Observations on Arithmetic Invariants and U-Duality Orbits in N =8 Supergravity

TL;DR

This paper explores the classification of black hole solutions in N=8 supergravity using U-duality invariants, linking orbit types to physical properties and entropy, and discusses implications for the theory's finiteness.

Contribution

It establishes a relation between different orbits of E7(7) symmetry and invariants, providing formulas for state degeneracy and entropy, and clarifies the role of light-like orbits in the theory.

Findings

U-duality invariants classify black hole orbits in N=8 supergravity.

A formula for state degeneracy reproduces Bekenstein-Hawking entropy.

Light-like orbits decouple from the index, with implications for black hole solutions.

Abstract

We establish a relation between time-like, light-like and space-like orbits of the non-compact E{7(7)}(R)symmetry and discrete E{7(7)}(Z) invariants. We discuss the U-duality invariant formula for the degeneracy of states d(Q) which in the approximation of large occupation numbers reproduces the Bekenstein-Hawking entropy formula for regular black holes with AdS_2 horizon. We explain why the states belonging to light-like orbits, corresponding to classical solutions of "light black holes with null singularity", decouple from the corresponding index. We also present a separate U-duality invariant formula for the class of light-like orbits specified by discrete E{7(7)}(Z) invariants. We conclude that the present study of the non-perturbative sector of the theory does not reveal any contradiction with the conjectured all-loop perturbative finiteness of D=4 N=8 supergravity.