A `black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal.Comment: Revtex, 8 pages, 1 figur

Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum Chern-Simons theory of a U(1) connection on a punctured 2-sphere, the horizon. Subtle mathematical features of the quantum Chern-Simons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (non-gravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all non-rotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated.An effort has been made to make the paper self-contained by including short reviews of the background material.

Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4d Riemannian quantum gravity that generalises the well-known Barrett-Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4d gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al. which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi parameter into the framework of spin foam quantisation. We generalize some of our considerations to the Lorentzian setting and obtain a new spin foam model in that context as well.

A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are "isolated" near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary Killing field even in a neighborhood of the horizon; rather, the physical assumption is that neither matter fields nor gravitational radiation fall across the portion of the horizon under consideration. A precise notion of non-rotating isolated horizons is formulated to capture these ideas. With these boundary conditions, the gravitational action fails to be differentiable unless a boundary term is added at the horizon. The required term turns out to be precisely the Chern-Simons action for the self-dual connection. The resulting symplectic structure also acquires, in addition to the usual volume piece, a surface term which is the Chern-Simons symplectic structure. We show that these modifications affect in subtle but important ways the standard discussion of constraints, gauge and dynamics. In companion papers, this framework serves as the point of departure for quantization, a statistical mechanical calculation of black hole entropy and a derivation of laws of black hole mechanics, generalized to isolated horizons. It may also have applications in classical general relativity, particularly in the investigation of of analytic issues that arise in the numerical studies of black hole collisions.

Boulatov and Ooguri have generalized the matrix models of 2d quantum gravity to 3d and 4d, in the form of field theories over group manifolds. We show that the Barrett-Crane quantum gravity model arises naturally from a theory of this type, but restricted to the homogeneous space S 3 = SO(4)/SO(3), as a term in its Feynman expansion. From such a perspective, 4d quantum spacetime emerges as a Feynman graph, in the manner of the 2d matrix models. This formalism provides a precise meaning to the "sum over triangulations", which is presumably necessary for a physical interpretation of a spin foam model as a theory of gravity. In addition, this formalism leads us to introduce a natural alternative model, which might have relevance for quantum gravity.

We study holography for asymptotically AdS spaces with an arbitrary genus compact Riemann surface as the conformal boundary. Such spaces can be constructed from the Euclidean AdS 3 by discrete identifications; the discrete groups one uses are the so-called classical Schottky groups. As we show, the spaces so constructed have an appealing interpretation of "analytic continuations" of the known Lorentzian signature black hole solutions; it is one of the motivations for our generalization of the holography to this case. We use the semi-classical approximation to the gravity path integral, and calculate the gravitational action for each space, which is given by the (appropriately regularized) volume of the space. As we show, the regularized volume reproduces exactly the action of Liouville theory, as defined on arbitrary Riemann surfaces by Takhtajan and Zograf. Using the results as to the properties of this action, we discuss thermodynamics of the spaces and analyze the boundary CFT partition function. Some aspects of our construction, such as the thermodynamical interpretation of the Teichmuller (Schottky) spaces, may be of interest for mathematicians working on Teichmuller theory.

We propose a new systematic approach that allows one to derive the spin foam (state sum) model of a theory starting from the corresponding classical action functional. It can be applied to any theory whose action can be written as that of the BF theory plus a functional of the B field. Examples of such theories include BF theories with or without cosmological term, Yang-Mills theories and gravity in various spacetime dimensions. Our main idea is two-fold. First, we propose to take into account in the path integral certain distributional configurations of the B field in which it is concentrated along lower dimensional hypersurfaces in spacetime. Second, using the notion of generating functional we develop perturbation expansion techniques, with the role of the free theory played by the BF theory. We test our approach on various theories for which the corresponding spin foam (state sum) models are known. We find that it exactly reproduces the known models for BF and 2D Yang-Mills theories. For the BF theory with cosmological term in 3 and 4 dimensions we calculate the terms of the transition amplitude that are of the first order in the cosmological constant, and find an agreement with the corresponding first order terms of the known state sum models. We discuss implications of our results for existing quantum gravity models.e-print

It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higherdimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the "spin foam" quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions. * In three spacetime dimensions Einstein's general relativity becomes a beautiful and simple theory. There are no local degrees of freedom, and gravity is an example of topological field theory. Owing to this fact, a variety of techniques from TQFT can be used, and a great deal is known about quantization of the theory. More precisely, when written in the first order formalism, three-dimensional gravity is just the BF theory, whose action is given by:(1)Here M is the spacetime manifold, F is the curvature of the spin connection, and B is the frame field one-form. The trace is taken in the Lie algebra of the relevant gauge group, which in the case of 3D is given by SO(2, 1) for Lorentzian spacetimes and by SO (3) in the Euclidean case. The quantization of BF theory is well-understood, both canonically and by the path integral method, at least in the Euclidean case. This is one of the possible ways to construct quantum gravity in three spacetime dimensions: it exists as a topological field theory. 1 It is tempting to apply the beautiful quantization methods from TQFT to other, more complicated theories, including those with local degrees of freedom. An interesting proposal along these lines was made in a series of papers by Martellini and collaborators [2], who proposed to treat Yang-Mills theory as a certain deformation of the BF theory. This gives an interesting picture of the confining phase of Yang-Mills theory.Recently, a proposal was made suggesting a way to apply the ideas and methods from TQFT to four-dimensional gravity. The new approach to quantum gravity, for which the name "spin foam approach" was proposed in [3], lies on the intersection between TQFT and loop quantum gravity (see [4] for a recent review on "loop" gravity). As it was advocated by Rovelli and Reisenberger [5], the results of the "loop" approach suggest a possibility of constructing the partition function of 4D gravity as a "spin foam" model. The first "spin foam" model of 4D gravity was constructed by Reisenberger [6], and was intimately related to the self-dual canonical (loop) quantum gravity. Later, Barrett, Crane [7] and Ba...