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.
Highlights
The notion of entropy for a black hole (BH) associated to its area [1, 2] led to a holographic picture of the black hole horizon carrying some information with a density of approximately 1 bit per unit Planck sized area
The expectation that on one hand, the nature of these bits of information accounting for the entropy depends on the quantum structure of space-time and on the other, the finiteness of the black hole entropy hints at the discreteness of space-time at the Planck scale are both realised in the loop quantum gravity (LQG) framework
By properly taking into account the regularization in the form of introducing a new boundary for each puncture that is necessary in order to have a well defined quantization of Chern-Simons theory on a punctured 2-sphere, we have defined the Hilbert space of a quantum IH
Summary
The notion of entropy for a black hole (BH) associated to its area [1, 2] led to a holographic picture of the black hole horizon carrying some information with a density of approximately 1 bit per unit Planck sized area. The dimension of the resulting Chern-Simons Hilbert space of the punctured two-sphere has been computed in different statistical mechanical ensembles and shown to generate a leading term linear in the IH area [5, 6, 7, 8, 9] (see [10] for a review) which matches the Bekenstein-Hawking formula for a specific numerical real value γ0 of the Barbero-Immirzi parameter γ This feature of the standard LQG black hole entropy calculation has raised concern in the past years. We show that the generators of this affine algebra contains both the gravitational degrees of freedom, previously taken into account in the LQG counting, and some new ones which furnish a representation in terms of free matter fields In this way, a dual local CFT description is explicitly realised at each puncture on the horizon. By momentarily focusing on a single puncture, we show how this implies the existence of a local conformal symmetry
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