Infinite Black Hole Entropies at Infinite Distances and Tower of States

TL;DR

This paper explores the relationship between black hole entropy, infinite distance in moduli space, and the emergence of a tower of light states, proposing a new black hole entropy distance conjecture within the swampland framework.

Contribution

It introduces the black hole entropy distance conjecture, linking infinite entropy limits to towers of states, and analyzes potential candidates for these states in string theory and quantum models.

Findings

Infinite entropy limits are at infinite distance in black hole geometry space.

Kaluza-Klein modes cannot always serve as the tower of states due to decoupling.

Goldstone-like modes may act as the tower of states in non-extremal black holes.

Abstract

The aim of this paper is to elucidate a close connection between the black hole area law and the infinite distance conjecture in the context of the swampland. We consider families of black hole geometries, parametrized by their event horizon areas or by the values of their entropies, and show that the infinite entropy limit is always at infinite distance in the space of black hole geometries. It then follows from the infinite distance conjecture that there must be a tower of states in the infinite entropy limit, and that ignoring these towers on the horizon of the black hole would invalidate the effective theory when the entropy becomes large. We call this the black hole entropy distance conjecture. We then study two candidates for the tower of states. The first are the Kaluza-Klein modes of the internal geometry of extremal ${\cal N}=2$ black holes in string theory, whose masses on the horizon are fixed by the ${\cal N}=2$ attractor formalism, and given in terms of the black hole charges similarly to the entropy. However, we observe that it is possible to decouple their masses from the entropy, so that they cannot generically play the role of the tower. We thus consider a second kind of states: inspired by N-portrait quantum models of non-extremal black holes, we argue that the Goldstone-like modes that interpolate among the black hole microstates behave like the expected light tower of states.