CFT/gravity correspondence on the isolated horizon

Abstract

A quantum isolated horizon can be modelled by an SU(2) Chern-Simons theory on a punctured 2-sphere. We show how a local 2-dimensional conformal symmetry arises at each puncture inducing an infinite set of new observables localised at the horizon which satisfy a Kac-Moody algebra. By means of the isolated horizon boundary conditions, we represent the gravitational flux degrees of freedom in terms of the zero modes of the Kac-Moody algebra defined on the boundary of a punctured disk. In this way, our construction encodes a precise notion of CFT/gravity correspondence. The higher modes in the algebra represent new nongeometric charges which can be represented in terms of free matter field degrees of freedom. When computing the CFT partition function of the system, these new states induce an extra degeneracy factor, representing the density of horizon states at a given energy level, which reproduces the Bekenstein's holographic bound for an imaginary Immirzi parameter. This allows us to recover the Bekenstein-Hawking entropy formula without the large quantum gravity corrections associated with the number of punctures.

Horizon entropy with loop quantum gravity methods

“…The entropy result (36) matches exactly the one obtained in [19] by exploiting the local CFT symmetry introduced at each puncture by also blowing point particles to infinitesimal, but finite loops. In both cases, such a regularization procedure plays a fundamental role.…”

“…It would be interesting to study the algebra of observables that one could define living on these new boundaries, in analogy with the analysis of [19], and investigate if a Virasoro structure might emerge just from within the LQG formalism.…”

“…In Section 3 we show how the Ashtekar-Barbero connection on the IH becomes non-commutative and we introduce a second new non-commutative connection in order to be able to rely on techniques developed in [17,18] in the context of 2+1 gravity with non-vanishing cosmological constant to quantize the boundary theory using LQG techniques. The quantization is carried out in Section 4, where the IH quantum state is defined by regularizing point punctures with finite loops, as required by the extended nature of the LQG configuration variables and in analogy to the proposal of [19]; we then define the physical scalar product of the horizon theory, imposing the quantum version of the boundary conditions. In Section 5 we use the equivalence [18] between the Chern-Simons observables expectation values and the physical amplitudes of 2+1 canonical LQG to compute the number of IH dof by means of the physical scalar product previously defined.…”

Kinematical gravitational charge algebra

“…It seems natural to expect these magnetic edge modes to be encoded in a boundary connection field. Either by providing with a Chern-Simons-like boundary symplectic term [28][29][30] or with a mixed term coupling the boundary connection to the boundary frame field [31] as was derived in the case of isolated horizon boundary conditions.…”

Anyonic statistics and large horizon diffeomorphisms for loop quantum gravity black holes

“…In the large black hole limit the topological spin h j vanishes and the twist is equal to 1, rendering the ribbonlike nature unimportant. How these qualitative considerations can have a bearing on the system of quantum gravitational anyons of the horizon and how this might be related to [52] will be investigated elsewhere.…”