Aïda Feddaoui
email: aida-feddaoui@etud.univ-tln.fr

Nicolas Boizot
email: boizot@univ-tln.fr

Eric Busvelle
email: busvelle@univ-tln.fr

Vincent Hugel
email: vincent.hugel@univ-tln.fr

¶ A Feddaoui

A Kalman Filter for Linear Continuous-discrete Systems with Asynchronous Measurements

come

A Kalman filter for linear continuous-discrete systems with asynchronous measurements

Aïda Feddaoui, Nicolas Boizot, Eric Busvelle, Vincent

INTRODUCTION

This paper focuses on observers for linear sampled-data systems, i.e. control systems having continuous state dynamics and a discrete measurement procedure -see e.g. [START_REF] Ichikawa | Linear Time Varying Systems and Sampled-data Systems[END_REF]. In particular, we address the case where only parts of the output vector is available at a given sample time. This situation arises when the output vector is obtained through several sensors that do not have the same (possibly non-uniform) sampling rate. Such systems, also known as multi-rate sampled-data systems, or asynchronous continuousdiscrete systems are often met in practice, for instance in global positioning problems, as in [START_REF] Mayhew | Multirate sensor fusion for gps using kalman filtering, fuzzy methods, and map matching[END_REF], or in the field of drone control [START_REF] Bristeau | Hardware and software architecture for state estimation on an experimental low-cost small-scaled helicopter[END_REF]. Likewise, one can be confronted with such asynchronous systems in the field of submarine robotics, as can be seen from [START_REF] Barbalata | An adaptive controller for au-tonomous underwater vehicles[END_REF], [START_REF] Maalouf | L1 adaptive depth and pitch control of an underwater vehicle with real-time experiments[END_REF] and [START_REF] Blain | Implementation of a ROV navigation system using acoustic/doppler sensors and kalman filtering[END_REF].

Since we are considering systems having a non-uniform, multi-rate measurement procedure, and possibly a time-varying behaviour, the continuous-discrete framework and the Kalman filter appear as an adequate tool. Indeed, the correction gain of the Kalman filter is constantly updated, which appears preferable to observers having a fixed, offline computed, correction gain. Using a Kalman filter to perform such a task might not appear as a new idea, as it can be seen in e.g. [START_REF] Bristeau | Hardware and software architecture for state estimation on an experimental low-cost small-scaled helicopter[END_REF][START_REF] Durieu | Multisensory data fusion for localisation in mobile robotics[END_REF][START_REF] Mayhew | Multirate sensor fusion for gps using kalman filtering, fuzzy methods, and map matching[END_REF]. However, the present paper's contribution lies in the fact that we propose a mathematical framework that allows us to state the problem -and thus write the observer's equations-and actually prove its convergence in the deterministic setting.

The main difficulties are, on the one hand, dealing with several subdivisions of time in order to represent the asynchronous outputs, and on the other hand, proving that the observer's Riccati equation is bounded over time. This latter issue is handled by following the ideas developed in [START_REF] Boizot | On the stability of a differential riccati equation for continuous-discrete observers[END_REF], where only the synchronous setting is considered. Moreover, the present work is done with the purpose of expanding the results to the nonlinear setting (see e.g. [START_REF] Thach | Continuous-discrete time observer design for lipschitz systems with sampled measurements[END_REF][START_REF] Andrieu | Selftriggered continuous discrete observer with updated sampling period[END_REF]) and high-gain formalism (e.g. [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF][START_REF] Boizot | Adaptive-gain extended kalman filter: Extension to the continuous-discrete case[END_REF][START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF]).

Two recent articles are also worth mentioning, in which a similar task is performed, in the linear framework for [START_REF] Moarref | Observer design for linear multi-rate sampleddata systems[END_REF], and in the nonlinear, high-gain setting for [START_REF] Shen | Continuous observer design for a class of multi-output nonlinear systems with multi-rate sampled and delayed output measurements[END_REF]. There are two main differences between the present approach and those two works. First, they consider observers having a fixed, off-line computed, correction gain, or in other word, Luenberger-like observers. Second, the two proposed observers are set in the pure continuous setting, whereas we consider a continuous-discrete setting where the correction step is performed only when measurements are available.

The remaining of the article is divided into four parts. Section 2 presents the control system under consideration and the corresponding Kalman filter. Section 3 deals with the proof of the exponential convergence of this observer. This demonstration heavily relies on the existence of bounds for the Riccati matrix. The proof of this result is quite technical and long, and is therefore not detailed in this article. It basically follows the ideas developed in [START_REF] Boizot | On the stability of a differential riccati equation for continuous-discrete observers[END_REF], with an increased complexity coming from the asynchronicity of the measurements. However, this paper adresses the important issue of the preservation of observability under multi-rate sampling, which is a necessary condition in order to bound the Riccati matrix. An academic illustrative example is given in Section 4, and Section 5 concludes the article.

DEFINITION OF THE MULTI-RATE KALMAN FIL-TER

Throughout this article, we define a time subdivision {t k } k∈N as a strictly increasing sequence of real numbers with t 0 = 0 and t k → ∞ when k → ∞. We denote Id the identity matrix with appropriate dimensions, |.| is the cardinal of a set, and M n×m is the set of (n × m) matrices.

System Under Consideration

Let ẋ(τ) = A(τ)x(τ) + B(τ)u(τ), with x(0) = x 0 y(τ) = C(τ)x(τ) (Σ c )
be a linear continuous system where

τ ∈ R + , x(τ) ∈ R n , y(τ) ∈ R ny , u(τ) ∈ R nu and A ∈ M n×n , B ∈ M n×nu , C ∈ M ny×n .
We assume that (Σ c ) is observable in a classic sense (see e.g. [START_REF] Sontag | Mathematical Control Theory[END_REF]), that the elements of A(τ), B(τ) and C(τ) are in L ∞ (R + , R) and are uniformly bounded by a constant B > 0. Moreover, we suppose that all the elements of C(τ) are derivable at least once and have their derivative bounded over time1 .

To this plant, we associate the following linear continuous-discrete system with asynchronous measurements

ẋ(τ) = A(τ)x(τ) + B(τ)u(τ), with x(0) = x 0 y k = C σ k x k (Σ k cda )
where,

• to each element of y(τ), denoted2 y (i) (τ) for i ∈ {1, ..., ny}, we associate a time subdivision {s

(i) k } k∈N ; • the subdivision of time {τ k } k∈N is defined by {τ k } k∈N := ny i=1 {s (i) k } k∈N ,
where elements belonging to several subdivisions {s (i) k } k∈N appear only once;

• σ k is a set, and l

(i)
k is an index defined as follows

σ k = {i ∈ {1, ..., ny}|∃ l (i) k ∈ N such that s (i) l (i) k = τ k }; • x k = x(τ k ) and y k is defined with the help of the matrix C σ k =     . . . C (i) k . . .     i∈σ k
, where

C (i)
k is the i th row of C(τ k ), and therefore y

(i) k := C (i) k x k .
In other words, C σ k is made of the rows of C(τ k ) corresponding to the outputs measured at time τ k , and

y k ∈ R |σ k | .

Kalman Filter

The continuous-discrete asynchronous Kalman filter is defined in two parts for all k ∈ N:

1. two prediction equations defined for τ ∈ [τ k-1 , τ k [, with initial values xk-1 (+)
and S k-1 (+),

2. two correction equations at time τ k .

We denote by

• x(t) the estimated state for all τ ∈ [τ k-1 , τ k [,
• xk (-) the estimated state at the end of a prediction step and before a correction step,

• xk (+) is the estimated state after a correction step. Note that xk (+) is also the initial estimated state of a new prediction interval [τ k , τ k+1 [.

Prediction equations

ẋ(τ) = A(τ) x(τ) + B(τ)u(τ) Ṡ(τ) = -A(τ) S(τ) -S(τ)A(τ) -(SQS)(τ) (K 1 )
Correction equations

               xk (+) = xk (-) -S k (+) -1 ∑ i∈σ k C (i) k R (i) k -1 C (i) k xk (-) -y (i) k s (i) l (i) k -s (i) l (i) k -1 S k (+) = S k (-) + ∑ i∈σ k C (i) k R (i) k -1 C (i) k s (i) l (i) k -s (i) l (i) k -1 (K 2 )
In other words, the corrections are made with respect to each output y

(i)
k actually available at time τ k associated to a weighting factor equal to the time elapsed since the last time this specific output was measured.

We assume that Q and R are symmetric positive definite matrices, in particular R is a diagonal matrix 3 , and both are in compact subsets of the form:

q Id ≤ Q ≤ q Id , with 0 < q < q r i ≤ R (i) k ≤ r i , with 0 < r i < r i with R (i)
k being the i th diagonal component of R. Furthermore, we assume as well that S(0) is a symmetric positive definite matrix chosen inside a compact subset.

Remark 1 Let us note that this formulation is based on the classic continuous-discrete framework (see e.g. [START_REF] Deza | High-gain estimation for nonlinear systems[END_REF][START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF][START_REF] Jazwinski | Stochastic Processus and Filtered Theory[END_REF]). It is also particularly adapted to the problem here addressed.

The proof of convergence of this observer is quite immediate after proving Lemma 2 below, which gives positive lower and upper bounds for the matrix S(τ). It is actually close to the Kalman filter's convergence proof for continuous-discrete systems with synchronous outputs, which can be found in [START_REF] Boizot | Adaptive-gain extended kalman filter: Extension to the continuous-discrete case[END_REF].

For the present case, the main difficulties are, on the one hand, to properly define the system under consideration and to deal with two types of time subdivisions :

{s (i)
k } k∈N for each output y (i) , i ∈ {1, ..., ny} and {τ k } k∈N . On the other hand, we have to demonstrate the following lemma:

Lemma 2 Consider the prediction-correction equations (K 1 ) and (K 2 ) with the set of assumptions on Q, R, S(0), A(τ) and C(τ) given above. Then, there exist constants µ i > 0, i ∈ {1, ..., ny}, and two scalars 0 < α < β , such that, for all subdivisions with s

(i) k -s (i) k-1 ≤ µ i , k ∈ N, αId ≤ S(τ) ≤ β Id
for all τ ∈ R + , before or after a correction step.

Here, α and β are independent from the chosen subdivisions. Also, the constants µ i are independent from the initial conditions.

Convergence of the Kalman Filter

We will prove here that the estimated state x(τ) calculated by the Kalman filter (K 1 )-(K 2 ) converges to x(τ), the state of (Σ k cda ). To this end, let the error signal be defined as ε(τ) = ( xx)(τ), and consider the Lyapunov function V (ε) = (ε Sε)(τ). We calculate:

ε(τ k ) = ( ẋ -ẋ)(τ) = A(τ)ε(τ) (1)
ε k (+) = xk (+) -x(τ k ) = xk (-) -x(τ k ) -S k (+) -1 ∑ i∈σ k C (i) k R (i) k -1 C (i) k xk (-) -y (i) k s (i) l (i) k -s (i) l (i) k -1 = ε k (-) -S k (+) -1 ∑ i∈σ k C (i) k R (i) k -1 C (i) k ε k (-) s (i) l (i) k -s (i) l (i) k -1
This last equation can be rewritten in a more compact form with the help of the following notations. First, let I σ k be the following diagonal matrix -with m k = |σ k |:

I σ k :=       s (i 1 ) l (i 1 ) k -s (i 1 ) l (i 1 ) k -1 0 . . . 0 s (i m k ) l (im k ) k -s (i m k ) l (im k ) k -1      
Second, the matrix R σ k is defined the same way as C σ k , and we denote R-1 σ k := R -1 σ k I σ k . This latter matrix is well defined since I σ k is clearly invertible from the definition of a time subdivision. The above equation thus can be rewritten:

ε k (+) = ε k (-) -S k (+) -1 C σ k R-1 σ k C σ k ε k (-). (2)
Now, we need to compute the derivative of V (ε) over time, and also the expression of

V (ε) after a correction step. For τ ∈ [τ k-1 , τ k [ : d dτ V (ε) = dε Sε dτ = ε Sε + ε ( Ṡε + S ε) = ε A Sε + ε (-A S -SA -SQS) + ε SAε = -ε SQSε.
At time τ k , we follow the ideas of [START_REF] Jazwinski | Stochastic Processus and Filtered Theory[END_REF]:

(ε Sε) k (+) = ε k (-) S k (+) -2C σ k R-1 σ k C σ k + C σ k R-1 σ k C σ k S k (+) -1 C σ k R-1 σ k C σ k ε k (-) (3)
= ε k (-) S k (-) -1 S k (+)S k (-) -1 -1 ε k (-) (4)
where the transition between (3) and ( 4) is due to (K 2 ) rewritten as:

S k (+) -S k (-) = C σ k R-1 σ k C σ k .
Still using (K 2 )

S k (-) -1 S k (+)S k (-) -1 = S k (-) -1 + S k (-) -1 C σ k R-1 σ k C σ k S k (-) -1 .
The matrix inversion lemma leads to

S k (-) -1 S k (+)S k (-) -1 -1 = S k (-) -1 + S k (-) -1 C σ k R-1 σ k C σ k S k (-) -1 -1 = S k (-) -C σ k Rσ k +C σ k S k (-) -1 C σ k -1 C σ k Thus        (ε Sε) k (+) = (ε Sε) k (-) -ε k (-)C σ k Rσ k +C σ k S k (-) -1 C σ k -1 C σ k ε k (-) dε Sε dτ (τ) = -(ε SQSε)(τ)
At time τ k , the second term of the right hand side expression of ( 5) is semi-positive definite, since S(τ) is positive definite for all τ ∈ R + by Lemma 2. Then

(ε Sε) k (+) ≤ (ε Sε) k (-).
Since dV dτ < 0 is decreasing over [τ k-1 , τ k [ for all k ∈ N, and also after a discrete step, we conclude that dV dτ < 0 for all τ ∈ R + . As a consequence, ε = ( xx)(τ) = 0 is an asymptotically stable point. Moreover, this convergence is exponential. Indeed, with the help of Lemma 2:

dε Sε dτ (τ) = -(ε SQSε)(τ) ≤ -α q(ε Sε)(τ) ε Sε(τ) ≤ (ε Sε)(0)e -α qτ ε(τ) 2 ≤ β α ε(0) 2 e -α qτ .

Stability of the Riccati Equation

The present section is dedicated to the proof of Lemma 2. It follows the ideas exposed in [START_REF] Boizot | On the stability of a differential riccati equation for continuous-discrete observers[END_REF] for synchronous continuous-discrete systems, and expands them to the multi-rate setting. For the sake of brevity, this quite long proof is not reproduced here. We only sketch the main ideas, and discuss the main technical differences between the classic continuous-discrete case and the multi-rate one.

As it is done in [START_REF] Boizot | On the stability of a differential riccati equation for continuous-discrete observers[END_REF] -but also in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] for continuous systems-the existence of bounds for the Riccati equation is demonstrated in two parts:

1. the proof of the existence of an upper bound; 2. the proof of the existence of a lower bound.

Each part, consists of first exhibiting a bound in small-time, and second, finding a long-time bound.

The small-time bounds are easily obtained with the help of Gronwall's lemma. However, they become useless as time increases, since they come from an exponential term that increases with time.

The argument used to prove the existence of a long-time bound varies depending on wether the upper bound or lower bound is considered.

In the first case, the key point is the regularity of the Riccati equation. It requires a limiting condition on the maximum step size of time subdivisions. However, at this stage, this condition only applies to the subdivision {τ k } k∈N .

In the second case, the long-time lower bound depends on whether or not observability is preserved under multi-rate sampling. For this purpose, we prove that the Gramm matrix of (Σ k cda ) is always positive definite, provided that each subdivision {s (i) k } k∈N , i ∈ {1, ..., ny}, has a small enough maximum step size.

In the remaining part of this section, we discuss the following key point mentioned in the previous paragraph:

Let G cda (T ) denote the Gramm matrix of the system (Σ k cda ), associated to the fixed time T > 0.

If all the subdivisions {s (i)

k } k∈N , i ∈ {1, ..., ny}, have a small enough maximum step size, then there exists a > 0 such that aId < G cda (T ).

1. First of all, the Gramm matrix at time T , G cda (T ) is defined as

G cda (T ) = ny ∑ i=1 λ (i) k ∑ j=1 ϕ a (s (i) j , T )C (i) j C (i) j ϕ a (s (i) j , T )(s (i) j -s (i) j-1 ) = ny ∑ i=1 G (i) cda (T )
where

• the index λ (i)
k is such that:

λ (i) k = max l ∈ N such that s (i) l ≤ T ; • ϕ a (t, s) is the resolvent of (Σ c ), with ϕ a (s, s) = Id.
2. Second, let us state the following lemma about continuous systems.

Lemma 3 ([11], Ch. 6) Let G c (T ) be the continuous Grammian of (Σ c ) at time T : G c (T ) = T 0 ϕ a (v, T )C (v)C(v)ϕ a (v, T )dv.
If the elements of A(τ) and C(τ) are uniformly bounded in L ∞ (R + , R) by some constant B > 0, then there exist two positive scalars 0 < ᾱ < β , that depend on B and T only, such that

ᾱId ≤ G c (T ) ≤ β Id.
Remark 4 Notice that by the definition of the matrix product,

G c (T ) = ny ∑ i=1 G (i) c (T ) = ny ∑ j=1 T 0 ϕ a (v, T )C (i) (v)C (i) (v)ϕ a (v, T )dv
In order to use this lemma, let us write:

G c (T ) = G c (T ) -G cda (T ) + G cda (T ). (5)
Now, we want to combine Lemma 3 with the fact that G c (T ) -G cda (T ) can be made small enough. This is done with the help of the following lemma, which can be proved using similar arguments as the proof of Lemma 3.11 in [START_REF] Boizot | On the stability of a differential riccati equation for continuous-discrete observers[END_REF].

Lemma 5 Let m(t), t ∈ [0, T ], be a (n × n) symmetric matrix, at least differentiable once.

Let µ be a positive constant, and {t k } k∈{0,1,...,k} an arbitrary finite subdivision of [0, T ], such that t jt j-1 ≤ µ for all j ∈ N, with t 0 = 0 and t k , the maximal element of this subdivision, is such that Tt k ≤ µ.

We suppose that all the coefficients of m have their derivative bounded over time.

Then T 0 m(v)dv - k ∑ j=1 m(t j ) t j -t j-1 ≤ µ (KT + L) Id,
where L = sup t m(t) 2 , with . 2 the matrix norm induced by the euclidean norm and K = n 2 max k,l,t m k,l (t) , with m k,l (t) the element of the k th row and l th column of the matrix m (t).

3. By lemma 5, there are constants L > 0 and K i > 0 such that for all i ∈ {1, ..., ny} , we have at time T :

G (i) c (T ) -G (i) cda (T ) = T 0 ϕ a (v, T )C (i) (v)C (i) (v)ϕ a (v, T )dv - λ (i) k ∑ j=1 ϕ a (s (i) j , T )C (i) j C (i) j ϕ a (s (i) j , T )(s (i) j -s (i) j-1 ) ≤ µ i (K i T + L)Id (6)
Let us apply Lemma 3 on G c (T ), using the expression (5) and the inequality [START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF] :

αId ≤ G c (T ) = ny ∑ i=1 G (i) c (T ) - ny ∑ i=1 G (i) cda (T ) + ny ∑ i=1 G (i) cda (T ) ≤ ny ∑ i=1 G (i) cda (T ) + ny ∑ i=1 µ i (K i T + L)Id
It allows us to conclude that:

ᾱ - n y ∑ i=1 µ i (K i T + L) Id ≤ G cda (T ).
Finally, by choosing each µ i , i ∈ {1, ..., ny} sufficiently small to make ᾱ -∑ n y i=1 µ i (K i T + L) positive, then, the linear continuous-discrete system with multi-rate outputs (Σ k cda ) associated to (Σ c ) is observable as soon as it is the case for (Σ c ).

ILLUSTRATIVE EXAMPLE

The behaviour of the Kalman filter defined in (K 1 )-(K 2 ) is illustrated with the following autonomous time-varying system, where x(τ) ∈ R4 :

ẋ(τ) =     -1 1 0 0 0 cos(t) 0 sin(t) 0 0 -1 1 0 -sin(t) 0 cos(t)     x(τ) (7) = A(τ)x(τ).
This system is an academic, time-varying, constantly oscillating system, as it is illustrated in Figure 1. For simplicity, we do not consider control inputs since, as it is well known from linear systems control theory, they do not play a critical role.

The output vector y(τ) ∈ R 2 is given by:

y(τ) = 1 0 0 0 0 0 1 0 x(τ) (8)
= Cx(τ).

According to [START_REF] Sontag | Mathematical Control Theory[END_REF][START_REF] Shieh | Transformations of a class of time varying multivariable control systems to block companion forms[END_REF] it is immediate to check that the continuous system made of ( 7)-( 8) is observable 4 . However, observability is lost as soon as C(τ) is restricted to only one of its two lines.

Figure 2: A dot represents a measured output. For the run here displayed, the maximum step sizes are, 0.48 for {s

k }, 0.37 for {s

k } and 0.37 for {τ k }. In this example, the two outputs y (1) (τ) = C (1) x(τ) and y (2) (τ) = C (2) x(τ) are measured using non-uniform subdivisions of time. Each one of those two subdivisions are obtained by randomly modifying a uniform one -starting with an initial period of 0.3 time units for y (1) (t), and 0.25 time units for y (2) (t). This is illustrated in Figure 2 where the three subdivisions {s

(1) k }, {s (2)
k } and {τ k } together with the sets σ k clearly appear. The output trajectories are displayed in Figure 3.

We performed a simulation of the proposed observer following the procedure below.

The initial state of the system is x(0) = (1, -1, 1, -1). The initial state of the observer is set to x(0) = (-3, 4, -5, 6). The matrices Q and R, are simply equal to identity.

In order to select a good initial value for the Riccati matrix, we first run the observer with S(0) = Id, while forcing the error term to 0 in the correction equations. This procedure provides a stabilised Riccati matrix, which is used as S(0) to run our experiment.

Finally the results of this simple simulation are displayed in Figure 4.

CONCLUSIONS AND FUTURE WORKS

In this paper, an adaptation of the Kalman filter to continuous-discrete systems with multi-rate sampled outputs has been proposed. To this end, each correction step involves a weighted sum of the output errors calculated on the basis of the measurements available at this sample time. Moreover, we showed the observability of (Σ k cda ) provided that (Σ c ), the associated continuous system, is observable. Some improvements are left for the future. To begin with, we plan to consider ns the number of sensors instead of ny the number of outputs. Two different outputs correspond to the same sensor if their associated subdivisions of time are the same. By doing so, larger R (i) sub-matrices can be considered and cross-correlations between outputs delivered by a same sensor can be taken into account.

Furthermore, the presence of redundant sensors can lead to an improved version of Lemma 2. Indeed, the maximum step size condition on the time subdivision of a given sensor could be relaxed provided there is an active redundant sensor.

Figure 1 :Remark 6

 16 Figure 1: For proper display purposes, the other graphs are shown on smaller time windows.

k

  } k∈N , i ∈ {1, ..., ny} in (K 2 ). It is unfortunately not the case. Actually, considering the subdivision {τ k } k∈N instead of {s

k

  } k∈N leads, when computing iteratively S k (+), either to overlapping intervals of time or disjointed intervals of time.

Figure 3 :

 3 Figure 3: Measured outputs (continuous outputs are shown for information).

Figure 4 :

 4 Figure 4: Convergence of the estimated state toward the real state.

This condition may appear restrictive but is necessary in order to apply Lemma (5) to a time-varying matrix C.

This notation is not ambiguous since we only use order 1 derivatives.

In fact, this hypothesis can be relaxed when several outputs come from the same sensor. This topic is raised in the conclusion.

In particular, following the notations of[START_REF] Shieh | Transformations of a class of time varying multivariable control systems to block companion forms[END_REF], it's observability index equals 2.

ACKNOWLEDGMENTS


Introduction

String theory has proven to be extremely successful in reproducing the entropy of supersymmetric black holes, such as the three-charge black hole in five dimensions [START_REF] Strominger | Microscopic origin of the Bekenstein-Hawking entropy[END_REF] and the four-charge black hole in four dimensions [START_REF] Johnson | Entropy of 4-D extremal black holes[END_REF][START_REF] Maldacena | Statistical entropy of four-dimensional extremal black holes[END_REF][START_REF] Maldacena | Black hole entropy in M theory[END_REF]. The entropies of these black holes scale like the square root of the product of their charges (or some some duality-invariant form thereof [START_REF] Kallosh | E(7) symmetric area of the black hole horizon[END_REF]), and in the microscopic counting this square root comes from using Cardy's formula to count the states of a certain 1+1 dimensional system of strings and branes.

However, in certain four-dimensional compactifications of string theory one can construct three-charge black holes whose entropy scales with these charges like S ∼ p1 p 2 p 3 2 3 [START_REF] Bueno | Quantum Black Holes in Type-IIA String Theory[END_REF]. The curvature at the horizon of these black holes is small, precisely as one would expect from the fact that their entropy grows like the square of the charge. These black holes cannot be constructed in "normal" four-dimensional supergravity, where the horizon curvature of three-charge black hole is Planckian, but exist if one adds to the supergravity Lagrangian certain terms coming from perturbative String-Theory α corrections at tree level in g S . Since these black holes do not have a regular limit when these correction terms are removed 1 they are called "quantum black holes."

The purpose of this letter is to try to understand the microscopic entropy of the type IIA quantum black holes constructed in [START_REF] Bueno | Quantum Black Holes in Type-IIA String Theory[END_REF]. The first step in this direction is to uplift these black hole solutions to eleven dimensions and to propose an M-theory interpretation in terms of three mutually-orthogonal intersecting stacks of M2-branes at the tip of a fourdimensional Gibbons-Hawking-like base. Since the α corrections to Type-IIA Calabi-Yau compactifications come from higher-derivative terms in String Theory [START_REF] Antoniadis | R**4 couplings in M and type II theories on Calabi-Yau spaces[END_REF][START_REF] Antoniadis | String loop corrections to the universal hypermultiplet[END_REF], the uplifted black hole will be a solution of the equations of motion of eleven-dimensional supergravity modified by the addition of certain higher-derivative terms.

The second step is to argue that if the Calabi-Yau manifold can be written as a (possibly singular) elliptic fibration, the branes that make the eleven-dimensional solution can be dualized to a configuration of two intersecting mutually-supersymmetric D3 branes carrying momentum along their common direction. By counting the possible way of carrying this momentum and by remembering that these D3 branes sit on top of a conical singularity that effectively enhances the central charge of the 1+1 dimensional theory on their worldvolume, we are able to reproduce the peculiar charge dependence, S ∼ p 1 p 2 p 3 2 3 , of the entropy of the quantum black holes.

In section 2 we introduce the effective theory corresponding to Type-IIA String Theory compactified on a Calabi-Yau manifold as well as the corresponding Type-IIA quantum black hole solutions. In section 3 we argue that the M-theory uplift of quantum black holes can be interpreted as arising from three stake of intersecting M2 branes and in section 4 we use this to propose a microscopic description of their entropy. Section 5 contains conclusions and future directions. In the appendices (A) and (B) we review the construction of Type-IIA quantum black holes as well as the H-FGK formalism [START_REF] Mohaupt | Non-extremal Black Holes, Harmonic Functions, and Attractor Equations[END_REF][START_REF] Mohaupt | The Hesse potential, the c-map and black hole solutions[END_REF][START_REF] Meessen | Non-Extremal Black Holes of N=2,d=5 Supergravity[END_REF][START_REF] Galli | Non-extremal black holes of N=2, d=4 supergravity[END_REF][START_REF] Galli | The Freudenthal gauge symmetry of the black holes of N=2,d=4 supergravity[END_REF][START_REF] Meessen | Black holes and black strings of N=2, d=5 supergravity in the H-FGK formalism[END_REF][START_REF] Galli | Black hole solutions of N=2, d=4 supergravity with a quantum correction[END_REF][START_REF] Bueno | Black holes and equivariant charge vectors in N=2,d=4 supergravity[END_REF][START_REF] Bueno | Non-perturbative black holes in Type-IIA String Theory vs. the No-Hair conjecture[END_REF] and the structure of N = 2 four-dimensional ungauged supergravity coupled to vector multiplets that were used in their construction.

2 Type-IIA Quantum Black Holes Quantum black holes [START_REF] Bueno | Quantum Black Holes in Type-IIA String Theory[END_REF] are solutions of the effective theory corresponding to Type-IIA String Theory compactified on a Calabi-Yau manifold in the presence of perturbative corrections to the Special Kähler geometry of the vector multiplet sector. Despite having only three charges, these four-dimensional supersymmetric black holes have a macroscopically large horizon area. It is not hard to see either from this or from their explicit construction that these black holes do not have a macroscopic horizon in the "classical" limit, when the perturbative corrections are turned off, which justifies calling them "quantum black holes".

Type-IIA String Theory on a Calabi-Yau manifold

Type-IIA String Theory compactified to four-dimensions on a Calabi-Yau three-fold, with Hodge numbers (h 1,1 , h2,1 ), is described, up to two derivatives, by a N = 2 fourdimensional supergravity coupled to vector-and hyper-multiplets. As explained in appendix (A), we are going to truncate the hyperscalars to a constant value, and therefore we will only be concerned about the vector-multiplet sector of the theory. The corresponding prepotential can be written as an infinite series around mz i → ∞ 2 , and it is given by [START_REF] Candelas | Moduli space of Calabi-Yau manifolds[END_REF][START_REF] Candelas | A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory[END_REF][START_REF] Candelas | An Exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds[END_REF][START_REF] Mohaupt | Black hole entropy, special geometry and strings[END_REF]

F = - 1 3! κ 0 ijk z i z j z k + ic 2 + i (2π) 3 {d i } n {d i } Li 3 e 2πid i z i , (2.1)
where z i , i = 1, ..., n v = h 1,1 , are the scalars in the vector multiplets, κ 0 ijk are the classical intersection numbers, d i ∈ Z + is a h 1,1 -dimensional summation index and Li 3 (x) is the third polylogarithmic function. The constant c is proportional to the Euler characteristic of the Calabi-Yau three-fold, χ multiplied by the Riemann zeta function: c ≡ χζ (3) (2π) 3 . The first two terms in the prepotential correspond to tree level and fourth-loop perturbative (which is the only non-vanishing one) contributions in the α -expansion, respectively [START_REF] Mohaupt | Black hole entropy, special geometry and strings[END_REF]

F P = - 1 3! κ 0 ijk z i z j z k + ic 2 , (2.2)
while the third term comes from non-perturbative corrections produced by world-sheet instantons. In this paper we will focus on large-volume compactifications where these corrections can be safely ignored, and hence focus on the quantum black holes obtained from the prepotential (2.2). The most general quantum black hole solutions, that are governed by the prepotential (2.1) have been constructed in [START_REF] Bueno | Non-perturbative black holes in Type-IIA String Theory vs. the No-Hair conjecture[END_REF].

Our starting point is the prepotential (2.2), which in homogeneous coordinates X Λ , Λ = (0, i), can be written as

F (X ) = - 1 3! κ 0 ijk X i X j X k X 0 + ic 2 X 0 2 . (2.3)
The scalars z i are given by

z i = X i X 0 . (2.4)
Adding the constant term c to the prepotential modifies the geometry of the scalar manifold, which is no longer homogeneous, and therefore it is said that the geometry has been corrected by quantum effects. The scalar geometry defined by (2.3) is hence the so-called quantum corrected d-SK geometry [START_REF] De Wit | Special geometry, cubic polynomials and homogeneous quaternionic spaces[END_REF][START_REF] De Wit | Symmetry structure of special geometries[END_REF]. The attractor points of (2.3) have been extensively studied in [START_REF] Bellucci | On Quantum Special Kaehler Geometry[END_REF].

The Type-IIA quantum black holes belong to a particular class of purely magnetic black hole solutions of the theory defined by (2.3). We review them in appendix (2) and refer the reader to [START_REF] Bueno | Quantum Black Holes in Type-IIA String Theory[END_REF][START_REF] Bueno | Non-perturbative black holes in Type-IIA String Theory vs. the No-Hair conjecture[END_REF] for more details.

The quantum-corrected STUmodel

Type-IIA quantum black holes exist in any Type-IIA Calabi-Yau compactification with h 1,1 > h 2,1 . To make contact to a description of these black holes in terms of intersecting branes we have to choose a particular model and the best candidate is the popular STU model, whose black hole solutions and attractors have been extensively studied. Since the hypermultiplets are truncated, the specific value of h 2,1 is irrelevant as long as it is smaller than h 1,1 . The values of h 1,1 and κ 0 ijk are

h 1,1 = 1 , κ 0 123 = 1 (2.5)
and therefore the prepotential (2.3) becomes

F (X ) = X 1 X 2 X 3 X 0 + ic 2 X 0 2 . (2.6)
The "normal" four-and five-dimensional black holes and attractors of this quantumcorrected STU model been previously considered in [7-10, 19, 28], but here we focus on the black holes that do not have a classical (c → 0) limit. The solution corresponding to these black holes (discussed in detail in Appendix B) has

e -2U = 3c 1 3 H 1 H 2 H 3 2/3 , (2.7
)

z i = ic 1 3 H i (H 1 H 2 H 3 ) 1/3 . (2.8)
The space-time metric is therefore

ds 2 4 = -3 -1 c -1 3 H 1 H 2 H 3 -2/3 dt 2 + 3c 1 3 H 1 H 2 H 3 2/3 d y 2 , (2.9)
where d y 2 = dy 1 2 + dy 2 2 + dy 3 2 is the Euclidean metric on R 3 . Since in the H-FGK formalism the H-variables correspond to the imaginary part of the covariantly holomorphic symplectic section appropriately weighted to be Kähler neutral, supersymmetry require them to be harmonic functions on the transverse space R 3 . A single-center black hole has then

H i = a i + p i √ 2 1 r , i = 1, 2, 3 , (2.10)
where r 2 = y 1 2 + y 2 2 + y 3 2 , the p i are the three charges of the black hole (A.12) and the a i are arbitrary constants that can be written in terms of the asymptotic value of the scalars at spatial infinity z i ∞ as

a i = -s p i Im z i ∞ √ 3c , (2.11)
where s p i is the sign of the charge p i . The entropy of this black hole is

S = 3c 1 3 2 π p 1 p 2 p 3 2/3 (2.12)
and its mass is the sum of the three charges:

M = 3c 2 a 2 a 3 p 1 + a 1 a 3 p 2 + a 1 a 2 p 3 . (2.13)
It is easy to see that each term that contributes to the mass is positive definite since Sign a i = Sign p i and a 1 a 2 a 3 > 0. In the next section we will try to describe this threecharge four-dimensional black hole in terms of intersecting branes in M-theory.

The M-theory configuration

In order to see whether Type-IIA quantum black holes have an interpretation via intersecting branes it is desirable to have the precise ten-dimensional configuration corresponding to the four-dimensional solution. For tree-level Type-IIA Calabi-Yau compactifications the map between the ten-dimensional and the four-dimensional fields is known [START_REF] Ferrara | Dimensional Reduction of Type II Superstrings[END_REF][START_REF] Cecotti | Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories[END_REF][START_REF] Ferrara | Quaternionic Manifolds for Type II Superstring Vacua of Calabi-Yau Spaces[END_REF][START_REF] Bodner | 2,2) vacuum configurations for type IIA superstrings: N=2 supergravity Lagrangians and algebraic geometry[END_REF], but since we are considering four-dimensional solutions to the prepotential that includes perturbative corrections in α at tree level in g S , and since these correction come from an R 4 -like term in ten dimensions, no explicit map is known. However, as we will show below, we will still be able to control the behavior of the dilaton and the Calabi-Yau volume, which will allow us to obtain the higher-dimensional configuration corresponding to Type-IIA quantum black holes.

The String Theory dilaton

As explained in appendix (A), for black hole solutions of ungauged four-dimensional supergravity, the hyperscalars are truncated to a constant value. In principle, the dilaton belongs to the universal hypermultiplet, and therefore one may naively conclude that it should be constant for every black hole solution. However, in the process of obtaining the effective N = 2 four-dimensional supergravity in its standard form (A.1), several rescalings and redefinitons are performed on the original ten-dimensional fields. In addition, since we are considering all the perturbative corrections to the Special Kähler sector, the corresponding N = 2 ungauged supergravity action is not the effective compactification theory of Type-IIA String Theory at tree level, so we should expect more intrincate redefinitions. Our purpose is to show now that Type-IIA quantum black holes have a constant ten-dimensional dilaton and a constant Calabi-Yau manifold volume, and we will do this in two steps:

Tree level: At tree level, the Type-IIA dilaton φ is related to the four-dimensional dilaton q as follows [START_REF] Gurrieri | N=2 and N=4 supergravities as compactifications from string theories in 10 dimensions[END_REF][START_REF] Grimm | The Effective action of type II Calabi-Yau orientifolds[END_REF] e -q = e -φ Vol 6 ,

where

Vol 6 = 1 6 J ∧ J ∧ J , (3.2)
is the volume of the Calabi-Yau manifold, J being the corresponding Kähler form. The Kähler potential of the N = 2 Special Kähler manifold is related to the volume of the compactification Calabi-Yau manifold by [START_REF] Gurrieri | N=2 and N=4 supergravities as compactifications from string theories in 10 dimensions[END_REF][START_REF] Grimm | The Effective action of type II Calabi-Yau orientifolds[END_REF] e

-K = e 3φ 2 6 J ∧ J ∧ J = e 3φ 2 Vol 6 . (3.3)
Since for Type-IIA quantum black holes both e -K and q are constants, this implies that both e -φ Vol 6 = const. and e

3φ 2 Vol 6 = const. (3.4)
and hence at tree level both the dilaton φ and the Calabi-Yau volume are constant.

Perturbative corrections: It is also easy to see that this tree level result is not changed when including perturbative corrections. Indeed, [START_REF] Antoniadis | R**4 couplings in M and type II theories on Calabi-Yau spaces[END_REF][START_REF] Antoniadis | String loop corrections to the universal hypermultiplet[END_REF] have shown that the loop corrections in ten dimensions that give rise to the perturbative corrections of the prepotential from the four-dimensional point of view, only mix the dilaton and the volume among themselves. Therefore, since they were constant at the tree level, they continue to be constant after the loop corrections have been taken into account. We thus conclude that for Type-IIA quantum black holes the dilaton and the volume of the Calabi-Yau manifold are constant.

M2-branes

In order to argue that the M-theory uplift of quantum black holes can be interpreted as coming from a superposition of M2 branes on a conical singularity it is useful to recall the usual supersymmetric solution corresponding to three stacks of M2 branes on a six-torus [START_REF] Tseytlin | Harmonic superpositions of M-branes[END_REF][START_REF] Gauntlett | Overlapping branes in M theory[END_REF]]

ds 2 = (H 1 H 2 H 3 ) 1 3 -(H 1 H 2 H 3 ) -1 dt 2 + H -1 1 dx 2 1 + dx 2 2 + H -1 3 dx 2 3 + dx 2 4 +H -1 2 dx 2 5 + dx 2 6 + g mn dy m dy n , (3.5)
where m, n = 1, . . . , 4 and the M2-branes are respectively located along the x 1 , x 2 , x 3 , x 4 and x 5 , x 6 directions. Supersymmetry requires the transverse metric, g mn , to be Hyper-Kähler and the M2 warp factors to be harmonic in this metric: ∆ y H i (y) = 0 , i = 1, 2, 3. To compare with the uplifted quantum black holes we will focus on solutions where the Hyper-Kähler space is a Gibbons-Hawking (Taub-NUT) space:

ds 2 GH = V (y) -1 dΨ + A i (y)dy i 2 + V (y)δ ij dy i dy j , i = 1, 2, 3 , (3.6)
where * dA(y) = dV (y) ⇒ ∆ y V (y) = 0 .

(3.7)

The eleven-dimensional metric is therefore given by

ds 2 = (H 1 H 2 H 3 ) 1 3 -(H 1 H 2 H 3 ) -1 dt 2 + H -1 1 dx 2 1 + dx 2 2 + H -1 3 dx 2 3 + dx 2 4 +H -1 2 dx 2 5 + dx 2 6 + ds 2 GH , (3.8)
where now the H i are harmonic in R 3 .

M2-branes beyond classical supergravity

In order to interpret Type-IIA quantum black-holes as composed of three stacks of orthogonally intersecting M2-branes, one should uplift their solution to M-theory. Since the four-dimensional theory where these black holes are constructed includes all the perturbative corrections in α , at tree level in g S , the uplifted black holes should be solutions of the standard eleven-dimensional supergravity to which one has added the higher-derivative terms that give rise to the 4D perturbative α -corrections.

There are two features of the quantum black hole metric (2.9) that will guide us to obtain this solution. The first is that the volume of the six-dimensional torus/Calabi-Yau manifold is constant to all levels in the corrections and hence, rescaling the time to t → 3c 1 3 t , the eleven-dimensional solution can be put into an M2-brane form:

ds 2 = (H 1 H 2 H 3 ) 1 3 -3c 1 3 H 1 H 2 H 3 -1 dt 2 + H -1 1 dx 2 1 + dx 2 2 + H -1 3 dx 2 3 + dx 2 4 +H -1 2 dx 2 5 + dx 2 6 + ds 2 (3.9)
The four-dimensional base metric ds 2 is no longer Gibbons-Hawking but becomes:

ds 2 = 3 -1 c -1 3 H 1 H 2 H 3 -1 3 dΨ 2 + 3c 1 3 H 1 H 2 H 3 1 3 δ ij dy i dy j , i = 1, 2, 3 . (3.10)
This metric is not Ricci flat and it does not even have constant curvature. This is to be expected, since the solution (3.9) solves the equations of motion of eleven-dimensional supergravity modified by appropriate higher curvature terms, which modify in turn the Gibbons-Hawking character of the base. Indeed, if one tries to compare this metric to a Gibbons-Hawking one, one finds that on one hand the gauge field A i (y), corresponding to D6 brane charge in ten dimensions, is zero, but that on the other hand the corresponding warp factor is not constant by rather has the form:

V (y) = 3c 1 3 H 1 H 2 H 3 1 3 , (3.11)
which has the same behavior at infinity and near the black hole as the warp factor of a Taub-NUT space with a nontrivial charge:

lim r→0 V (y) ∼ 1 r , (3.12)
where r = (y 1 ) 2 + (y 2 ) 2 + (y 3 ) 2 . Furthermore, when the three warp factors become equal the function V becomes harmonic throughout the space

∆ y V (y) = 0 , (3.13)
despite the absence of a Gibbons-Hawking (D6) charge. It important to notice that in the M-theory uplift the term ds 2 ΨΨ is constant, which is consistent with the ten-dimensional dilaton of the quantum black hole solution being constant. This is an nontrivial check that the eleven-dimensional brane configuration we propose gives the fundamental constituents of Type-IIA quantum black holes.

The microscopic entropy

Having obtained an eleven-dimensional metric that resembles that of three stacks of coincident M2 branes, we can easily compactify it along one of the torus directions to obtain a D2-D2-F1 metric, which upon a further T-duality along the F1 direction becomes a D3-D3-P metric, where the momentum P runs along the direction common to the two D3 branes. This duality chain transforms the quantum eleven-dimensional black hole into a type IIB D3-D3-P black hole, whose microscopic entropy can be reproduced straightforwardly by Strominger-Vafa-type arguments. Indeed, if the numbers of the two types of D3 branes are N 1 and N 2 and if N P units of momentum are running along the common directions of these branes, the most efficient way to carry this momentum when N 1 and N 2 are co-prime is to use the strings stretched between the two stacks of D3 branes. These "bi-fundamental" strings have a mass gap equal to

1 N 1 N 2 R
, where R is the radius of the common direction of the branes, and hence the entropy of the system comes from partitioning the N P units of momentum between modes that carry integer multiples of 1 N 1 N 2 , or otherwise from the number of integer partitions of N 1 N 2 N P . By taking into account the fact that there are four bosonic species of bi-fundamentals as well as their fermionic partners, this gives the entropy 2π √ N 1 N 2 N P . The argument above reproduces the entropy of a D3-D3-P black hole in five dimensions, which is sourced by a stack of branes in R 4 . One can write this R 4 as a Gibbons-Hawking space with V = 1/r and a nontrivial fiber satisfying dA = dV . If we focus instead on a stack of D3-D3-P branes in a Taub-NUT space with Kaluza-Klein monopole charge N , the corresponding warp factor is V = 1 + N r and we obtain a regular four-dimensional black hole with entropy 2π

√ N 1 N 2 N P N [2, 3
]. We would like to use a similar argument to explain microscopically the entropy of our quantum black holes. However, at first glance there are two problems with this: The first is that our base, (3.11), is no longer Gibbons-Hawking but has A = 0 and V = 3c

1 3 (H 1 H 2 H 3 ) 1/3
. Nevertheless, near the branes the warp factor behaves like that of a Gibbons-Hawking space. Hence, even if the branes do not sit on top of an A N singularity, they sit on some other conical singularity whose effect on the central charge of the D3-D3 CFT one can calculate.

The second problem is that a generic eleven-dimensional uplift of a quantum black hole will not be a six-torus but a more complicated CY manifold. The key ingredient needed to relate the quantum black holes to the D3-D3-P system is the presence of two U (1) isometries, one of which is used for reducing to a ten-dimensional Type IIA black hole, and the other for T-dualizing3 . This can be easily done for any CY manifold that has a T 2 fiber. Nevertheless, in order for our construction of quantum black holes to yield regular solution, this elliptic fibration must be singular: CY manifolds with regular fibration have zero Euler characteristic and hence c vanishes which makes the black hole horizon singular. This singularity can be cured by including non-perturbative α effects, again at tree-level in g S [START_REF] Bueno | Non-perturbative black holes in Type-IIA String Theory vs. the No-Hair conjecture[END_REF], but the resulting solutions involve the Lambert W function and are much harder to manage.

The way out is to focus on singular elliptic fibrations, which give CY manifolds with nonzero Euler number. The places where the fibration degenerates become seven-branes upon dualization to the type IIB duality frame. The presence of these seven-branes does not affect the entropy counting, because this entropy comes from D3-D3 strings carrying momentum, which do not see the seven-branes.

There are two ways to take into account the effect of the conical singularity on the entropy. The first is to compare this singularity with a Gibbons-Hawking solution, and determine its effective Gibbons-Hawking charge. The second is to focus on the near-horizon geometry of the black hole and to compute the corresponding Brown-Henneaux central charge [START_REF] Brown | Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity[END_REF], which determines how the central charge of the D3-D3 CFT increases when the D3 branes are placed on top of the conical singularity. As we will explain below, the two calculations are equivalent, but since the first is more intuitive we will present it here.

Near the tip of a Gibbons-Hawking metric

ds 2 GH = V (y) -1 dΨ + A i (y)dy i 2 + V (y) dr 2 + dΩ 2 (2) , i = 1, 2, 3 , (4.1)
with

V = 1 + N r (4.2) the metric becomes that of R 4 /Z N ds 2 GH ∼ dρ 2 + ρ 2 d Ω2 (3) , (4.3)
with ρ = 2 √ r and d Ω2

(3) the standard metric on S 3 /Z N . When the D3-D3 system is placed at the tip of this space its central charge increases by a factor of N . This is a well-known phenomenon for the D1-D5-P black hole in Taub-NUT [START_REF] Kutasov | String theory in magnetic monopole backgrounds[END_REF], and our system is just its Tdual. Now, given a conical metric of the type (4.3), there is a way to extract directly this factor:

N = V S 3 V S 3 /Z N , (4.4)
where V S 3 is the volume of the three-sphere S 3 and V S 3 /Z N is the volume of the S 3 /Z N , at the same radius. In the Brown-Henneaux formalism this ratio of the volumes also gives the decrease of the effective three-dimensional Newton's constant, and hence the increase of the central charge of the corresponding CFT.

For the quantum black hole metric we discussed in section 3.3, the base metric

ds 2 = V (r) -1 dΨ 2 + V (r) dr 2 + r 2 dΩ 2 (2) , V (y) = 3c 1 3 H 1 H 2 H 3 1 3 , i = 1, 2, 3 (4.5)
has a conical singularity in the near-tip region:

ds 2 ∼ √ 2r 3 (cp 1 p 2 p 3 ) 1/3 dΨ 2 + 3 cp 1 p 2 p 3 1/3 √ 2r dr 2 + r 2 dΩ 2 (2) . (4.6)
Upon defining (with hindsight)

N E ≡ √ 3 √ c p 1 p 2 p 3 1/3 √ 2 , (4.7)
and changing coordinates to (ρ = 2 √ N E r), the metric near the conical singularity becomes

ds 2 ∼ dρ 2 + ρ 2 dΨ 2 4N 2 E + dΩ 2 (2) 4 ≡ dρ 2 + ρ 2 dΩ transverse . (4.8)
Hence, the singularity will increase the central charge of the D3-D3-P CFT by a factor given by the ratio of the volume of S 3 and the volume of the transverse space, which is nothing but N E .

V S 3 V Ωtransverse = 2π 2 2π 2N E 4π 2 = N E . (4.9)
Hence, the microscopic entropy of the quantum black holes will be given by

S = 2π N 1 N 2 N P N E , (4.10)
where N 1 and N 2 are the numbers of D3 branes and N P is the number of momentum quanta. Since the supergravity charges are proportional to these numbers, it is clear that this microscopic entropy count reproduces the correct charge growth of the quantum black hole

S = 3 2 c 1 3 π p 1 p 2 p 3 2/3 , (4.11)
which is already an important confirmation that our strategy is correct. Of course, the ideal would be to find exactly the coefficients that relate the supergravity charges to the quantized ones, and therefore establish that the macroscopic and the microscopic entropies are identical. However, since we know neither the eleven-dimensional uplifts of the Maxwell fields of the four-dimensional quantum black hole, nor the volume of the Calabi-Yau three-fold and its submanifolds, we cannot determine these coefficients from first principles. However, what we can do is to use the symmetry of the STU model in order to argue that in the M2-M2-M2 duality frame the supergravity charges are related to the quantized brane numbers via the same proportionality constant, γ:

p1 = γN 1 , p2 = γN 2 , p3 = γN P , (4.12)
where we have defined pi ≡ p i √ 2 to ease the presentation. We can now ask what is the value of γ that makes the microscopic and the macroscopic entropies agree. Upon using (4.12), equation (4.11) becomes:

S = 2π 3 3 2 4 c 1 2 p1 p2 p3 3 1 2 c 1 6 (p 1 p2 p3 ) 1 3 = 2π 3 3 2 4 c 1 2 γ 3 N 1 N 2 N P N E (4.13)
and therefore the desired value of γ is:

γ ≡ 4 1 3 3 1 2 c -1 6 . (4.14)
There is a nontrivial check that this value satisfies: The number N E , which gives the increase of the CFT central charge is expected to be a natural number, at least for some values of N 1 , N 2 and N P . However, N E is defined in terms of c, and therefore contains both a factor of ζ(3) as well as square and cubic roots:

N E = √ 3 √ c p1 p2 p3 1/3 . (4.15)
Hence one may naively infer that N E can never be a natural number. However, it turns out that when expressing N E in terms of quantized charges using (4.14) both the ζ(3) and the √ 3 drop out:

N E = (4 N 1 N 2 N P ) 1/3 (4.16)
and therefore N E can easily be an integer for a suitable choice of N 1 , N 2 and N P . The fact that the transcendental number in c drops out of this relation is a nontrivial check of our proposed microscopic description. Indeed, in [START_REF] Behrndt | Subleading contributions from instanton corrections in N=2 supersymmetric black hole entropy[END_REF] it was argued that a black hole entropy expression that contains such a transcendental number can never be reproduced from microscopic calculations, and our microscopic proposal evades this by absorbing this transcendental number into the central charge increase of the underlying CFT.

The puzzling aspect of equation (4.16) is that it makes the "classical" limit very hard to see. Indeed, as we explained in section (2) if one turns off the quantum corrections (c → 0) while keeping the supergravity charges of the black hole constant, the horizon becomes singular. This is reflected in our construction by the fact that our black hole has three charges and, when c → 0, the warping of the base space becomes trivial and the corresponding entropy becomes that of the three-charge system in four dimensions, which does not give rise to a macroscopic horizon.

However, one can also ask what happens if one takes the c → 0 limit while keeping the quantized black hole charges, N 1 , N 2 and N P constant. This does not affect N E , so it looks like in this limit the black hole entropy remains macroscopic. This is consistent with the fact that in this limit the four-dimensional supergravity charges blow up (4.14), but the factors of c cancel from the expression of the near-horizon limit of the metric (2.9), which remains a regular AdS 2 × S 2 . On the other hand, the expressions of the four-dimensional moduli diverge, and therefore it appears that in this limit the dictionary between ten-and four-dimensional solutions breaks down. It would clearly be interesting to try to derive equation (4.14) from first principles to see precisely how this breakdown occurs.

Conclusions

We have constructed an eleven-dimensional metric (3.9) that upon dimensional reduction gives the Type-IIA quantum black hole of the STU quantum-corrected model. Because the four-dimensional metric has constant dilaton and CY volume, this eleven-dimensional uplift can be interpretation as arising from three stacks of orthogonal M2 branes that sit at the apex of a cone in a four-dimensional transverse space. Because of the presence of correction terms in the Lagrangian, this space is not Gibbons-Hawking, although it has exactly the same kind of warping as a Gibbons-Hawking space. The strength of the conical singularity is proportional to the cubic root of the product of the three M2 charges When the dix-dimensional internal space of the compactification has a T 2 fiber we can dualize this solution to an asymptotically four-dimensional three-charge D3-D3-P solution in type IIB string theory. In flat space the microscopic entropy of this system is not enough to give rise to a regular horizon, but we have shown that the conical singularity enhances the central charge of this system, and the resulting microscopic entropy reproduces the entropy of the of the quantum black hole: S = 3c 2 π p 1 p 2 p 3 2/3 up to an overall coefficient which we could not determine. We have however been able to show that if this coefficient is such that the entropies match, a certain dependence of the entropy on transcendental numbers drops out, which we believe is a nontrivial check of our proposal.

Since we are working in a supergravity theory in the presence of quantum corrections to the geometry of the scalar manifold, our proposed microscopic description is not at the same level of rigor as the usual three-and four-charge black hole entropy counting. However, the fact that we have found a brane interpretation that reproduces the highly-unusual charge dependence of the entropy of quantum black holes makes us confident that we have identified the correct microscopic framework for understanding the entropy of all Type-IIA quantum black holes, which remains as an important open problem in String Theory.

RFP3-1321 to the Foundational Questions Institute.The work of CS was supported by the ERC Starting Independent Researcher Grant 259133, ObservableString.

A N = 2 four-dimensional supergravity and the H-FGK formalism Type-IIA quantum black holes are black hole solutions of Type-IIA String Theory compactified down to four dimensions on a Calabi-Yau three-fold, which is described, up to two derivatives, by a N = 2 four-dimensional ungauged supergravity. Therefore, it is convenient to review the basic formulation of the theory and its vector multiplet sector, since the hypermultiplets and the fermions can be always truncated for black hole solutions. The bosonic sector of any N = 2 four-dimensional supergravity coupled to vector multiplets can be written as follows [START_REF] De Wit | Lagrangians of N=2 Supergravity -Matter Systems[END_REF][START_REF] Andrianopoli | N=2 supergravity and N=2 superYang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map[END_REF]

S = d 4 x |g| R + G i j (z, z)∂ µ z i ∂ µ zj + 2I ΛΣ (z, z)F Λ µν F Σ µν -2R ΛΣ (z, z)F Λ µν F Σ µν (A.
1) The z i (i = 1, ...n v ) denote the n v complex scalar fields of the vector multiplets, which parametrize an n v -dimensional Special Kähler manifold with Kähler metric G i j (z, z). F Λ = dA Λ denote the field strengths of the Λ = 0, ..., n v one-form connections A i that belong to the vector multiplets, plus the graviphoton A 0 . The real matrices I ΛΣ ≡ Im N ΛΣ (z, z), R ΛΣ ≡ Re N ΛΣ (z, z) denote respectively the imaginary, negative definite, and real parts of the symplectic complex period matrix N . Hence, the period matrix determines the couplings of the one-form connections A Λ to the scalars z i of the vector multiplets. The equations of motion following from the action (A.1) are given by

G µν + 2G i j [∂ µ z i ∂ ν zj -1 2 g µν ∂ ρ z i ∂ ρ zj ] + 8Im N ΛΣ F Λ + µ ρ F Σ - νρ = 0 , (A.2) ∇ µ (G i j ∂ µ zj ) - 1 2 ∂ i G j k∂ ρ z j ∂ ρ zk + 1 2 ∂ i [G Λ µν * F Λ µν ] = 0 , (A.3) ∇ ν * G Λ νµ = 0 , (A.4)
where we have defined

G Λ ≡ - 1 4 √ -g δS δ * F Λ = Re N ΛΣ F Σ + Im N ΛΣ * F Σ . (A.5)
The Maxwell equations for the field strengths F Λ together with the corresponding Bianchi identities can be written in terms of differential forms as follows

dG Λ = 0 , dF Λ = 0 . (A.6)
Notice that, since G Λ is a closed two-form, it can be written locally as the exterior derivative of a one-form A Λ (A.11) Maxwell's equations can be integrated explicitly, in such a way that the complete electric connection one-form A Λ is given in terms of the time component A Λ t of the electric connection one-form and the time component of the magnetic connection one-form A Λ t . Indeed, let Σ M ≡ (A Λ t , A Λ t ) T be a symplectic vector made from the time components of the electric A Λ and magnetic A Λ connection one-forms. Then, it can be shown that

Σ M = 1 2 e 2U M M N Q N dτ , (A.13)
where M M N is a symplectic and symmetric matrix constructed from the couplings of the scalars and the vector fields as

M M N (N ) ≡    I + RI -1 R ΛΣ -RI -1 Λ Σ -I -1 R Λ Σ I -1 ΛΣ    . (A.14)
We choose to express all Maxwell field strengths in terms of the time components of the electric and the magnetic connection one-forms. For the electric field strengths this gives:

F Λ tτ = -∂ τ A Λ t , F Λ θφ = sin θ e -2U I -1 ΛΣ dA Σ t dτ -I -1 R Λ Σ dA Σ t dτ , (A.15)
and the expression for the magnetic field strengths G Λ can be similarly obtained from equation (A.15) using equation (A.5). Since the connection one-forms can be explicitly integrated, they can be eliminated from the action. The four-dimensional N = 2 ungauged supergravity action coupled to vector multiplets can then be shown to be completely equivalent, assuming the space-time background given (A.11) and radial dependence for all the fields, to the one-dimensional effective FGK action [START_REF] Ferrara | Black holes and critical points in moduli space[END_REF] for the 2n v complex fields z i (τ ) and the real field U (τ )

S FGK [U, z] = dτ U 2 + G i j żi żj -e 2U V bh (z, z, Q) , (A.16)
together with the Hamiltonian constraint,

U 2 + G i j żi żj + e 2U V bh (z, z, Q) = r 2 0 . (A.17)
Here V bh is the so-called black hole potential, which is given by [START_REF] Ferrara | Black holes and critical points in moduli space[END_REF] V

bh (z, z, Q) ≡ 1 2 M M N (N )Q M Q N . (A.18)
We are now ready to introduce the H-FGK formalism. The H-FGK formalism [START_REF] Mohaupt | The Hesse potential, the c-map and black hole solutions[END_REF][START_REF] Galli | Non-extremal black holes of N=2, d=4 supergravity[END_REF][START_REF] Galli | The Freudenthal gauge symmetry of the black holes of N=2,d=4 supergravity[END_REF][START_REF] Bueno | Black holes and equivariant charge vectors in N=2,d=4 supergravity[END_REF][START_REF] Meessen | H-FGK formalism for black-hole solutions of N=2, d=4 and d=5 supergravity[END_REF] consists of a particular change of variables from the (2n v + 1)-real U, z i to a new set of (2n v +2)-real variables H M (τ ) which transform as a symplectic, linear, representation the U-duality group of the theory, and become harmonic functions in Euclidean R 3 in the supersymmetric solution. The equations of motion in the new variables H M (τ ) can be written as

1 2 ∂ P M N log W ḢM ḢN - 1 2 Q M Q N + ∂ P M log W ḦM - d dτ ∂Λ ∂ ḢP + ∂Λ ∂H P = 0 (A.19)
together with the Hamiltonian constraint

- 1 2 ∂ M N log W ḢM ḢN - 1 2 Q M Q N + ḢM H M W 2 - Q M H M W 2 -r 2 0 = 0(A.20)
where

Λ ≡ ḢM H M W 2 + Q M H M W 2 , (A.21)
and

e -2U = W(H) ≡ HM (H)H M , HM + iH M = V M X , (A.22)
with V M being the covariantly holomorphic symplectic section that determines the vectormultiplet sector of N = 2 supergravity, and X a complex variable with the same Kähler weight as V M , making the quotient V M /X Kähler invariant. The symplectic vector HM (I) ≡ HM (H) stands for the real part of V M written as a function of the imaginary part, H M ; this can always be done by solving the stabilization equations. The function W(H) is usually known in the literature as the Hesse potential.

The effective theory is now expressed in terms of 2 (n v + 1) variables H M . The solution depends on 2 (n v + 1) + 1 parameters, namely the 2 (n v + 1) charges Q M = p Λ , q Λ T and the non-extremality parameter r 0 , from which it is always possible to reconstruct the complete solution in terms of the four-dimensional fields of the theory. The H-FGK formalism introduces an extra real degree of freedom. Hence the H-FGK action enjoys an extra gauge symmetry which, by gauge fixing, allows to get rid of the extra degree of freedom [START_REF] Galli | The Freudenthal gauge symmetry of the black holes of N=2,d=4 supergravity[END_REF].

B A quantum class of black holes

In this appendix we present the solution of the equations (A. [START_REF] Galli | Black hole solutions of N=2, d=4 supergravity with a quantum correction[END_REF]) and (A.20) that correspond to the quantum black holes of Type-IIA String Theory. Type-IIA quantum black holes are based on the following truncation of the H-variables and the charges H 0 = H 0 = H i = 0 , p 0 = q 0 = q i = 0 . where c = χζ(3) (2π) 3 . From equation (B.2) it follows that, in order to have a non-singular metric, c must be positive, that is, h 1,1 > h 2,1 is a necessary condition in order to obtain an admissible solution. There are plenty of Calabi-Yau manifolds that satisfy this condition, so we will not worry any more about it. The scalar fields, purely imaginary, are

z i = i (3!) 1 3 c 1 3 H i κ 0 ijk H i H j H k 1/3 , (B.3)
It is easy to see that the solution is not consistent in the classical limit c → 0, and also that no classical limit can be assigned to it, since when c = 0 the model is already singular before solving the equations of motion. Hence, we conclude that the corresponding solutions are genuinely quantum solutions, i.e., they only exist when the perturbative quantum corrections are incorporated into the action, and thus they are called Type-IIA quantum black holes.

Of course, we still have to solve the H i , i = 1, • • • , n v , as functions of the radial coordinate τ . Since in this letter we are interested only in supersymmetric solutions, we automatically know that [START_REF] Gauntlett | All supersymmetric solutions of minimal supergravity in five-dimensions[END_REF][START_REF] Meessen | The Supersymmetric configurations of N=2, D=4 supergravity coupled to vector supermultiplets[END_REF][START_REF] Huebscher | Supersymmetric solutions of N=2 D=4 sugra: The Whole ungauged shebang[END_REF]

H i = a i - p i √ 2 τ , r 0 = 0 , (B.4)
that is, the H i , i = 1, • • • , n v , are given by harmonic functions on R 3 . For supersymmetric solutions we have to take r 0 → 0 and therefore the general metric (A.11) simplifies to As we explained in section A, the connection one-forms A Λ can be explicitly obtained using equation (A.13). For Type-IIA quantum black holes we obtain

R 00 = 0 , I 0i = I i0 = 0 , R ij = 0 , (B.8)
which in turn implies that the following components of M M N are zero This implies that the connection one-forms A i have only magnetic components, which give rise to the magnetic charges p i of the black hole solution (A.12). Notice however that the time component of graviphoton A 0 is non-zero, although the corresponding charges are zero: the magnetic one because i * F is identically zero and the electric one thanks to a precise cancellation in the corresponding formula for q 0 : [START_REF] Mohaupt | Non-extremal Black Holes, Harmonic Functions, and Attractor Equations[END_REF]

M 0i = M i0 = 0 , M 0 0 = M 0 0 = M i j = M i j = 0 , M 0i = M i0 = 0 . (B.
q 0 = S 2 ∞ i * G 0 . (B.

(B. 1 )

 1 Using now equation (B.1) together with equations (A.[START_REF] Candelas | Moduli space of Calabi-Yau manifolds[END_REF]) and (2.3) we finde -2U = W(H) = (3!c) 1/3 2 κ 0 ijk H i H j H k 2/3 , (B.2)

ds 2 4 = 4 κ 0 ijk p i p j p k 2/3 τ 2 δ

 442 -e 2U (τ ) dt 2 + e -2U (τ ) δ mn dx m dx n , (B.[START_REF] Kallosh | E(7) symmetric area of the black hole horizon[END_REF] where δ is the Euclidean metric on R 3 . The near horizon τ → -∞ limit of the metric (B.5) is given bylim τ →-∞ ds 2 4 = -4 (3!c) 1/3 κ 0 ijk p i p j p k -2/3 τ -2 dt 2 + (3!c) 1/3 mn dx m dx n .(B.6) The entropy of the Type-IIA quantum black holes is given by S = (3!c) 1/3 4 π κ 0 ijk p i p j p k 2/3 . (B.7)

9 ) 2 e 2 e

 922 From equations (A.[START_REF] Mohaupt | Non-extremal Black Holes, Harmonic Functions, and Attractor Equations[END_REF]) and (B.1) we obtainΨ M = 1 2U M M i p i dτ , (B.10)and sinceΨ M = A Λ t , -A Λ t T , (B.11)we conclude using (B.9) that the only non-zero components of Ψ M areA i t = 1 2U M ij p j dτ ,

Unlike other black hole solutions constructed in these α -corrected theories[START_REF] Behrndt | Subleading contributions from instanton corrections in N=2 supersymmetric black hole entropy[END_REF][START_REF] Behrndt | Quantum N=2 supersymmetric black holes in the S-T model[END_REF][START_REF] Gaida | Gauge symmetry enhancement and N=2 supersymmetric quantum black holes in heterotic string vacua[END_REF][START_REF] Gaida | N=2 supersymmetric quantum black holes in five-dimensional heterotic string vacua[END_REF]

Actually, the prepotential obtained in a Type-IIA Calabi-Yau compactification is symplectically equivalent to the prepotential (2.1).

If these two isometries are not present our microscopic description does not work, but neither does the microscopic description of "normal" M2-M2-M2 black holes.

Acknowledgments

We would like to thank T. Ortín for useful discussions and comments. The work of IB was supported in part by the ERC Starting Independent Researcher Grant 240210, String-QCD-BH, by the John Templeton Foundation Grant 48222: "String Theory and the Anthropic Universe" and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-

where A Λ is the so-called magnetic dual of A Λ and both sets of connection one-forms can be arranged into a symplectic vector A M = A Λ , A Λ T . Supersymmetry constrains the couplings of all the fields of the theory in a very precise way which, for the vector-multiplet sector, is elegantly encoded in the language of Special Kähler Geometry [START_REF] De Wit | Lagrangians of N=2 Supergravity -Matter Systems[END_REF][START_REF] De Wit | Gauge and Matter Fields Coupled to N=2 Supergravity[END_REF]. In fact, the bosonic Lagrangian of N = 2 four-dimensional supergravity coupled to vector multiplets is determined by choosing a holomorphic section Ω ∈ Γ (SV) or, equivalently (when it happens to exist), a homogeneous function F (X ) of degree two, the N = 2 prepotential, from which G i j and N ΛΣ can be easily obtained as

)

Here X Λ denote the homogeneous coordinates on the scalar manifold, related to the scalar fields z i via

Therefore, choosing a second-degree homogeneous function F (X ) automatically determines an N = 2, four-dimensional, ungauged supergravity theory coupled to vector multiplets, which has the appropriate matter content for constructing black hole solutions. The most general static and spherically symmetric metric that solves the equations of motion (A.1) is given by [START_REF] Shahbazi | Black Holes in Supergravity with Applications to String Theory[END_REF][START_REF] Ferrara | Black holes and critical points in moduli space[END_REF][START_REF] Meessen | H-FGK formalism for black-hole solutions of N=2, d=4 and d=5 supergravity[END_REF] ds 2 4 = -e 2U (τ ) dt 2 + e -2U (τ ) γ mn dx m dx n ,

where τ is the radial coordinate. When (A.11) describes a physical black hole solution, r 0 is the non-extremality parameter, the exterior of the event horizon corresponds to τ ∈ (-∞, 0); the event horizon is at τ = -∞ and spatial infinity corresponds to τ → 0 -. The inner part of the Cauchy horizon corresponds to τ ∈ (τ s , ∞), with the inner horizon at τ → ∞ and the singularity at τ = τ s for a certain positive and finite real number τ s [START_REF] Galli | Non-extremal black holes of N=2, d=4 supergravity[END_REF]. Since the metric is spherically symmetric, we will assume that all the fields of the theory depend exclusively on the radial coordinate τ . We define the black hole charges as

where S 2 ∞ denotes an space-like two-sphere at spatial infinity τ → 0, p Λ correspond to the magentic charges and q Λ correspond to the electric charges of the black hole, which can be together arranged into a symplectic vector Q M ≡ p Λ , q Λ T . In the background given by


Natuf Topluluklarinda

Diş Çekimi Ve

Yerel Kimlik

Fanny Bocquentin

AVULSIONS DENTAIRES ET IDENTITé REGIONALE CHEZ LES NATOUFIENS

Keywords: Natufian, Epi-Palaeolithic, tooth evulsion, burial cluster, regional identity Anahtar Sözcükler: Natuf, Epi-Paleolitik, diş çekme, toplu gömüt, yerel kimlik, diğer verilerle

Since the earliest stages of anthropological studies, researchers have suspected that the Natufian populations practiced tooth evulsion. A comprehensive study of the Epipalaeolithic Levantine dental collections does indeed show an abnormal rate of ante mortem loss of the central upper incisors, which supports this cultural interpretation. Tooth removal is found in roughly 20% of the adult and subadult populations (sometimes as early as adolescence), and seems to have been practiced equally on females and males. For all Natufian groups in which this practice has been found, the removal seems to be concentrated primarily on the right upper incisor. The left upper incisor may also be removed, but this appears to be a secondary option. Combining this information with contextual data provides insight into the meanings of this practice. First, the individuals concerned are sometimes buried next to each other, which reinforces the idea that they are of comparable social statuses. Second, the practice seems to spread into the Galilee's major sites only at the very end of the Natufian period, and thus acts as a testimony to the evolution of a regional identity overtime.

ÖZET

İlk antropolojik incelemelerden bu yana araştırmacılar, Natuf topluluklarında diş çekimi uygulaması olduğunu gösteren verilerin bulunduğunu öngörmüşlerdir. Levant'ın Epi-Paleolitik dönemine ait dişler üzerine yapılan kapsamlı bir çalışma, gerçekten de orta üst kesicilerde ante mortem kayıp oranının anormal biçimde yüksek olduğunu göstererek bu kültürel yorumu desteklemiştir. Erişkin ve erişkin olmayan nüfusta (bazen ergenliğe kadar inen) diş çekimi kabaca %20 oranında bulunmaktadır ve bu durum erkeklerde ve kadınlarda eşit olarak görülmektedir. Bu uygulamanın görüldüğü bütün Natuf topluluklarında, diş çekimi öncelikle sağ üst kesicide yoğunlaşmaktadır. Sol üst kesicinin de çekilmiş olduğu örnek-

INTRODUCTION

La pratique de l'extraction volontaire d'une ou de plusieurs incisives supérieures chez les natoufiensderniers chasseurs-cueilleurs et premiers sédentaires du Levant -a été suggérée dès les premières études anthropologiques menées sur cette population épipaléolithique [START_REF] Keith | New Discoveries Relating to the Antiquity of Man[END_REF][START_REF] Keith | The late Paleolithic Inhabitants of Palestine[END_REF]. L'auteur signalait sa présence à Shukbah (5 cas), El Wad (7 cas) et Kebara (sans précision) [START_REF] Keith | New Discoveries Relating to the Antiquity of Man[END_REF][START_REF] Keith | The late Paleolithic Inhabitants of Palestine[END_REF]. Ces cas ont été confirmés [START_REF] Mccown | The Natufian crania from Mount Carmel, Palestine, and their inter-relationship[END_REF][START_REF] Smith | Dental morphology and pathology in the Natufians: the dental evidence for dietary specializations[END_REF][START_REF] Smith | Paleonutrition and subsistence patterns in the Natufians[END_REF][START_REF] Smith | The Dental evidence for nutritional status in the Natufians[END_REF] et d'autres ont été découverts par la suite à Mallaha [START_REF] Ferembach | Squelettes du Natoufien d'Israël, étude anthropologique[END_REF][START_REF] Valla | Le Natoufien final de Mallaha (Eynan), deuxième rapport préliminaire: les fouilles de 1998 à 1999[END_REF][START_REF] Valla | Les fouilles à Mallaha en 2000 et 2001: 3ème rapport préliminaire[END_REF]. Ces découvertes font écho à des travaux ethnologiques qui signalent l'existence, dans différentes régions du monde, d'une telle pratique afin d'affirmer l'appartenance de l'individu au groupe ou le partage d'un statut donné, parfois lors de rites de passage (e.g. [START_REF] Baudoin | Les mutilations labiales et dentaires[END_REF][START_REF] Campbell | Dentition and palate of the Australian Aboriginal[END_REF][START_REF] Bourhis | Note sur les mutilations dentaires chez les indigènes de la circonscription de Yaoundé (Cameroun)-Race Bantou[END_REF]; références dans [START_REF] Boule | L'Homme fossile d'Asselar (Sahara)[END_REF]. De nombreuses découvertes archéologiques confirment la pratique de l'avulsion des dents antérieures sur les cinq continents à travers l'Histoire et la Préhistoire (e.g. [START_REF] Merbs | Patterns of activity-induced pathology in a Canadian Iniut population[END_REF][START_REF] Milner | Teeth as Artifacts of human Behavior: Intentional Mutilation and Accidental Modification[END_REF][START_REF] Kusaka | Strontium isotope evidence of migration and diet in relation to ritual tooth ablation: a case study from the Inariyama Jomon site, Japan[END_REF]) et, en Afrique du nord-ouest, dès le Paléolithique supérieur (synthèse dans: Verger-Pratoucy 1968; [START_REF] Humphrey | Tooth evulsion in the Maghreb: chronological and geographical patterns[END_REF].

Afin de mieux cerner et de caractériser cette pratique au sein de la population natoufienne, nous avons mené une étude systématique des pertes dentaires ante mortem sur un corpus qui rassemble la quasi-totalité des squelettes natoufiens disponibles à l'étude (environ 370 squelettes) dont notamment ceux issus des quatre sites majeurs du Natoufien: El Wad (N = 70), Nahal-Oren (N = 44), Mallaha (N = 111), Kebara (N = 48) et la grotte d'Hayonim (N = 58). Ce corpus couvre les 3 phases, ancienne, récente et finale de la culture natoufienne datée entre 13000 et 9500 cal. BC (Fig. 1). L'identité biologique de ces individus (inventaire, Nombre Minimal d'Individu, âge, sexe, morphométrie, caractères discrets, hygiènes dentaire, stress de croissance) ainsi que leur traitement funéraire et leur attribution chronologique a fait l'objet, à partir de données de première main, d'une révision synthétique récente (Bocquentin 2003;[START_REF] Bocquentin | Affinités et diversité des premiers groupes sédentaires du Proche-Orient[END_REF]) à laquelle se réfère le présent article.

AVULSIONS OU EXFOLIATIONS?

Il n'existe aucune preuve anthropologique absolue que la perte de la dent soit le résultat d'un acte délibéré [START_REF] Merbs | Patterns of activity-induced pathology in a Canadian Iniut population[END_REF][START_REF] Milner | Teeth as Artifacts of human Behavior: Intentional Mutilation and Accidental Modification[END_REF][START_REF] Robb | Intentional tooth removal in Neolithic Italian women[END_REF]). Les pertes dentaires, en effet, peuvent intervenir accidentellement à la suite d'un choc ou bien par exfoliation naturelle lorsque la dent atteint un degré d'usure extrême ou qu'elle présente une pathologie sévère (carie, infection) [START_REF] Clarke | Tooth dislocation: The relationship with tooth wear and dental abscesses[END_REF]. Néanmoins, l'extraction volontaire est une hypothèse qu'on ne saurait négliger dans certains contextes. À l'échelle d'une population, la récurrence des pertes ante mortem d'une catégorie spécifique de dents, alors que les autres sont moins affectées, est à prendre en considération. À l'échelle individuelle, l'absence de traumatisme ou d'infection observée sur l'alvéole oblitéré est une condition nécessaire pour supposer un cas d'avulsion. Mais ce critère ne permet pas d'exclure les cas de pertes naturelles dues à l'usure de la dent. Il reste donc indispensable de s'assurer que la perte de la dent a été précoce, alors que le reste de la denture était encore en très bonne condition, pour conclure à un cas d'avulsion. Lorsque le sujet est âgé, le degré d'usure des dents antagonistes permet, néanmoins, d'évaluer la précocité ou non des pertes observées. Nous avons donc retenu comme cas probables d'avulsion les pertes ante mortem des incisives centrales supérieures chez des individus qui présentent, par ailleurs, des dentures complètes et saines ou bien chez des individus dont la denture est plus éprouvée mais dont les incisives inférieures présentent une usure a minima indiquant ainsi une perte précoce des dents supérieures. Toute extraction volontaire ayant eu lieu tardivement ne peut, malheureusement, être reconnue.

Un simple inventaire de toutes les pertes ante mortem, quelle que soit leur origine présumée, permet de confirmer le statut particulier des incisives centrales supérieures (Tab. 1). En effet, alors que les molaires sont les dents les plus sujettes aux pertes ante mortem [START_REF] Powell | The analysis of dental wear and caries for dietary reconstruction[END_REF][START_REF] Larsen | Bioarchaeology. Interpreting behavior from the human skeleton[END_REF][START_REF] Hillson | Recording dental caries in archaeological human remains[END_REF], ce sont les incisives centrales supérieures qui sont le plus souvent absentes en contexte natoufien. D'autre part, contrairement à toutes les autres catégories de dent, le taux de perte des incisives centrales est plus important sur le maxillaire que sur la mandibule. De ce fait, il nous paraît justifié de considérer a priori les pertes ante mortem de cette catégorie de dents comme le résultat d'une extraction délibérée et de vérifier au cas par cas si cette hypothèse est acceptable. D'autres catégories de dents ont pu faire l'objet d'une extraction volontaire occasionnelle mais l'absence de récurrence rend difficile la reconnaissance de telles pratiques.

FREQUENCE ET CARACTERISTIQUES DES AVULSIONS NATOUFIENNES

Selon les critères définis précédemment, nous considérons qu'un minimum de 20 sujets natoufiens a subi une ablation volontaire d'une ou des deux incisives centrales supérieures. Il s'agit d'hommes et de femmes; l'individu le plus jeune est une adolescente dont les alvéoles sont complètement résorbés (Fig. 2), ce qui tend à confirmer l'hypothèse d'extractions parfois précoces [START_REF] Mccown | The Natufian crania from Mount Carmel, Palestine, and their inter-relationship[END_REF][START_REF] Smith | The Dental evidence for nutritional status in the Natufians[END_REF]. La ligne occlusale des mandibules des sujets adultes présente, en vue antérieure, une forme en ogive marquée, qui témoigne également d'extractions relativement précoces. L'ogive est plutôt symétrique dans le cas d'extraction des deux incisives centrales supérieures et asymétrique dans le cas où une seule dent a été extraite (Fig. 3). Il semble que deux phénomènes complémentaires soit à l'origine de cette forme en ogive: d'une part, la minimisation de la vitesse d'usure des dents mandibulaires qui ne sont plus soumises à l'attrition (usure par frottement des dents antagonistes) et, d'autre part, "l'éruption continue" des dents inférieures, un phénomène qui intervient lorsque les dents antagonistes n'exercent plus de force opposée [START_REF] Danenberg | Continuous tooth eruption in Australian Aboriginal skulls[END_REF][START_REF] Schwartz | Skeleton Keys: An Introduction to Human Skeletal Morphology, Development, and Analysis[END_REF][START_REF] Humphrey | Tooth evulsion in the Maghreb: chronological and geographical patterns[END_REF]. Les dents inférieures ne présentent pas encore d'aspect en ogive typique chez les plus jeunes individus (Fig. 2). Notre inventaire tient compte de deux individus (EWH36z2 et NOH42-48) dont le maxillaire n'est pas conservé mais dont la mandibule montre une forme en ogive caractéristique.

Parmi les 18 maxillaires retenus, nous avons observé 3 cas d'ablation partielle (l'un provient du site de Shukbah et les deux autres sont issus du site d'El Wad) pour lesquels un fragment de racine est encore en place dans l'alvéole oblitéré (Figs. 23). Dans ces trois cas, nous avons pu faire une observation directe car le maxillaire est en partie érodé. Il est possible que d'autres cas semblables existent mais nous n'avons pas effectué de radiographies systématiques pour le vérifier. Sur les 5 maxillaires que nous avons radiographiés (NOH34, NOH48, EMH10, EMH156 et EMH168) les extractions étaient complètes.

IDENTITé REGIONALE

On constate que toutes les communautés natoufiennes ne pratiquent pas l'avulsion des incisives supérieures. Son absence, notamment, parmi les individus inhumés dans la grotte d'Hayonim (Natoufien ancien et récent, Galilée: Fig. 1) est remarquable car cette population est bien représentée (Tab. 3). Mais l'absence de cette pratique est également remarquable à Mallaha (Natoufien ancien, récent et final, Galilée; Fig. 1) durant les phases ancienne et récente de l'occupation du site. En effet, les 4 individus concernés à Mallaha appartiennent à la dernière phase d'occupation de ce site (Tab. 2). Il semble donc que l'avulsion, présente dès le Natoufien ancien sur le mont Carmel, ne se soit pas répandue en Galilée avant la fin du Natoufien. Les populations de Judée (Shukbah et Erq-el-Ahmar) sont peu représentatives; l'avulsion y est présente au Natoufien récent mais n'est pas encore documentée au Natoufien ancien.

Lorsque le côté droit du maxillaire a pu être observé, nous avons constaté l'avulsion systématique de l'incisive centrale, ce qui n'est pas le cas à gauche. L'extraction de l'incisive droite apparaît donc comme une priorité partagée par toutes les communautés natoufiennes qui pratiquent l'avulsion, alors que l'extraction de l'incisive gauche est plus aléatoire. Lorsque celle-ci est néanmoins présente, la symétrie de l'ogive dessinée par les dents inférieures est en faveur de deux extractions contemporaines. À El Wad et à Shukbah, les avulsions simples et les avulsions doubles sont présentes et ne sont liées ni au sexe ni à la période. À Nahal-Oren, lorsque les deux côtés peuvent être observés, l'extraction est toujours double; à Kebara et Mallaha, en revanche, les extractions ne sont connues qu'à droite.

Le contexte funéraire des individus concernés n'est pas toujours documenté. À El Wad les quatre individus pour lesquels la situation sur le site est connue apparaissent indépendants les uns des autres. À Kebara, les trois maxillaires concernés sont tous issus de la sépulture secondaire découverte au fond de la grotte et dont les os sont brûlés (Bocquentin 2003); aucun individu inhumé à l'entrée de la grotte ne présente cette caractéristique. À Nahal-Oren, trois des quatre individus concernés sont des femmes qui ont été inhumées à proximité l'une de l'autre. Deux d'entre elles (NO H42-48 et NOH48) se trouvaient dans la même tombe et la troisième (NO H34) avait été inhumée au même endroit mais dans une fosse antérieure (Crognier and Dupouy-Madre 1974: 109). À Mallaha, deux des quatre individus concernés (EMH10 et EMH163) étaient inhumés ensemble. On observe donc une répartition des sépultures qui tend à ne pas être aléatoire. Il se pourrait que ces individus partagent une situation sociale commune. Il reste difficile, néanmoins, d'interpréter ce phénomène qui ne touche que 17% des adultes et des adolescents natoufiens et qui n'est pas spécifique à un sexe. D'autres dents ont été occasionnellement arrachées. Nous avons observé deux exemples. L'un concerne une seconde molaire déciduale dont un fragment de racine est encore en place dans le maxillaire (EM locus 9[2]). Il semble que cette extraction ait été thérapeutique étant donné le retard de calcification de la prémolaire permanente qui suggère que l'exfoliation de la dent déciduale sous-jacente ne s'est pas faite en temps voulu. On constate d'ailleurs la même situation du côté opposé avec la seconde molaire déciduale encore en place. Le second cas peut avoir une signification culturelle puisqu'il concerne un individu dont les deux incisives centrales supérieures ont été extraites (NO H48) et dont l'incisive latérale inférieure gauche est aussi manquante (Fig. 4a). La racine de cette dent est encore en place dans l'alvéole entièrement résorbé (Fig. 4b).

REMARQUES ET CONCLUSION

La pratique de l'avulsion dentaire est bien attestée au Natoufien mais reste peu répandue. Elle concerne moins de 20% de la population, sans distinction de sexe. Les enfants ne sont pas concernés; l'extraction semble se faire au plus tôt à l'adolescence. C'est également ce qui est observé parmi les cas ethnologiques car l'ablation des dents marque, dans bien des cas, le passage de l'enfance à la vie adulte et ne s'effectue pas avant l'âge de 10-12 ans.

Cela correspond aussi avec la maturité de la denture et l'éruption de la seconde molaire permanente. L'ablation des incisives avant cet âge peut s'avérer inefficace dans la mesure où les dents adjacentes viennent combler l'espace libéré par un phénomène de mésiogression [START_REF] Dervaux | Le sourire : ses symboles et sa marque d'appartenance à une ethnie ou à un groupe social[END_REF]. Les raisons invoquées par les populations subcontemporaines qui pratiquent l'avulsion sont souvent multiples: apprécier la résistance à la douleur de l'initié, inscrire son identité clanique définitivement, obligation esthétique. La distribution tant géographique que chronologique rend compte de motivations et de modalités d'exécution fort variées de sorte qu'un parallèle interprétatif strict apparaît peu pertinent. Il n'en reste pas moins que l'avulsion dentaire et, a fortiori, celle des incisives centrales est visible de tous et apparaît ainsi comme un signe distinctif par excellence. On note qu'il ne s'agit pas d'une opération bégnine: l'avulsion qui s'effectue par percussion directe ou indirecte de la dent, peut provoquer, notamment, une fracturation du maxillaire, une hémorragie, ou une infection pouvant mener à la mort du sujet; elle a également une incidence majeure sur l'occlusion, la mastication, et le langage.

De façon particulièrement intéressante, on constate que toutes les communautés natoufiennes ne sont pas concernées par cette pratique. Au début du Natoufien, elle est attestée seulement sur certains sites du mont Carmel puis, à la fin du Natoufien, elle se diffuse en Galilée dans la population de Mallaha. L'extraction de l'incisive centrale supérieure droite apparaît prioritaire par rapport au côté gauche quels que soient le site et la période. Les individus concernés peuvent être inhumés à proximité les uns des autres ce qui renforce l'idée qu'ils partagent un statut identitaire commun. L'avulsion apparaît donc comme une pratique culturelle codifiée ancienne mais dont l'adoption ne se fait que tardivement dans les populations natoufiennes les plus septentrionales du corpus actuel.

REMERCIEMENTS

Tableau 2 -

 2 Liste des dentures pour lesquelles la perte ante mortem de l'une ou des deux incisives centrales supérieures peut être considérée comme un cas d'avulsion probable. La mention «au moins» signifie que le côté opposé n'est pas observable. Les restes humains provenant de Shukbah sont probablement moins complets qu'ils ne l'étaient à l'époque où Sir A. Keith a effectué son étude (cf. Bocquentin 2003). Age et sexe d'après Bocquentin 2003. Tableau 3 -Fréquence des cas d'avulsions probables des incisives centrales supérieures par côté (D : droit ; G : gauche) et par site. Le nombre des observations correspond au nombre de dents ou d'alvéoles observés chez les adolescents et les adultes. Le taux d'avulsion n'a été calculé que pour les populations les mieux représentées en gras).

Fig. 1 -

 1 Fig. 1 -carte de répartition des sites natoufiens concernés par cette étude et phases culturelles auxquelles sont attribués les squelettes du corpus.

Fig. 2 -

 2 Fig. 2 -Avulsion probable des deux incisives centrales supérieures d'une adolescente du site d'El Wad (EW H28-32 [12]). a : maxillaire en vue inférieure ; b : maxillaire en vue antérieure, la flèche indique l'apex encore en place dans l'alvéole droit oblitéré, probablement fracturé lors de l'extraction ; c : mandibule en vue antérieure, la forme en ogive n'est pas encore présente mais on constate que l'incisive latérale gauche présente un début d'usure contrairement aux incisives centrales.

Fig. 3

 3 Fig.3-a: Avulsion probable de l'incisive centrale supérieure droite (SH 10334), noter la forme en ogive asymétrique de la mandibule. b : avulsion probable des deux incisives centrales (EW H17), noter la forme symétrique de l'ogive formée par les dents antagonistes et le fragment de racine en place dans l'alvéole oblitéré indiqué par la flèche.

Fig. 4 -

 4 Fig. 4 -Cas d'extraction d'une incisive latérale inférieure gauche (NO H48). b: adiographie de la mandibule où l'on distingue la racine encore en place dans l'alvéole résorbé.

  Je suis très reconnaissante à Yasemin Yılmaz, Aksel Tibet et Laurence Astruc de m'avoir invitée à participer à ce cahier spécial de la revue Tüba-Ar. Je tiens également à remercier O. Bar-Yosef, A. Belfer-Cohen, H. Khalaily, D. Nadel et F. Valla, pour m'avoir permis de faire mes observations sur des col-lections dont la publication est en cours. Ce travail n'aurait pu aboutir sans le concours du Département d'Anthropologie de la Faculté de Médecine Sackler de Tel Aviv et celui du Peabody Museum de l'Université d'Harvard où sont conservés les collections, ni les corrections avisées de Messieurs Pascal Murail et Pascal Sellier. Je remercie également Harris Greenberg pour les corrections apportées au résumé anglais et les relecteurs anonymes pour leurs suggestions pertinentes.

	Tableau 1 -Fréquence des
	pertes ante mortem par
	catégories de dents. La
	population natoufienne
	(adolescents et adultes) de
	tous sites et périodes
	confondus est prise en
	compte; N : nombre
	d'observations par
	catégories de dents, les
	côtés droit et gauche ont
	été additionnés.