BF description of higher-dimensional gravity theories

Abstract

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...

The strong and weak holographic principles

“…As topological quantum field theories are the only class of field theories that naturally yield finite dimensional Hilbert spaces, one may try to use them to construct examples of holographic theories [13]. Furthermore, as Crane pointed out, it may be possible to extend these structures to quantum theories of gravity because it is a fact that at both the classical and quantum mechanical level, and for any dimension [47], general relativity and supergravity can be understood as deformed or constrained topological quantum field theories [49,23,13,19,51,50,14,15].…”

Group Field Theory: An Overview

“…The Barrett-Crane prescription was understood to be linked to the Plebanski formulation of gravity where the Einstein action is written as a BF theory subject to constraints [19]. This formulation and the corresponding spin foam models were extended to gravity in any dimensions [20].…”

Black hole entropy from loop quantum gravity in higher dimensions

“…The bulk simplicity constraint together with the boundary condition (5) restricts us to simple SO(D + 1) representations on H, which are labelled by a single nonnegative integer λ [13,18]. This is consistent with the fact that we could have imposed diagonal horizon simplicity constraints on the horizon degrees of freedom.…”

“…We thus introduce the bi-vectors L IJ n per face together with the diagonal simplicity con- [18]. The symplectic structure of the bivectors is again similar to (6) and agrees with the one calculated from n I n ,s J n on the constraint surface of the simplicity constraints.…”

“…≈ means equality on the constraint surface, defined by the SO(D+1) Gauß law and the simplicity constraint π a[IJ π b|KL] = 0. The simplicity constraint ensures that π aIJ ≈ 2/β n [I e J] b q ab √ q, that is, it is the product of a normal n I and a densitized D + 1-bein orthogonal to n I , see [11,12,18] for further details, e.g. a topological sector in 3 + 1 dimensions.…”