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CFT/Gravity Correspondence on the Isolated Horizon

Amit Ghosh, Daniele Pranzetti

TL;DR

The paper demonstrates that quantum isolated horizons in loop quantum gravity admit a local two-dimensional conformal symmetry at each puncture, realized through an SU(2) affine Kac-Moody algebra whose zero modes encode horizon fluxes. Via the Sugawara construction, a Virasoro algebra arises, and holonomies are mapped to vertex operators, with a Wakimoto free-field realization giving rise to non-geometric, matter-like degrees of freedom encoded by the higher KM modes. The CFT partition function acquires a horizon-degeneracy factor, and for an imaginary Barbero-Immirzi parameter $\gamma=i$ this reproduces the holographic Bekenstein bound and recovers the Bekenstein-Hawking entropy without large puncture-counting corrections. This work proposes a concrete CFT/gravity dual perspective at the horizon, suggesting a mechanism to couple matter to LQG through horizon conformal degrees of freedom and inviting extensions to non-spherical isolated horizons.

Abstract

A quantum isolated horizon can be modeled 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 localized at the horizon which satisfy a Kac-Moody algebra. By means of the isolated horizon boundary conditions, we represent the gravitational fluxes 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.

CFT/Gravity Correspondence on the Isolated Horizon

TL;DR

The paper demonstrates that quantum isolated horizons in loop quantum gravity admit a local two-dimensional conformal symmetry at each puncture, realized through an SU(2) affine Kac-Moody algebra whose zero modes encode horizon fluxes. Via the Sugawara construction, a Virasoro algebra arises, and holonomies are mapped to vertex operators, with a Wakimoto free-field realization giving rise to non-geometric, matter-like degrees of freedom encoded by the higher KM modes. The CFT partition function acquires a horizon-degeneracy factor, and for an imaginary Barbero-Immirzi parameter this reproduces the holographic Bekenstein bound and recovers the Bekenstein-Hawking entropy without large puncture-counting corrections. This work proposes a concrete CFT/gravity dual perspective at the horizon, suggesting a mechanism to couple matter to LQG through horizon conformal degrees of freedom and inviting extensions to non-spherical isolated horizons.

Abstract

A quantum isolated horizon can be modeled 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 localized at the horizon which satisfy a Kac-Moody algebra. By means of the isolated horizon boundary conditions, we represent the gravitational fluxes 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.

Paper Structure

This paper contains 10 sections, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. CFT/Gravity
  3. Kac-Moody algebra
  4. Virasoro algebra
  5. Holonomies as Vertex Operators
  6. Free Field Representation
  7. Partition Function
  8. Conclusions and Comments
  9. Free Boson
  10. Kac Character Formula

Figures (2)

  • Figure 1: Hole $H$ around the source $p$. $D$ is a disk around $H$ where the connection is flat. Its boundary $\partial D$ consists of two components, an inner boundary $\partial H$ and an outer boundary $B$.
  • Figure 2: An highest weight state can be constructed from the holonomy defined on the line $e$.