Microscopic entropy of black holes : a two-dimensional approach M. Cadoni, Capri 2004 Abstract Two-dimensional gravity models allow in many situations to compute the microscopic entropy of black holes and black branes in higher dimensions. We present recent results achieved using this approach. The relevance of these results for the AdS/CFT correspondence (or more in general for the holographic principle) is also discussed.

Introduction The problem: Microscopic derivation of the Bekenstein-Hawking area law S=A/4 for black holes (BH) and black branes (BB) Solution of the problem relevant also for related topics Solution of the problem relevant also for related topics  Holographic principle ( Fundamental or Emergent?)  Information loss for black holes

Two different strategies 1. Detailed Knowledge of the fundamental dynamics (e.g string theory) controlling the BH microscopic degrees of freedom is necessary 2.Area law should have some explanation also at the level of the low-energy effective theory describing the BH Usually 1. uses non-perturbative solutions of string theory and some duality (e.g AdS/CFT) to map the BH into a weak-coupled system Nice example: Strominger and Vafa calculation of the entropy of the 5D extremal RN black hole. Bad features:  Rely on SUSY  Works only for BPS states (extremal BH and BB)

It is not easy to implement strategy 2. It is not easy to implement strategy 2.  Because of no-hair theorems it seems impossible to reproduce the huge degeneracy of BH states using low-energy gravity theory We can circumvent this difficulty using an effective 2D description of the BH or BB We can circumvent this difficulty using an effective 2D description of the BH or BB  For spherical symmetric BH and BB thermodynamics seems to be determined completely by the 2D (r,t) sections of the spacetime. Only spherical perturbations should enter in the entropy computation  2D gravity is a topological theory ( as 3D gravity)  Pure gauge degrees in the bulk become physical on the boundary  We can calculate the entropy of the 2D BH first finding a duality (e.g ADS/CFT) between bulk gravity and a boundary theory then counting states in the 1D boundary theory

Summary Summary 1. Microscopic entropy of 2D AdS BH 2. Microscopic entropy of non-dilatonic Black branes a) Thermodynamics of non-dilatonic BB b) Dimensional reduction c) Statistical entropy of the BB 3. Conclusions

1.Microscopic entropy of 2D AdS BH The simplest 2D gravity model allowing for AdS solutions is The simplest 2D gravity model allowing for AdS solutions is The BH solutions are (M= BH mass )

Thermodynamical entropy of the BH is To compute the microscopic entropy of the 2D BH we use a method that works for 3D ( Brown-Hennaux, Strominger) and 2D ( M.C. and S. Mignemi) BH: BH entropy is computed by considering the deformation algebra generated on the 1D boundary of AdS by the action of the 2D bulk diffeos

 This defines a CFT 1 living on the boundary of AdS 2  Can be considered as a AdS 2 /CFT 1 correspondence In more detail 1.Choose suitable boundary conditions for the metric (r=  ) 2.Identify the group of asymptotic symmetries (ASG) preserving the r=  behaviour of the metric 3.Show that the ASG is generated by a Virasoro algebra with (possible) central extension

4. Compute the central charge c using a canonical realisation of the ASG (Regge, Teitelboim): H is the Hamiltonian,  are the Killing vector associated with the ASG and J a surface term (charge) 5. Use the Cardy formula To compute the entropy of the CFT 1

The orthogonality problem (the bundary of AdS 2 is a point!) can be solved introducing time-integrated charges  Going through the various steps of the calculations one finds perfect agreement between the entropy of the CFT 1 and the thermodynamical entropy of the AdS BH Origin of the non-vanishing central charge: breaking of the conformal symmetry of the ASG due to a non constant scalar field :

The method can be used to calculate the entropy of 4D (or higher-D) BHs, which admit AdS 2 as effective description  Example : The BH (zero branes) solutions of the model in the near-horizon regime One can also identify the boundary conformal theory (M.C., P. Carta, D. Klemm, S. Mignemi): DFF conformal mechanics with external source  :

2. Microscopic entropy of non-dilatonic Branes The 2D approach can be used to calculate the entropy of non-dilatonic p-branes of SUGRA theories (M. C., Class. Quant. Grav. 21(2004)251, M.C. and N. Serra, hep- th/ )  is the dilaton, F n is the RR field strength of a n-1-form potential, n=p+2

We consider only non dilatonic branes 1. M-branes (2-brane and 5-brane in D=11) (no dilaton) 2. Self dual dyonic branes ( 1-brane in D=6 and 3- brane in D=10) (constant dilaton) Non dilatonic branes play a crucial role in the AdS p+2 /CFT p+1 correspondence  In the extremal limit the near-horizon geometry of the p-brane becomes AdS p+2  S D-p-2

 Maldacena conjecture Duality between type II string theory on AdS p+2  S D-p-2 and CFT p+1 Most of the progress about the string/CFT duality has come from comparing the two theories at zero temperature. For the gravity side this means extremal branes Finite temperature effects are important both for testing the duality and for discussing the thermodynamics of the brane from a microscopic point of view  They can be discussed considering the near-extremal brane

Excitations near extremality break conformal invariance of the AdS p+2 background  the brane acquires finite temperature We are interested in the near-horizon, near-extremal regime: r  0, u,E=fixed (E= energy above extremality) Using Bekenstein-Hawking area law we get the entropy as a function of the brane volume V and temperature T

N= number of coincident p-branes (related with the RR charge) In the spirit of the AdS/CFT correspondence S brane should be matched by the entropy of the dual field theory at finite temperature Klebanov et al tried to do this using a gas of weak interacting brane excitation. The scaling with V and T was the right one but the coefficients a p turned out to depend on n, the number of fields of the model

For the 3- and 1-brane the AdS/CFT duality allows an identification of n  For p=3 the dual CFT is N=4, U(N) SYM. This allows the identification n=8N 2. We have still a mismatch of a 3/4 factor, which is only qualitatively understood  gauge theory computation performed at weak ‘t Hooft coupling, gravity description holds at strong coupling  For p=1 brane becomes the BTZ BH times a 3- sphere, the dual theory is a 2D CFT, whose entropy has been calculated by Strominger. We get n=2N 2, matching exactly S brane with S CFT

For the two M-branes the situation is more involved  In this case AdS/CFT is poorly understood  It is required n ~ N 3/2 and n ~ N 3 (for p=2,5 respectively), hard to achieve with a field theory  Despite some progress using D-brane anti-D-brane systems (Danielssons et al) remains a puzzling point Let us use the 2D approach

Dimensional reduction As a first step we performe the dimensional reduction D  2 in the D-dimensional gravity action. We use the ansatz The scalar field  parametrizes the volume W of the brane embedded in the p+2-dim spacetime, W=  V The 2D, dimensionally reduced gravity model, is

k and  are function of the brane parameters N,V and of the radius of AdS R p k and  are function of the brane parameters N,V and of the radius of AdS R p The model has 2D BH solutions which are asymptotically AdS The model has 2D BH solutions which are asymptotically AdS The mass and entropy of the BH are

The 2D gravity model gives an effective description of the spherical excitations of the near-extremal, near- horizon brane solution. After identification of the integration constant A:  2D sections of the brane solutions match the 2D BH solution  The thermodynamical parameters of the BH reproduce exactly those of the near-extremal brane with T BH =T brane, S BH =S brane, M BH = E ( energy of excitations above extremality)

Statistical entropy Let us now compute the entropy of the 2D BH (or equivalently the entropy of the boundary CFT) using the canonical realization of the ASG In principle we just need to repeat the calculations of the pure AdS BH for the 2D model under consideration Problem: we get divergent charges  need a renormalization procedure To separate a finite from a divergent part in the charges we change the radial coordinate

 is an arbitrary dimensionless renormalisation parameter  the final result (entropy) will depend on this parameter We can now go through the various steps of the calculations : 1. Define appropriate boundary conditions for the metric 2. Identify the ASG 3. Show that it is generate by a Virasoro algebra 4. Compute the associated charges J The charges J are divergent, we can eliminate its divergent part subtracting the contribution of the AdS background (A=0)

We can now define renormalized charges Using J R in the canonical realization of the ASG we find the central charge The central charge depends on the arbitrary dimensionless parameter . We expect  to be a rational number ( c is a rational function of the conformal weights of the boundary fields)

We choose for the renormalization parameter the value Using Cardy formula we find the entropy of boundary thermal CFT Expressing  0 and A in terms of the brane parameters T, V, N, we reproduce exactly (including the right factor a p ) the brane thermodynamical entropy S= a p VT p Notice: even though we leave the parameter  unfixed, we can reproduce exact scaling of S brane on V, T, N  S will be determined up to a dimensionless numerical factor

Conclusions  The main question raised at the beginning was: does the AdS/CFT correspondence survive finite temperature effects ( breaking of the conformal symmetry)?  Answering to this question may be crucial also for related problems (e.g. use of the correspondence for describing the non-perturbative phase of QCD)  Although the results for the entropy for the 1- and the 3-brane (AdS 3 /CFT 2 and AdS 5 /CFT4) give some hope that the answer may be yes, the results for the 2- and 5 brane indicate that this in general may not be the case

There is also a general scaling argument that points against a yes answer:  If the dual theory is ( at least in some regime) a weak coupled field theory, the energy E(S,V) must have an extensive (or sub-extensive if it has a Casimir part) behaviour under S  S, V  V  This scaling behaviour can be achieved only if we interpret V as the intrinsic volume of the brane, which it is OK for the extremal brane but unnatural for the BB (V should be taken as the volume of the brane embedded in AdS)

On the other hand the results our 2D approach points in a completely different direction Finite temperature effects in higher dimensional AdS/CFT dualities can be described by an effective AdS 2 /CFT 1 duality endowed with a scalar field that breaks the conformal symmetry and produces a non- vanishing central charge in the conformal algebra Moreover, the thermodynamics of the 2D models has a nice scaling behaviour and can be classified in terms of t violation of Eulero identity (extensivity) E=TS, through E=kTS From this point of view the holographic principle looses any fundamental character - becomes emergent- due to a fundamental feature of the gravitational interaction: its non- extensivity