Automorphic black holes as probes of extra dimensions

Abstract

Recent progress in the understanding of the statistical nature of black hole entropy shows that the counting functions in certain classes of models are determined by automorphic forms of higher rank. In this paper we combine these results with Langlands' reciprocity conjecture to view black holes as probes of the geometry of spacetime. This point of view can be applied in any framework leading to automorphic forms, independently of the degree of supersymmetry of the models. In the present work we focus on the class of Chaudhuri-Hockney-Lykken compactifications defined as quotients associated to Z N groups. We show that the black hole entropy of these CHL N models can be derived from elliptic motives, thereby providing the simplest possible geometric building blocks of the Siegel type entropy count.

A general framework of automorphic inflation

“…[30,31] contain reviews). Such forms can be obtained by lifting classical modular forms to Siegel forms and for the models considered so far the origin can in fact be traced to elliptic modular forms [32].…”

Black Hole Probes of Automorphic Space

“…The lift for the class of Maaß-Skoruppa roots with complex multiplication derives from the existence of algebraic Hecke characters ψ H of weight one. The L-functions of powers ψ w+1 H of these characters are the inverse Mellin transforms of the forms f w+2 [12].…”

“…al. of the Taniyama-Shimura-Weil conjecture [13,14,15], but no such heavy machinery is necessary for the concrete cases based on the CHL N models, where the elliptic curves E N can be constructed explicitly for each order N [12].…”

Emergent spacetime and black hole probes from automorphic forms

“…This raises the question of how one can identify the motives of the variety which support the automorphic forms that appear in the entropy results. This problem was addressed in [3], on which the discussion of this section is based.…”

“…The modularity of the entropy implies that if one were to experiment with such black holes in the laboratory, a finite number of measurements would suffice to completely determine these functions. The review of this problem is based on [3].…”