New variables for classical and quantum gravity in all dimensions: I. Hamiltonian analysis

Abstract

Loop Quantum Gravity heavily relies on a connection formulation of General Relativity such that 1. the connection Poisson commutes with itself and 2. the corresponding gauge group is compact. This can be achieved starting from the Palatini or Holst action when imposing the time gauge. Unfortunately, this method is restricted to D + 1 = 4 spacetime dimensions. However, interesting String theories and Supergravity theories require higher dimensions and it would therefore be desirable to have higher dimensional Supergravity loop quantisations at one's disposal in order to compare these approaches.In this series of papers we take first steps towards this goal. The present first paper develops a classical canonical platform for a higher dimensional connection formulation of the purely gravitational sector. The new ingredient is a different extension of the ADM phase space than the one used in LQG which does not require the time gauge and which generalises to any dimension D > 1. The result is a Yang -Mills theory phase space subject to Gauß, spatial diffeomorphism and Hamiltonian constraint as well as one additional constraint, called the simplicity constraint. The structure group can be chosen to be SO(1, D) or SO(D +1) and the latter choice is preferred for purposes of quantisation.

Loop Quantum Cosmology, Modified Gravity and Extra Dimensions

“…The connection dynamics of n+1 dimensional gravity (n ≥ 3) with a gauge group SO(n + 1) or SO(1, n) is obtained in [38]. The Ashtekar connection formalism of n + 1 dimensional gravity constitutes an SO(1, n) (or SO(n + 1)) connections A I J a and a group value densitized vector π b I J defined on an oriented n dimensional manifold S, where a, b = 1, 2 .…”

“…A a are n dimensional spin connection and extrinsic curvature, respectively. Besides the Simplicity constraint, the n+1 dimensional gravity has three constraints similar to 3+1 dimensional general relativity [38,40]:…”

Black hole entropy from loop quantum gravity in higher dimensions

“…≈ 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.…”

“…The classical part of this derivation was extended to higher dimensions in [9] and to LanczosLovelock gravity in [10], where the recently introduced connection variables for higher-dimensional general relativity [11][12][13][14] were employed. While the horizon degrees of freedom can be rewritten in a form similar to a higherdimensional Chern-Simons theory, it turns out to be more economical to use a canonically conjugate pair of normals n I ands I as horizon degrees of freedom [9].…”

Alternative self-dual gravity in eight dimensions

“…Higher dimensional Ashtekar formalism has shown to be a promising proposal [1][2][3][4][5] (see also Ref. [6][7][8][9][10]).…”