Title A note on black hole entropy in loop quantum gravity Permalink

Several recent results have hinted that black hole thermodynamics in loop quantum gravity simplifies if one chooses an imaginary Barbero-Immirzi parameter γ = i. This suggests a connection with SL(2,C) or SL(2,R) conformal field theories at the “boundaries” formed by spin network edges intersecting the horizon. I present a bit of background regarding the relevant conformal field theories, along with some speculations about how they might be used to count black hole states. I show, in particular, that a set of unproven but plausible assumptions can lead to a boundary conformal field theory whose density of states matches the BekensteinHawking entropy. email: carlip@physics.ucdavis. edu The computation of black hole entropy [1–3] has been one of the triumphs of loop quantum gravity. The achievement has come at a price, however: the area operator in loop quantum gravity depends on the Barbero-Immirzi parameter γ, and one must tune γ to a rather peculiar value, fixed by an obscure combinatorial problem, to obtain the correct Bekenstein-Hawking entropy [4, 5]. This tuning only has to be done once—if γ is adjusted to fit, say, the Schwarzschild black hole, the theory gives the correct entropy for all black holes—but the physical meaning of this choice remains enigmatic. Over the past few years, some interesting new hints have emerged. First, it has been observed that in a grand canonical ensemble, the dependence of entropy on γ can be shifted to a chemical potential term for the number of “punctures” where a spin network intersects the horizon [6]. In the presence of suitable holographic matter fields, it may be possible to set this chemical potential to zero [7], and steps have been taken to develop a conformal field theory description [8]. At the same time, it has been noted that if one analytically continues the Barbero-Immirzi parameter to the value γ = i, certain partition functions automatically give the correct Bekenstein-Hawking entropy [9, 10], and certain spin foam amplitudes acquire a correct imaginary part [11]. The choice γ = ±i is the one for which Ashtekar variables were first introduced [12], and it is a natural one: the loop quantum gravity connection is self-dual at this value, and is the only value for which the Barbero-Ashtekar connection is a diffeomorphism-invariant spacetime gauge field [13, 14]. Unfortunately, it is not clear how to consistently implement reality conditions when γ is complex; at least for practical purposes, a real value seems to be needed. I do not have the answers to these problems, and it remains possible that a correct thermodynamic treatment could give a value of the entropy that is independent of γ [15]. But in this note I will suggest some ingredients that might be included in the answers, including a few elements of conformal field theory that may not be familiar to most people working in loop quantum gravity. In particular, there are interesting hints that coset constructions, Liouville theory, and holomorphic Chern-Simons theories may be relevant. 1. Chern-Simons and WZNW theory A question about a black hole in quantum gravity is a question about conditional probability: one must impose the condition that the desired black hole is present. One way to do so is to demand that a hypersurface ∆ be a horizon. Such a horizon is not a physical boundary, but it is a place where one is imposing “boundary conditions,” and as a consequence, the Einstein-Hilbert action acquires an added boundary term. In the first-order formalism of loop quantum gravity, this term is a three-dimensional Chern-Simons action [2, 19,20],