forms $E_4(Z)$ and $E_8(Z)=(E_4(Z))^2$ of $E_{7(-25)}(Z)$. The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on the embeddings of cubic rings $A$ into the exceptional Jordan algebra and their actions on the 24 dimensional orthogonal quadratic subspace of $J_3^{\mathbb{O}}(\mathcal{R})$, we show that the quantum degeneracies of rank one black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms of $SL(2,A)$. If the discriminant of the cubic ring $A$ is $D=p^2$ with $p$ a prime number then the isotropic lines in the 24 dimensional quadratic space define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. For $p=7$ they are the Leech lattice with no roots and the lattice $A_6^4$ with 168 root vectors. We also review the current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity.