Holography Principle and Arithmetic of Algebraic Curves

According to the holography principle (due to G. ‘t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non–commutative space. §0. Introduction 0.1. Holography principle. Consider a manifold M (“bulk space”) with boundary N. The holography principle postulates the existence of strong ties between certain field theories onM and N respectively. For example, in the actively discussed Maldacena’s conjecture ([Mal], [Wi]), M is the anti de Sitter space AdSd+1 (or AdSd+1 × S), N its conformal boundary. On the boundary one considers the large N limit of a conformally invariant theory in d dimensions, and on the bulk space supergravity and string theory (cf. e.g. [AhGuMOO], [Mal], [Suss], [’tH], [Wi], [WiY]). The holography principle was originally suggested by ‘t Hooft in order to reconcile unitarity with gravitational collapse. In this case M is a black hole and N is the event horizon. Thus the bulk space should be imagined as (a part of) space–time. There are other models where the boundary can play the role of space–time (Plato’s cave picture), with the bulk space involving an extra dimension (e. g. the renormalization group scale) and a Kaluza–Klein type reduction [AlGo], and “brane world scenarios” where one models our universe as a brane in higher dimensional space–time, with gravity confined to the brane. In this paper we consider first of all a class of Euclidean AdS3 bulk spaces which are quotients of the real hyperbolic 3–space H by a Schottky group. The boundary (at infinity) of such a space is a compact oriented surface with conformal structure, which is the same as a compact complex algebraic curve. Such spaces are analytic continuations of known (generally rotating) Lorentzian signature black