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 N=2 black holes in string theory, whose masses on the horizon are fixed by the 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.
Highlights
In this paper we want to generalize the infinite distance conjecture of the swampland scenario [1,2,3] to black hole geometries with horizon
For non-extremal black holes we argue that the associated light tower of states is related to Goldstone-like modes that interpolate among the black hole microstates
We have argued that besides the physical Kaluza-Klein modes and string excitations there exists another kind of tower of modes that can be determined by the distance to the large entropy limit in the space of non-extremal black hole geometries
Summary
In this paper we want to generalize the infinite distance conjecture of the swampland scenario [1,2,3] to black hole geometries with horizon. The attractor equations of supersymmetric extremal black holes [59, 60] imply that large horizons and large entropies follow from large values of internal moduli or dilaton fields on the horizon The latter are accompanied by towers of Kaluza-Klein modes from the internal geometry, whose masses on the horizon go to zero in the infinite entropy limit. The emergence of this massless tower of modes in the infinite entropy limit of non-extremal black holes means that the effective horizon field theory must contain these states. For large moduli at infinite distance, it follows that the large entropy is at large distance in the space of N = 2 black hole string backgrounds and the associated tower of states is well understood in terms of the Kaluza-Klein states of the internal manifold. We will see that the large entropy limit of non-extremal (and extremal) black holes - a priori not being linked to large internal moduli fields - is at infinite distance in the space of four-dimensional black hole metrics
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