Motivated by the analogy proposed by Witten between Chern-Simons and
Conformal Field Theories, we explore an alternative way of computing the
entropy of a black hole starting from the isolated horizon framework in Loop
Quantum Gravity. The consistency of the result opens a window for the interplay
between Conformal Field Theory and the description of black holes in Loop
Quantum Gravity.Comment: 9 page

A. Ashtekar

A. Corichi

C. Rovelli

Enrique F Borja

I. Agullo

Iván Agulló

Jacobo Díaz-Polo

K.A. Meissner

M. Bojowald

M. Domagala

P. Di Francesco

T. Thiemann

Journal of Cosmology and Astroparticle Physics

English

arXiv

Motivated by the analogy proposed by Witten between Chern-Simons and conformal field theories, we explore an alternative way of computing the entropy of a black hole starting from the isolated horizon framework in loop quantum gravity. The consistency of the result opens a window for the interplay between conformal field theory and the description of black holes in loop quantum gravity.We thank V. Aldaya, A. Ashtekar, J. A. de Azcarraga, R. Coquereaux and L. Freidel for
interesting discussions and suggestions. We specially thank F. Barbero and E. Villasenor
for many discussions, strong ecouragement, and a careful reading of the manuscript. I. A.
thanks R. Wald for his kind hospitality at the University of Chicago where part of this work
was done. J. D. thanks A. Ashtekar for his kind hospitality at the IGC at Penn State during
the realization of part of this work.
This work was in part supported by Spanish grants FIS2008-06078-C03-02, FIS2008-
01980 and ESP2005-07714-C03-01, the NSF grant PHY0854743 and the Eberly research
founds of Penn State, and the NSF grant PHY04-56619 to the University of Chicago. I. A.
and J. D. acknowledge financial support provided be the Spanish Ministry of Science and
Education under the FPU program.Peer reviewe

Agulló, Iván

Fernández Borja, Enrique

Díaz-Polo, Jacobo

Digital.CSIC

arXiv:0903.1667v1  [hep-th]  10 Mar 2009Computing Black Hole entropy in Loop Quantum Gravity from aConformal Field Theory perspectiveIva´n Agullo´∗Enrico Fermi Institute and Department of Physics,University of Chicago, Chicago, IL 60637 USA andDepartamento de F´ısica Teo´rica and IFIC,Centro Mixto Universidad de Valencia-CSIC. Facultad de F´ısica,Universidad de Valencia, Burjassot-46100, Valencia, SpainEnrique F. Borja†Departamento de F´ısica Teo´rica and IFIC,Centro Mixto Universidad de Valencia-CSIC. Facultad de F´ısica,Universidad de Valencia, Burjassot-46100, Valencia, SpainJacobo Dı´az-Polo‡Institute for Gravitation and the Cosmos, Physics Department,Penn State, University Park, PA 16802, U.S.A. andDepartamento de Astronomı´a y Astrof´ısica,Universidad de Valencia, Burjassot-46100, Valencia, Spain(Dated: March 10, 2009)Motivated by the analogy proposed by Witten between Chern-Simons and Con-formal Field Theories, we explore an alternative way of computing the entropy of ablack hole starting from the isolated horizon framework in Loop Quantum Gravity.The consistency of the result opens a window for the interplay between ConformalField Theory and the description of black holes in Loop Quantum Gravity.PACS numbers: 04.70.Dy, 11.25.Hf, 04.60.Pp∗Ivan.Agullo@uv.es†Enrique.Fernandez@uv.es‡Jacobo.Diaz@uv.es2I. INTRODUCTIONEver since the pioneering work of Bekenstein [1] about the physical entropy of black holes,one of the main challenges of quantum gravity has been to describe the microscopic degrees offreedom responsible for this entropy. At the present time there are several proposals that, inspite of their totally different motivations, have reproduced the Bekenstein-Hawking law atthe leading order, furthermore showing an agreement in the first order logarithmic correction.This proliferation of different alternatives raises the puzzle of understanding the underlyingreason of this broad agreement. On the basis of the observation that, in addition, most ofthese approaches involve Conformal Field Theory (CFT) techniques at some stage, it hasbeen suggested that conformal symmetry could play a fundamental role in this scenario (see[2] and references therein).Loop Quantum Gravity (LQG) [3, 4] offers a detailed description of the black hole horizonquantum states [5]. In the isolated horizon framework, a black hole is introduced as aninner boundary of the spacetime manifold. Over this boundary, constraints implementingthe isolated horizon properties are imposed. They reduce, already at the classical level, theSU(2) gauge symmetry of the theory to a U(1) gauge symmetry on the horizon. TheseU(1) degrees of freedom, that at the quantum level fluctuate independently from the ones ofthe bulk, are described by a Chern-Simons (CS) theory and are responsible for the horizonentropy. By counting these CS degrees of freedom, a robustly verified [6, 7, 8] linear behaviorof entropy as a function of the horizon area is obtained at the leading order, showing inaddition the existence of a first order logarithmic correction.On the other hand, E. Witten proposed [9] the correspondence between the Hilbert spaceof generally covariant theories and the space of conformal blocks of a conformally invarianttheory. This idea was applied in [10] to the computation of the entropy for a horizondescribed by a SU(2)-CS theory, by putting its Hilbert space in correspondence with thespace of conformal blocks of a SU(2)-Wess-Zumino-Witten (WZW) model.The purpose of the present paper is to make use of Witten’s correspondence for theU(1)-CS theory describing the black hole horizon in LQG, looking for some hints on therole of CFT techniques in this framework. Taking into account the fact that this U(1)group arises as the result of a geometric symmetry breaking from the SU(2) symmetryin the bulk, one can still make use of the well established correspondence between SU(2)3Chern-Simons and Wess-Zumino-Witten theories. However, in this case it will be necessaryto impose restrictions on the SU(2)-WZW model in order to implement the symmetryreduction. Through this procedure we expect to eventually reproduce the counting ofdimensions of the U(1)-CS Hilbert space.II. BLACK HOLE ENTROPY COUNTINGLet us summarize the main features and results of the black hole entropy counting inLQG in the isolated horizon framework [5]. On a space-like slice Σ, the geometry of thebulk is described, as usual, by a spin network. Some of the spin network edges, however,end at the horizon surface S (the intersection of Σ and the isolated horizon), endowing itwith an area given byA = 8πγℓ2PN∑I=1√jI(jI + 1) , (1)where jI ∈ N/2 label the SU(2) irreducible representations corresponding to the N edgespiercing the horizon, γ denotes the Barbero-Immirzi parameter and ℓP is the Planck length.These edges carry an additional label mI ∈ {−jI ,−jI + 1, ..., jI} (the corresponding spinprojection) characterizing their intersection with the horizon (punctures).On the other hand, the horizon geometry is described by a U(1)Chern-Simons theorydefined over a sphere with N distinguishable topological defects (corresponding with thepunctures).1 The states of this theory are characterized by labels aI ∈ Zκ (κ being the levelof the CS theory) quantifying the angle deficits that give rise to the distributional curvatureof the horizon concentrated at each puncture. The spherical topology of the horizon impliesthat these aI labels must satisfy the so called projection constraint∑I aI = 0. The matchingof both (bulk and horizon) geometries through the boundary conditions gives rise to arelation between aI and mI labels, that reads2mI = −aI mod κ . (2)For a given value A of area, the entropy can be computed as S(A) = kB log n(A), being1 The fact that punctures are distinguishable is related to the action of diffeomorphisms during the quan-tization procedure [3, 5], and plays a key role in the entropy counting.4kB the Boltzman constant and n(A) the number of independent Chern-Simons states com-patible with the above constraints, taking into account the distinguishable character of thepunctures. This is to say, n(A) is the number of different aI-labeled horizon states (satisfyingthe projection constraint) such that, for each of them, there exists (at least) one (jI , mI)-labeled piercing from the bulk compatible with it and with the value A of the horizon area.The relation between mI and aI labels allows us then to formulate the entropy counting asa well defined combinatorial problem in terms only of the mI labels as in [11]. Then, n(A)can be rewritten as: n(A) = 1 +∑A′≤A d(A′), where d(A) is the number of all the finite,arbitrarily long, ordered sequences ~m = (m1, ..., mN) of non-zero half-integers, such thatN∑I=1mI = 0 ,N∑I=1√|mI |(|mI |+ 1) =A8πγℓ2P. (3)Explicit expressions for the solution of this combinatorial problem were obtained in [8, 12].If we define kI = 2|mI | and the occupancy numbers nk as the number of punctures carryinga label value m such that k = 2|m|, then a set of numbers {nk}, k = 1, 2, ... characterizesa ~m sequence up to reorderings and sign assignments for mI = ±12kI . Thus, d(A) can beexpressed in terms of the set C of all the {nk} sets compatible with a given area A byassociating two sources of degeneracy to each of these sets {nk}. The first is the numberR({nk}) of different ways of reordering the kI labels in order to obtain all the correspondingordered sequences ~k = (k1, ..., kN). The second source of degeneracy is the number P ({nk})of different sing assignments for the associatedmI numbers, in such a way that the projectionconstraint is satisfied. With thisd(A) =∑{nk}∈CR({nk})× P ({nk}), (4)where the sum is extended over all the sets {nk} in C.This set C of all {nk} configurations compatible with a given area eigenvalue can be com-puted analytically [8] using number-theory related techniques, through an exact charac-terization of the horizon area spectrum of LQG. The factor R({nk}) has its origin in thedistinguishable character of punctures (acquired in the process of quantization of geometry)and can be obtained from basic combinatorics as R({nk}) = (∑k nk)!/∏k nk!, where thesum and product are extended to all values of k (note that, in practice, for a finite value Aof area all the sums and products are always finite). Finally, the factor P ({nk}) accounts for5the dimensionality of the Hilbert space of the U(1)-CS theory once the boundary conditionshave been fixed and was obtained in [8, 12] to be:P ({nk}) =12π∫ 2pi0dθ∏knk2 cos (kθ) . (5)III. IMPLEMENTING THE ANALOGY BETWEEN CHERN-SIMONS ANDWESS-ZUMINO-WITTEN.Let us begin by recalling the classical scenario and how the symmetry reduction takesplace at this level. The geometry of the bulk is described by a SU(2) connection, whoserestriction to the horizon S gives rise to a SU(2) connection over this surface. As a conse-quence of imposing the isolated horizon boundary conditions this connection is reduced toa U(1) connection. In [5] this reduction is carried out, at the classical level, just by fixinga unit vector ~r at each point of the horizon. By defining a smooth function r : S → su(2)a U(1) sub-bundle is picked out from the SU(2) bundle. This kind of reduction can bedescribed in more general terms as follows (see, for instance, [13]). Let P (SU(2), S) be aSU(2) principal bundle over the horizon, and ω the corresponding connection over it. Ahomomorphism λ between the closed subgroup U(1) ⊂ SU(2) and SU(2) induces a bundlereduction form P (SU(2), S) to Q(U(1), S), being Q the resulting U(1) principal bundle withreduced U(1) connection ω′. This ω′ is obtained, in this case, from the restriction of ω toU(1).All the conjugacy classes of homomorphisms λ : U(1) → SU(2) are represented in the setHom(U(1), T (SU(2))), where T (SU(2)) = {diag(z, z−1)|z = eiθ ∈ U(1)} is the maximaltorus of SU(2). The homomorphisms in Hom(U(1), T (SU(2))) can be characterized byλp : z 7→ diag(zp, z−p) , (6)for any p ∈ Z. However the generator of the Weyl group of SU(2) acts on T (SU(2)) bydiag(z, z−1) 7→ diag(z−1, z). If we divide out by the action of the Weyl group we are just leftwith those maps λp with p a non-negative integer, p ∈ N0, as representatives of all conjugacyclasses. These λp characterize then all the possible ways to carry out the symmetry breakingfrom the SU(2) to the U(1) connection that will be quantized later.However, one can follow the alternative approach of first quantizing the SU(2) connection6on S and imposing the boundary conditions later on, at the quantum level. This would giverise to a SU(2)-CS theory on the horizon to which the boundary conditions have to beimposed. The correspondence with conformal field theories can be used at this point tocompute the dimension of the Hilbert space of the SU(2)-CS as the number of conformalblocks of the SU(2)-WZW model, as it was done in [10]. It is necessary to require, then,additional restrictions to the SU(2)-WZW model that account for the symmetry breaking,and consider only the degrees of freedom corresponding to a U(1) subgroup.Let us briefly review the computation in the SU(2) case, to later introduce the symmetryreduction. The number of conformal blocks of the SU(2)-WZW model2, given a set ofrepresentations P = {j1, j2, ..., jN}, can be computed in terms of the so-called fusion numbersN ril [14] asNP =∑riN r1j1j2Nr2r1j3...N jNrN−2jN−1 . (7)These N ril are the number of independent couplings between three primary fields, i.e. themultiplicity of the r-irreducible representation in the decomposition of the tensor productof the i and l representations [ji] ⊗ [jl] =⊕rNril[jr]. This expression is known as a fusionrule. NP is then the multiplicity of the SU(2) gauge invariant representation (j = 0) inthe decomposition of the tensor product⊗Ni=1[ji] of the representations in P . The usualway of computing NP is using the Verlinde formula [14] to obtain the fusion numbers. Butalternatively one can make use of the fact that the characters of the SU(2) irreduciblerepresentations, χi = sin [(ji + 1)θ]/ sin θ, satisfy the fusion rules χiχj =∑rNrijχr. Takinginto account that the characters form an orthonormal set with respect to the SU(2) scalarproduct, 〈χi|χj〉 = δij , one can obtain the number of conformal blocks just by projectingthe product of characters over the gauge invariant representationNP = 〈χj1...χjN |χ0〉 =∫ 2pi0dθπsin2 θN∏I=1sin [(jI + 1)θ]sin θ. (8)This expression is equivalent to the one obtained in [10] using the Verlinde formula; it givesrise to the same result for every set of punctures P.To implement, now, the symmetry breaking we have to restrict the representations in Pto a set of U(1) representations. This corresponds in the case of Chern-Simons theory to2 Notice that, though we are omitting the κ subindex, the group of the WZW theory is in fact the quantumgroup SU(2)κ. The κ dependence is implicit in the allowed sets of representations P .7performing a symmetry reduction locally at each puncture. It is known that each SU(2)irreducible representation j contains the direct sum of 2j + 1 U(1) representations eijθ ⊕ei(j−1)θ ⊕ ... ⊕ e−ijθ. One can make an explicit symmetry reduction by just choosing oneof the possible restrictions of SU(2) to U(1) which, as we saw above, are given by thehomomorphisms λp. This corresponds here to pick out a U(1) representation of the formeipθ ⊕ e−ipθ with some p ≤ j. The fact that we will be using these reducible representations,consisting of SU(2) elements as U(1) representatives, can be seen as a reminiscence fromthe fact that the U(1) freedom has its origin in the reduction from SU(2).Having implemented the symmetry reduction, let us compute the number of independentcouplings in this U(1)-reduced case. Of course, we are considering now U(1) invariant cou-plings, so we have to compute the multiplicity of the m = 0 irreducible U(1) representationin the direct sum decomposition of the tensor product of the representations involved. Asin the previous case, this can be done by using the characters of the representations and thefusion rules they satisfy. These characters can be expressed as η˜pI = eipIθ+e−ipIθ = 2 cos pIθ.Again, we can make use of the fact that the characters ηi of the U(1) irreducible represen-tations are orthonormal with respect to the standard scalar product in the circle. Then, thenumber we are looking for is given byNPU(1) = 〈η˜p1...η˜pN |η0〉 =12π∫ 2pi0dθN∏I2 cos pIθ , (9)where η0 = 1 is the character of the U(1) gauge invariant irreducible representation. Wecan see that this result is exactly the same as the one obtained for P ({nk}) in Eq. (5),coming from the U(1)-CS theory, just by identifying the pI with kI labels.IV. REMARKS AND CONCLUSIONSLet us put this result in context with the entropy counting. As explained above, in com-puting the entropy of a black hole within LQG, there are several contributions involved.Some of them are related with the LQG framework, like the computation of C by character-izing the black hole area spectrum, or R({nk}) due to the distinguishability of the puncturesoriginated in the quantization process. The term P ({nk}), however, is related with the CStheory on the horizon. Once one has introduced all the conditions imposed by the LQG8framework, what is left is the counting of states of a CS theory subject to some externalinputs. If there is any connection between this CS theory and a Conformal Field Theory,one should expect this CFT to reproduce precisely this term P ({nk}), subject to the sameexternal inputs. This is exactly what we observe here by identifying the pI and kI labels. Weare, thus, proposing a precise implementation of Witten’s analogy through this symmetryreduced counting that yields the expected result.From the physical point of view, the main change we are introducing, besides using theCS-CFT analogy, is to impose the isolated horizon boundary conditions at the quantumlevel, instead of doing it prior to the quantization process. This is a preliminary step in thedirection of introducing a quantum definition of isolated horizons.It is very interesting to observe that, as shown in [15], the contribution to entropy ofP ({nk}) has a linear growth with area, including a logarithmic correction. The R({nk})term, on the other hand, introduces very particular quantum effects, that are specific fromLQG. In particular, the stair-like behavior appearing at the Planck scale in the entropy-arearelation [7, 8], has its origin in this factor. Thus, the picture obtained here seems to becompatible with the proposed role of conformal symmetry on the first order behavior ofblack hole entropy.AcknowledgmentWe thank V. Aldaya, A. Ashtekar, J. A. de Azcarraga, R. Coquereaux and L. Freidel forinteresting discussions and suggestions. We specially thank F. Barbero and E. Villasenorfor many discussions, strong ecouragement, and a careful reading of the manuscript. I. A.thanks R. Wald for his kind hospitality at the University of Chicago where part of this workwas done. J. D. thanks A. Ashtekar for his kind hospitality at the IGC at Penn State duringthe realization of part of this work.This work was in part supported by Spanish grants FIS2008-06078-C03-02, FIS2008-01980and ESP2005-07714-C03-0, the NSF grant PHY0854743 and the Eberly research founds ofPenn State, and the NSF grant PHY04-56619 to the University of Chicago. I. A. and J. D.acknowledge financial support provided be the Spanish Ministry of Science and Education9under the FPU program.[1] J.D. Bekenstein, Phys. Rev. D 9, 3292-3300 (1974).[2] S. Carlip, Phys. Rev. Lett. 82, 2828 (1999); Phys. Rev. Lett. 99, 021301 (2007); Gen. Rel.Grav. 39, 1519 (2007).[3] T. Thiemann. Modern canonical quantum general relativity, CUP, Cambridge (2007).[4] C. Rovelli. Quantum Gravity, CUP, Cambridge (2004).[5] A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Phys. Rev. Lett. 80, 904 (1998); A. Ashtekar,J. Baez, K. Krasnov, Adv. Theor. Math. Phys. 4, 1 (2000).[6] K. A. Meissner, Class. Quantum Grav. 21, 5245 (2004).[7] A. Corichi, J. D´ıaz-Polo, E. Ferna´ndez-Borja, Class. Quantum Grav. 24, 243 (2007); Phys.Rev. Lett. 98, 181301 (2007).[8] I. Agullo, J.F. Barbero G., J. Diaz-Polo, E. F. Borja, E. J. S. Villasen˜or, Phys. Rev. Lett.100, 211301 (2008).[9] E. Witten, Commun. Math. Phys 121, 351 (1989).[10] R. Kaul and P. Majumdar, Phys. Lett. B 439, 267 (1998); Phys. Rev. Lett. 84 5255 (2000).[11] M. Domagala, J. Lewandowski, Class. Quantum Grav. 21, 5233 (2004).[12] J.F. Barbero G., E. J. S. Villasen˜or, Phys. Rev. D, 77, 121502(R) (2008).[13] M. Bojowald, H.A.. Kastrup, Class. Quantum Grav. 17, 3009 (2000).[14] P. Di Francesco, P. Mathieu and D. Senechal Conformal Field Theory. Springer, NewYork(1997).[15] I. Agullo, J.F. Barbero G., J. Diaz-Polo, E. F. Borja, E. J. S. Villasen˜or, in preparation.

Computing black hole entropy in loop quantum gravity from a conformal field theory perspective

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Computing Black Hole entropy in Loop Quantum Gravity from a Conformal
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Computing Black Hole entropy in Loop Quantum Gravity from a Conformal Field Theory perspective

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