Sylvain Ribault
email: ribault@lpta.univ-montp2.fr

A family of solvable non-rational conformal field theories

We find non-rational conformal field theories in two dimensions, which are solvable due to their correlators being related to correlators of Liouville theory. Their symmetry algebra consists of the dimension-two stress-energy tensor, and two dimension-one fields. The theories come in a family with two parameters: the central charge c and a complex number m. The special case m = 0 corresponds to Liouville theory (plus two free bosons), and m = 1 corresponds to the H + 3 model. In the case m = 2 we show that the correlators obey third-order differential equations, which are associated to a subsingular vector of the symmetry algebra.

Introduction

Conformal field theories in two dimensions have many applications in string theory and condensedmatter physics. In some applications the system under consideration, say strings in a given spacetime, dictates the theory to be studied. In some other cases however, a CFT is used as a phenomelogical theory for a system without a known first-principles description. The relevant theory must then be chosen among the known CFTs according to the desired properties of the system. This approach relies on the existence of a large enough number of CFTs among which to choose, and on a good enough knowledge of their properties. Many rational CFTs are known, and their properties are often well-understood. However most CFTs are non-rational, and only a few non-rational CFTs have been solved so far, among them Liouville theory [START_REF] Teschner | Liouville theory revisited[END_REF] and the H + 3 model [START_REF] Gawedzki | Noncompact WZW conformal field theories[END_REF]. It is therefore interesting to expand the small family of solvable non-rational CFTs.

The inspiration for the present article comes from the relation [START_REF] Ribault | H + 3 correlators from Liouville theory[END_REF] between the correlation functions of the H + 3 model and correlation functions of Liouville theory. Namely, correlators of generic H + 3 primary fields are related to correlators involving both generic Liouville primary fields, and the particular field V -1 2b where b is related to the central charge of Liouville theory by c = 1 + 6(b + b -1 ) 2 . This field is known to be a degenerate field with a null vector at level two, which implies that Liouville correlators involving this field obey second-order differential equations [START_REF] Belavin | Infinite conformal symmetry in two-dimensional quantum field theory[END_REF]. But there is an infinite series of degenerate fields in Liouville theory, which lead to higher-order differential equations. It is therefore natural to wonder whether Liouville correlators involving such degenerate fields -or even generic fields -can be related to correlators in some models which would be generalizations of the H + 3 model. We will propose a positive answer to this question, and sketch some properties of the resulting models. In contrast to the usual situation in the study of conformal field theory, we start with the solution of the models, namely an ansatz for their correlators in terms of Liouville correlators. The problem is then to reconstruct the models, and in particular their symmetry algebra, from their correlators. This is a priori a non-trivial problem, because our ansatz gives us only correlators of primary fields, but the notion of a primary field is not even defined if we do not know the symmetry algebra. Fortunately, the correct symmetry algebra will be suggested by the Lagrangian description of the models, which is very similar to the free-field description of the H + 3 model. (The new Lagrangians however do not correspond to sigma models.) We will then check that this symmetry algebra is consistent with the properties of the correlators, in particular in the case when they obey third-order differential equations.

An ansatz for the correlation functions

Our ansatz will be directly inspired by the H + 3 -Liouville relation, we will therefore begin by a reminder of this relation. More details and references can be found in [START_REF] Ribault | H + 3 correlators from Liouville theory[END_REF].

Reminder of the H + 3 -Liouville relation

In the conformal bootstrap approach, the H + 3 model is defined by its symmetry algebra, and primary fields which transform in a certain way with respect to this algebra. Solving the model means finding the correlation functions of the primary fields; these correlation functions are subject to symmetry and consistency conditions. Let us briefly review these objects. The chiral symmetry algebra of the H + 3 model is the affine Lie algebra sℓ 2 , which is generated by three currents J 3 (z), J + (z), J -(z) with the following operator product expansions (OPEs):

J 3 (z)J 3 (w) = - k 2 (z -w) 2 + O(1) , (2.1)
J 3 (z)J ± (w) = ± J ± (w) z -w + O(1) , (2.2)
J + (z)J -(w) = - k (z -w) 2 + 2J 3 (w) z -w + O(1) , (2.3)
where k > 2 is called the level of the affine Lie algebra. As a consequence, the H + 3 model is a conformal field theory with central charge c = 3k k-2 ; the stress-energy tensor which generates the conformal transformations can be deduced from the currents by the Sugawara construction [START_REF] Di Francesco | Conformal field theory New[END_REF]

T = 1 2(k -2) (J + J -) + (J -J + ) -2(J 3 J 3 ) , (2.4)
where we use the following definition of the normal-ordered product of operators:

(AB)(z) = 1 2πi z dx dx x -z A(x)B(z) . (2.5)
Affine primary operators Φ j (µ|w) with spin j, isospin µ and worldsheet position w are defined by their OPEs with the currents, from which their OPEs with the stress-energy tensor follows:

J -(z)Φ j (µ|w) = µ z -w Φ j (µ|w) + O(1) , (2.6)
J 3 (z)Φ j (µ|w) = µ ∂ ∂µ z -w Φ j (µ|w) + O(1) , (2.7)
J + (z)Φ j (µ|w) = µ ∂ 2 ∂µ 2 -j(j+1) µ z -w Φ j (µ|w) + O(1) , (2.8)
T (z)Φ j (µ|w) = ∆ j (z -w) 2 Φ j (µ|w) + 1 z -w ∂ ∂w Φ j (µ|w) + O(1) , (2.9)
where the conformal dimension of the field Φ j (µ|w) is

∆ j = - j(j + 1) k -2 .
(2.10)

The fields Φ j (µ|w) have similar OPEs with the antiholomorphic currents Ja (z) and stress-energy tensor T (z), but their dependences on z and μ will be kept implicit. The spin takes values j ∈ -1 2 + iR, corresponding to continuous representations of the algebra sℓ 2 .

In the H + 3 model all fields are either affine primaries or affine descendents thereof, and correlators of affine descendents can be deduced from correlators of affine primaries by the Ward identities. Therefore, in order to solve the model on the Riemann sphere with coordinates z, z, it is enough to determine the n-point functions of affine primaries for all n:

Ω n ≡ n i=1 Φ j i (µ i |z i ) .
(2.11) Now all such correlators of affine primaries are related to Liouville theory correlators as follows:

Ω n = δ (2) ( n i=1 µ i )|u| 2 |Θ n | 1 b 2 n i=1 V α i (z i ) n-2 a=1 V -1 2b (y a ) , (2.12)
where we consider Liouville theory at parameter

b = 1 √ k -2 , (2.13)
with primary operators V α (z) with momenta

α = b(j + 1) + 1 2b , (2.14)
and conformal dimensions

∆ α = α(b + b -1 -α) .
(2.15)

The objects u and Θ are defined by

u = n i=1 µ i z i , Θ n = i<j z ij a<b y ab n i=1 n-2 a=1 (y a -z i ) , ( 2
µ i t -z i = u n-2 a=1 (t -y a ) n i=1 (t -z i )
.

(2.17)

Motivation of the ansatz

Let us replace the fields V -1 2b

in the H + 3 -Liouville relation (2.12) with more general fields V -m 2b with m ∈ C, and modify the prefactors as well:

Ω (m) n = δ (2) ( n i=1 µ i )|u| 2λ |Θ n | 2θ n i=1 V α i (z i ) n-2 a=1 V -m 2b (y a ) , (2.18)
where λ and θ are numbers to be determined. Can we still interpret Ω (m) n as an n-point function (2.11) of fields Φ j (µ|z) in some conformal field theory? In the process we keep the definitions of u, Θ and y a eqs. (2.16,2.17), but the nature of the operator Φ j (µ|z) will be modified: it will now be a primary field with respect to an unknown symmetry algebra, with an unknown conformal dimension ∆ (m) j with respect to a Virasoro algebra with an unknown central charge.

We hope that this interpretation of Ω (m) n exists for any choice of m, but we expect the values of the extra parameters λ, θ in (2.18) to be determined in terms of m by symmetry and other requirements. Let us first consider the constraints on the three-point function due to global conformal invariance. This requires that the dependence of Ω (m) 3 on the worldsheet coordinates z 1 , z 2 , z 3 should be

Ω (m) 3 ∝ |z 12 | -2 ∆ (m) j 1 +∆ (m) j 2 -∆ (m) j 3 |z 13 | -2 ∆ (m) j 1 +∆ (m) j 3 -∆ (m) j 2 |z 23 | -2 ∆ (m) j 2 +∆ (m) j 3 -∆ (m) j 1 ,(2.19)
with an arbitrary prefactor depending on µ 1 , µ 2 , µ 3 . According to our ansatz, Ω (m) 3 is actually related to a four-point Liouville correlator, which behaves as

V α 1 (z 1 )V α 2 (z 2 )V α 3 (z 3 )V -m 2b (y 1 ) = |z 12 | 2∆ -m 2b -2∆ 12 |z 13 | -2∆ -m 2b -2∆ 13 |z 23 | 2∆ -m 2b -2∆ 23 |y 1 -z 2 | -4∆ -m 2b G(z) , (2.20)
where we use the notations ∆ 12 = ∆ α 1 + ∆ α 2 -∆ α 3 for the Liouville conformal dimensions, and G(z) for a known function of the cross-ratio z = z 32 (y-z 1 ) z 31 (y-z 2 ) whose precise form is not needed here. From the change of variables (2.17) we have u(yz 2 ) = µ 2 z 12 z 23 (in the limit t → z 2 ), and in addition z = 1 + µ 3 µ 2 actually does not depend on z 1 , z 2 , z 3 . Thus, the equations (2.18) and (2.20) imply

Ω (m) 3 ∝ |u| 2λ+6θ+4∆ -m 2b |z 12 | -2θ-2∆ -m 2b -2∆ 12 |z 13 | -2θ-2∆ -m 2b -2∆ 13 |z 23 | -2θ-2∆ -m 2b -2∆ 23 . (2.21)
Comparing with the condition (2.19) reveals that global conformal invariance requires

λ + 3θ + 2∆ -m 2b = 0 , ∆ (m)
j i = ∆ α i + ∆ -m 2b + θ . (2.22)
In order to derive the extra relation which we need for fully determining λ, θ and ∆ (m) j i , we will use a more heuristic reasoning. Consider the behaviour of our ansatz (2.18) near y a = y b . In the H + 3 case (m = 1), correlators are continuous at such points, a property which is particularly important in the study of the H + 3 model on the disc [START_REF] Hosomichi | Solution of the H + 3 model on a disc[END_REF]. More generally, in the cases when the Liouville field V -m 2b is degenerate, the leading behaviour of two such fields when coming close is

V -m 2b (y a )V -m 2b (y b ) ∼ y ab →0 |y ab | -4∆ -m 2b +2∆ -m b V -m b (y a ) , (2.23)
and the behaviour of our ansatz is thus

Ω (m) n ∼ y ab →0 |y ab | -4∆ -m 2b +2∆ -m b +2θ . (2.24)
Assuming the critical exponent -4∆ -m 2b + 2∆ -m b + 2θ to vanish, and solving eq. ( 2.22) as well, we obtain

θ = m 2 2b 2 , λ = m(1 + b -2 (1 -m)) , ∆ (m) j = ∆ α - m 4 (2 + 2b -2 -b -2 m) . (2.25)
These relations will be assumed to hold not only when the Liouville field V -m 2b is degenerate, but in all cases. Notice that it is possible to define a "spin" variable j such that

∆ (m) j = -(j + 1)(b 2 j + m -1) , (2.26)
in which case the momentum α of the corresponding Liouville field V α (z) is

α = b(j + 1) + m 2b .
(2.27)

In the following we will argue that the ansatz (2.18), together with the particular values (2.25) for the parameters λ, θ and the conformal weights of the fields Φ j (µ|z), has a meaning in terms of correlators in a conformal field theory.

Symmetry algebra of the new theories

We have checked that our ansatz is consistent with global conformal invariance, let us now study the rest of the conformal symmetry, and the possible extra symmetries which underlie our ansatz. To do this, it will be convenient to introduce a Lagrangian description of the corresponding theories. We will therefore propose such a description, check that it reproduces the relation (2.18) with Liouville theory, and then use it for deriving the symmetries.

Lagrangian description

We use the same bosonic fields (φ, β, γ) as in the H + 3 model [START_REF] Hosomichi | Free field approach to string theory on AdS(3)[END_REF], and we propose the following Lagrangian and stress-energy tensor:

L (m) = 1 2π ∂φ ∂φ + β ∂γ + β∂γ + b 2 (-β β) m e 2bφ , (3.1)
T (m) = -β∂γ -(∂φ) 2 + (b + b -1 (1 -m))∂ 2 φ , (3.2)
where -β β is assumed to be real positive.

The interaction term L (m) int = (-β β) m e 2bφ is marginal with respect to the stress-energy tensor T (m) . In the cases m = 0 (Liouville theory) and m = 1 (H + 3 model) it is known to be exactly marginal. This follows from the exponential dependence of the interaction term on φ, a feature which is still present for arbitrary values of m. In perturbative calculations, many terms in the series expansion of exp -L (m) int will then yield vanishing contributions due to φ-momentum conservation in the free theory. We therefore expect the interaction to be exactly marginal, which implies that the corresponding theories have conformal symmetry.

In the case m = 1 of the H + 3 model, integrating out the non-dynamical fields β, β in the path integral DφDβD βDγDγ e -d 2 w L (m) yields a sigma model, whose target space is indeed H + 3 . In the general case, it is still possible to integrate out β, β if ℜm > 1 2 , but the resulting Lagrangian has no sigma model interpretation because it involves higher powers of ∂γ∂γ.

The relation between H + 3 and Liouville correlators has been rederived by Hikida and Schomerus using a path-integral computation [START_REF] Hikida | H + 3 WZNW model from Liouville field theory[END_REF]. We will now emulate this calculation in order to rederive our ansatz (2.18) from the path-integral definition of a theory with the Lagrangian (3.1). Of course, this path-integral definition will be complete only after we specify the fields Φ j (µ|z) in terms of (φ, β, γ):

Φ j (µ|z) = |µ| 2m(j+1) e µγ-μγ e 2b(j+1)φ . (3.3)
The path-integral definition of the n-point function is then

Ω (m) n = DφDβD βDγDγ e -d 2 w L (m) n i=1 Φ j i (µ i |z i ) . (3.4)
Let us perform the integrations over γ and γ, and then β and β:

Ω (m) n = δ (2) ( n i=1 µ i ) Dφ e -1 2π d 2 w ∂φ ∂φ+b 2 u i µ i w-z i 2m e 2bφ n i=1 |µ i | 2m(j i +1) e 2b(j i +1)φ , (3.5)
where we still have u = n i=1 µ i z i . We then perform the change of integration variable φ → φmb -1 log |u| .

This produces a global factor |u| 2m(1+b -2 (1-m)) , due to an implicit worldsheet curvature term in the Lagrangian, which corresponds to the linear dilaton term (b + b -1 (1m))∂ 2 φ in the stress-energy tensor T (m) , eq. (3.2). (For simplicity we have used flat space as a model of the Riemann sphere and omitted the worldsheet curvature term. This subtlety is dealt with in [START_REF] Hikida | H + 3 WZNW model from Liouville field theory[END_REF].) Then, we define y a as the zeroes of n i=1 µ i w-z i as in eq. (2.17), and perform the change of integration variable

ϕ(w) = φ(w) + mb -1 log n-2 a=1 (w -y a ) n i=1 (w -z i ) . (3.7)
This yields

Ω (m) n = δ (2) ( n i=1 µ i ) |u| 2m(1+b -2 (1-m)) |Θ n | m 2 b 2 × Dϕ e -1 2π d 2 w(∂ϕ ∂ϕ+b 2 e 2bϕ ) n i=1 e (2b(j i +1)+ m b )ϕ(z i ) n-2 a=1 e -m b ϕ(ya) , (3.8)
where Θ n was defined in eq. (2.16). The second line of this formula is the path-integral version of the Liouville theory correlator which appears in our ansatz (2.18), with the expected values for the Liouville momenta α i = b(j i + 1) + m 2b . And the prefactors also agree with the expectations (2.25). Therefore, the path-integral calculation provides a Lagrangian definition (3.1) for the new theories whose correlators we conjectured. This definition will allow us to easily study the symmetries of these theories.

Symmetry algebra

Let us interpret the theories with Lagrangian (3.1) as free theories of the fields (φ, β, γ, β, γ) with contractions

φ(z)φ(w) = -log |z -w| , β(z)γ(w) = 1 w -z , (3.9)
deformed by the interaction term L (m) int = (-β β) m e 2bφ . The chiral symmetry algebra is then the set of holomorphic fields whose OPEs with L (m) int vanish up to total derivatives. Such chiral fields can be constructed from the basic holomorphic fields ∂φ, β, γ by using the normal-ordered product (2.5), see for instance [START_REF] Bouwknegt | W symmetry in conformal field theory[END_REF] for the rules of computing with this product.

We already know one chiral field, namely the stress-energy tensor T (m) (3.2), which generates a Virasoro algebra with central charge

c (m) = 3 + 6(b + b -1 (1 -m)) 2 .
(3.10)

But in the case of the H + 3 model (m = 1) we know that the creation modes of this Virasoro algebra are not enough for generating the full spectrum of the model from the fields Φ j (µ|z). These fields are indeed affine primaries, that is primary fields with respect to the much larger symmetry algebra sℓ 2 . Can we find a larger algebra for general values of m? Let us introduce the following holomorphic currents:

J -= β , (3.11)
J 3 = -βγ -mb -1 ∂φ , (3.12
)

J + = βγ 2 + 2mb -1 γ∂φ -(m 2 b -2 + 2)∂γ , (3.13)
where normal ordering is implicitly assumed when needed. The OPEs of these currents obey the relations (2.1-2.3) of an sℓ 2 algebra at level k = 2 + m 2 b -2 . However, only the currents J -and J 3 are symmetries of our model. The current J + indeed has a nontrivial OPE with

L (m)
int , and is no symmetry. And the stress-energy tensor which we could build from J -, J 3 , J + by the Sugawara construction (2.4) is therefore also no symmetry. It actually differs from the stress-energy tensor (3.2), with respect to which the interaction L (m) int is marginal. The chiral symmetry algebra is therefore generated by the three fields T, J 3 , J -(where from now on we omit the superscipt of T (m) ), and we compute their OPEs as

T (z)T (w) = 1 2 c (m) (z -w) 4 + 2T (w) (z -w) 2 + ∂T (w) z -w + O(1) , (3.14)
T (z)J -(w) = J -(w) (z -w) 2 + ∂J -(w) z -w + O(1) , (3.15)
T (z)J 3 (w) = (1 -m)(1 -mb -2 ) (z -w) 3 + J 3 (w) (z -w) 2 + ∂J 3 (w) z -w + O(1) , (3.16)
J -(z)J -(w) = O(1) , (3.17)

J 3 (z)J -(w) = J -(w) w -z + O(1) , (3.18)
J 3 (z)J 3 (w) = - 1 + 1 2 m 2 b -2 (z -w) 2 + O(1) . (3.19)
This looks very much like the subalgebra of the affine algebra sℓ 2 obtained by removing the current J + , except that there is now a central term in the T J 3 OPE, so that J 3 is no longer a primary field. Notice that this algebra, unlike the Lagrangian L (m) , is invariant under the duality

b → b -1 , m → mb -2 . (3.20)
Notice also that setting J -= 0 yields a smaller algebra generated by T and J 3 , which might be interesting as well. 1 It can be checked that the fields Φ j (µ|z) are primary with respect to our chiral algebra. We can indeed compute the OPEs of their free-field realization (3.3) with J -, J 3 and T , which respectively reproduces the OPEs (2.6), (2.7) and (2.9) with however the conformal dimension ∆ (m) j (2.26). And we can check that the correlators (2.18) have the correct behaviour under global symmetry transformations. We already performed this analysis in the case of the global conformal transformations of the three-point function Ω (m) 3 , this is how we found ∆ (m) j in the first place. The global Ward identity for J -is ( n i=1 µ i )Ω (m) n = 0, which is obviously satisfied. We furthermore

compute n i=1 µ i ∂ ∂µ i Ω (m) n = -(1 -m)(1 -mb -2 )Ω (m)
n . This is what is expected knowing that J 3 is no longer primary, see the T J 3 OPE (3.16).

These simple consistency checks leave two questions about our determination of the symmetry algebra:

1. Is the symmetry algebra large enough? We still have to show that the descendent fields obtained by repeatedly acting on the primaries Φ j (µ|z) with the creation modes of T, J 3 , J -do span the spectrum. In particular, can the Φ j 1 (µ 1 |z 1 )Φ j 2 (µ 2 |z 2 ) OPE be written as a sum over such descendents? In principle, we could address this issue by studying the z 12 → 0 expansion of our ansatz (2.18) for the correlators Ω (m)

n . This would however be tedious. Instead, let us focus on the m = 1 case. The spectrum of the H + 3 model is known to be generated by the 1 We could even consider a three-parameter family of T, J 3 algebras with arbitrary central terms T (z)T (w) =

1 2 c (z-w) 4 + • • • , T (z)J 3 (w) = Q (z-w) 3 + • • • and J 3 (z)J 3 (w) = - k 2 (z-w) 2 + • • • . The relation c = 1 -6 Q 2
k defines a subfamily where the identification T = -1 k (J 3 J 3 ) + Q k ∂J 3 is allowed. After introducing a boson ϕ such that J 3 = ∂ϕ this corresponds to a linear dilaton theory. Our parameters m, b parametrize a different subfamily. modes of J 3 , J + , J -, but we will now argue that the modes of T, J 3 , J -are actually enough. Let us define these modes by T (z) = n∈Z L n z -n-2 and J a (z) = n∈Z J a n z -n-1 ; creation modes are J a n<0 and L n<0 , while primary fields Φ j (µ|z) correspond to primary states |p such that L n>0 |p = J a n>0 |p = 0. Any level one J + -descendent state can by definition be written as |d = J + -1 |p ′ with |p ′ an affine primary state in the H + 3 spectrum. The spectrum being made of continuous representations, this primary can be rewritten as |p ′ = J - 0 |p with |p another primary. Using the Sugawara construction (2.4), we then have

|d = J + -1 J - 0 |p = -J - -1 J + 0 + 2J 3 -1 J 3 0 + (k -2)L -1 |p , (3.21)
which is manifestly a combination of L -1 , J 3 -1 , J - -1 -descendents of affine primary states. This reasoning can be iterated to higher level J + -descendent states. This proof is special to m = 1 and cannot be generalized, but it demonstrates that our T, J 3 , J -symmetry algebra is likely large enough.

Are the chiral fields algebraically independent?

In the H + 3 model, the fact that the chiral fields T, J + , J 3 , J -are not independent but related by the Sugawara construction implies that the correlators obey Knizhnik-Zamolodchikov differential equations. In our generalized theories with parameter m, correlators (2.18) do obey differential equations for values of m such that the field V -m 2b is a Liouville degenerate field. This happens if

m = p + b 2 q , p, q = 0, 1, 2 • • • (3.22)
We therefore expect that the structure of our chiral algebra becomes in some sense reducible for these values of m, so that differential equations can be derived for the correlators. This is what we will explicitly demonstrate in the case m = 2.

Differential equation in the case m = 2

We will first derive the third-order differential equations satisfied by the correlator Ω

n from the Belavin-Polyakov-Zamolodchikov equations satisfied by the corresponding Liouville correlators, and then check that these equations can be recovered from our symmetry algebra.

Third-order BPZ equation

The Virasoro module generated by the Liouville field V -1 b has a null vector at level three [START_REF] Belavin | Infinite conformal symmetry in two-dimensional quantum field theory[END_REF]:

χ 3 = (-1 + 2b -2 )L -3 + 2L -1 L -2 + 1 2 b 2 L 3 -1 V -1 b . (4.1)
Assuming that this null vector vanishes implies that correlators involving V -1 b obey third-order BPZ differential equations. In particular, let us write the BPZ equation associated to a degenerate field V -1 b (y) in the Liouville correlator which appears in our ansatz (2.18). We call y b = y the insertion points of the other degenerate fields so that y a = (y, y b ); moreover we call z I = (z i , y b ) the positions of all fields except V -1 b (y). The BPZ equation then involves the following differential operator:

D = b 2 2 ∂ 3 ∂y 3 + I ∂ ∂y 2 y -z I ∂ ∂z I + 2∆ α I (y -z I ) 2 ∂ ∂y + I ∆ -1 b + 2 (y -z I ) 2 ∂ ∂z I + 2∆ -1 b ∆ α I (y -z I ) 3 . (4.2)
This BPZ equation implies

D ′ Ω (2)
n = 0 where D ′ = Θ 2 b 2 n DΘ -2 b 2
n , with Θ n defined by eq. (2.16). We thus wish to compute D ′ ; at the same time we should perform the change of variables (z i , y a ) → (z i , µ i ) defined by eq. (2.17). In particular we should rewrite

∂ ∂z i = ∂ ∂z i ya in terms of δ δz i ≡ ∂ ∂z i µ j
. This will be done thanks to the identity [3]

n i=1 1 y -z i ∂ ∂z i + b 1 y -y b ∂ ∂y b -X ∂ ∂y = n i=1 1 y -z i δ δz i , (4.3)
where we defined

X ≡ b 1 y -y b - n i=1 1 y -z i = - n i=1 µ i (y-z i ) 3 n i=1 µ i (y-z i ) 2 . ( 4.4)
We also use

∂ ∂y = n i=1 µ i y -z i ∂ ∂µ i . (4.5)
Explicit calculations yield

D ′ = b 2 2 ∂ 3 ∂y 3 -X ∂ 2 ∂y 2 + 2 ∂ ∂y L -2 (y) -4b -2 XL -2 (y) + (2b -2 -1) L -3 (y) - i µ i (y -z i ) 3 ∂ ∂µ i -X i µ i (y -z i ) 2 ∂ ∂µ i , (4.6)
where we defined

L -2 (y) = i 1 y -z i δ δz i + ∆ (2) j i y -z i , (4.7)
L -3 (y) = - i 1 (y -z i ) 2 δ δz i + 2∆ (2) j i y -z i , (4.8)
where the conformal dimensions ∆

(2) j i and ∆ α i are related by eq. (2.25). The operator D ′ can be understood as a generalization of the sℓ 2 Knizhnik-Zamolodchikov differential operator, which in our notations can be written as [3]

D KZ = b 2 ∂ 2 ∂y 2 + L -2 (y) . (4.9)

Reformulation of the BPZ equation in terms of symmetry generators

If our identification of the symmetry algebra is correct, the differential equation

D ′ Ω (2)
n = 0 should have a reformulation in terms of the chiral fields T, J 3 , J -. This is a non-trivial requirement on D ′ , and we will now show that it is satisfied.

Let us denote Q = Φ j i (µ i |z i ) . The actions of differential operators ∂ ∂y , L -2 (y) and L -3 (y) on Q have immediate interpretations as insertions of the chiral fields J 3 (y), T (y) and ∂T (y), for instance

∂ ∂y Q = i z i dt J 3 (t) y -t Q = y dt J 3 (t) t -y Q = J 3 (y)Q , (4.10)
where we used the formula (4.5) for ∂ ∂y and the OPE (2.7) of J 3 with Φ j (µ|z). Iterating the actions of such differential operators yields results like

∂ ∂y L -2 (y) Q = ∂ ∂y T (y)Q = ∂T Q + (J 3 T )(y)Q , (4.11)
where the normal-ordered product (J 3 T ) is defined as previously, see eq. (2.5). Moreover, we have

i µ i (y -z i ) 3 ∂ ∂µ i Q = 1 2 i z i dt ∂ 2 J 3 (t) y -t Q = 1 2 ∂ 2 J 3 (y)Q . (4.12)
Finally, the factors of X in D ′ can be related to insertions of ∂J -and ∂ 2 J -, thanks to eq. ( 4.4) and identities of the type

i µ i (y -z i ) 3 P(y)Q = 1 2 y dt ∂ 2 J -(t) t -y P(y)Q = 1 2 (∂ 2 J -P)(y) Q , (4.13)
which is valid for any operator P. Therefore, the equation

D ′ Ω (2)
n = 0 with the operator D ′ given in eq. (4.6) can be rewritten as R(y) Φ j i (µ i |z i ) = 0, where

R = 1 2 b 2 (∂J -(J 3 (J 3 J 3 ))) + 2(∂J -(J 3 T )) + [2b -2 + 1](∂J -∂T ) + 3 2 b 2 (∂J -(J 3 ∂J 3 )) + 1 2 [b 2 + 1 -2b -2 ](∂J -∂ 2 J 3 ) -1 2 (∂ 2 J -(J 3 J 3 )) + [-1 + b -2 ](∂ 2 J -∂J 3 ) -2b -2 (∂ 2 J -T ) . (4.14)
We can already conjecture that, for all the values (3.22) of m such that the fields V -m 2b is degenerate, the BPZ equations can similarly be rewritten in terms of operators T, J -, J 3 . It is however not clear how to deduce our operator R from a null vector like (4.1), without performing the explicit calculations as we did.

The equation R(y) Φ j i (µ i |z i ) = 0 of course does not mean that the operator R should be set to zero, because this equation is valid only at special points y = y 1 • • • y n-2 . These points were defined (2.17) as the zeroes of ϕ(t) = n i=1 µ i t-z i ; they can be characterized in terms of the operator J -by

J -(y) n i=1 Φ j i (µ i |z i ) = 0 . (4.15)
Actually, for any operator P we have (J -P)(y) n i=1 Φ j i (µ i |z i ) = 0. Therefore, the operator R is expected to vanish modulo operators of the type (J -P).

In other words, we expect that this operator corresponds to a subsingular vector R|0 in the vacuum module of our symmetry algebra. That is, if we would set the singular vector J -|0 to zero, then R|0 would become a singular vector in the resulting coset module. We are actually not setting J -|0 to zero, but the presumptive subsingular vector R|0 is nevertheless associated to differential equations satisfied by the correlation functions. Relations between subsingular vectors and differential equations were found previously by Dobrev [START_REF] Dobrev | Subsingular vectors and conditionally invariant (q-deformed) equations[END_REF] in the context of finite-dimensional symmetry algebras, but they do not seem widespread in conformal field theory so far. 2

Subsingular vectors of the symmetry algebra

In order to show that the third-order differential equation can be deduced from our symmetry algebra, we still have to prove that the operator R (4.14) corresponds to a subsingular vector. We will actually investigate the more general operator

R {λ i } = λ 1 (∂J -J 3 J 3 J 3 ) + λ 2 (∂J -J 3 T ) + λ 3 (∂J -∂T ) + λ 4 (∂J -J 3 ∂J 3 ) + λ 5 (∂J -∂ 2 J 3 ) + λ 6 (∂ 2 J -J 3 J 3 ) + λ 7 (∂ 2 J -∂J 3 ) + λ 8 (∂ 2 J -T ) , (4.16)
which depends on arbitrary coefficients λ 1 • • • λ 8 . Here and in the following we use the shorthand notation ABCD = (ABCD) = (A(B(CD))) for multiple normal orderings, see [START_REF] Bouwknegt | W symmetry in conformal field theory[END_REF] for more details on calculations involving such expressions.

We will perform this investigation with the help of the free fields (φ, β, γ). Since the map from (J -, J 3 , T ) to (φ, β, γ) defined by the equations (3.11), (3.12) and (3.2) is a morphism of algebras, it can indeed be used for determining whether R {λ i } generates a nontrivial ideal of the coset algebra obtained by modding out the ideal generated by J -. (This is the algebraic formulation of our subsingular vector problem.) So let us compute the operator R {λ i } in terms of the fields (φ, β, γ), modulo operators of the type (J -P) = (βP). We will denote by ≃ the equality of operators modulo such terms. For example, we find

(J 3 (J 3 J 3 )) ≃ 2∂β∂γ -1 2 ∂ 2 βγ -3mb -1 ∂βγ∂φ -m 3 b -3 ∂φ 3 , (4.17)
(J 3 T ) ≃ ∂β∂γ + mb -1 ∂φ 3 -m(1 + b -2 (1 -m))∂φ∂ 2 φ , (4.18)
(J 3 ∂J 3 ) ≃ 1 2 ∂ 2 βγ + mb -1 ∂βγ∂φ + m 2 b -2 ∂φ∂ 2 φ , (4.19)
(∂T ) ≃ -∂β∂γ -2∂φ∂ 2 φ + (b + b -1 (1 -m))∂ 3 φ , (4.20) (∂ 2 J 3 ) ≃ -∂ 2 βγ -2∂β∂γ -mb -1 ∂ 3 φ . ( 4

.21)

We then compute

R {λ i } ≃ [2λ 1 + λ 2 -λ 3 -2λ 5 ] ∂β 2 ∂γ + -3mb -1 λ 1 + mb -1 λ 4 ∂β 2 γ∂φ + -m 3 b -3 λ 1 + mb -1 λ 2 ∂β∂φ 3 + -m(1 + b -2 (1 -m))λ 2 -2λ 3 + m 2 b -2 λ 4 ∂β∂φ∂ 2 φ + (b + b -1 (1 -m))λ 3 -mb -1 λ 5 ∂β∂ 3 φ + -1 2 λ 1 + 1 2 λ 4 -λ 5 + λ 6 -λ 7 ∂β∂ 2 βγ + m 2 b -2 λ 6 -λ 8 ∂ 2 β∂φ 2 + -mb -1 λ 7 + (b + b -1 (1 -m))λ 8 ∂ 2 β∂ 2 φ . (4.22)
Let us solve the equation R {λ i } ≃ 0, which when satisfied implies that R {λ i } corresponds to a subsingular vector. This equation leads to a system of 8 linear equations for the 8 unknowns λ 1 • • • λ 8 . Actually, the first five equations form a closed subsystem of equations for λ 1 • • • λ 5 . This subsystem has a nonzero solution only if its determinant vanishes, that is if

(m -1)(m -2)(m -b 2 )(m -2b 2 ) = 0 . (4.23)
The four solutions of this equation correspond to the Liouville operators V -m 2b being degenerate, leading to BPZ differential equations of order 2 or 3. This is therefore a strong check of our claim that the ansatz (2.18) does define conformal field theories with symmetry algebras generated by T, J 3 , J -. Let us perform a more detailed check in the case m = 2, by explicitly computing the nonzero solution to the system R {λ i } = 0:

λ 1 = 1 2 b 2 , λ 2 = 2 , λ 3 = 2b -2 + 1 , λ 4 = 3 2 b 2 , λ 5 = 1 2 (b 2 + 1 -2b -2 ) , λ 6 = -1 2 , λ 7 = -1 + b -2 , λ 8 = -2b -2 . (4.24)
With these values of of λ i , the operator R {λ i } (4.16) does agree with the operator R (4.14), which we found by reformulating the third-order BPZ equation.

Concluding remarks

We have argued that for any choice of m and the central charge c, the object Ω (m) n of eq. ( 2.18) can be interpreted as an n-point correlation function in a conformal field theory with a chiral symmetry algebra generated by the fields T, J 3 , J -with OPEs (3.14-3.19). This conformal field theory is solvable in the sense that its correlation functions are known in terms of Liouville correlation functions. Liouville theory itself has been solved in the sense that its three-point function on the sphere is explicitly known, while its n-point functions on arbitrary Riemann surfaces can in principle be deduced thanks to the conformal symmetry of the theory. (See [START_REF] Teschner | Liouville theory revisited[END_REF] for a review.) The solution of the new theory is therefore not very explicit, and no closed formula can be written for its three-point function except for some "degenerate" values of m (3.22) when it satisfies a differential equation.

This differential equation provided the most non-trivial test of our claims. In the case m = 2, we compared the differential equation deduced from our chiral symmetry algebra with the thirdorder BPZ equation, and found explicit agreement. We expect such an agreement to hold for all degenerate values of m, and it would be interesting to check this beyond case-by-case calculations.

There may exist other theories based on the same symmetry algebra. For instance, in the case m = 1, the non-rational, non-unitary H + 3 model shares its symmetry algebra with the unitary AdS 3 WZW model and with the rational SU 2 WZW models. For general values of m, our model remains non-rational and non-unitary. (The lack of unitarity still follows from Gawedzki's H + 3model argument [START_REF] Gawedzki | Noncompact WZW conformal field theories[END_REF], although it is not quite clear what the scalar product on the spectrum is, as J - has no conjugate field.) It would be interesting to construct rational or unitary theories based on the same symmetry algebra. Note however that our reasoning around eq. (3.21), which showed that in the case m = 1 the modes of T, J -and J 3 did generate the whole H + 3 spectrum, relied on the fact that the spectrum is purely continuous. But the AdS 3 and SU 2 WZW models involve discrete and finite-dimensional representations respectively, which presumably cannot be generated from the modes of T, J -and J 3 . Therefore, a rational theory based on the (T, J 3 , J -) algebra would probably differ from the SU 2 WZW model and have a smaller spectrum.

We have not assumed any restrictions on the choice of the parametrs m and c. For some purposes it might be useful to restrict say the central charge of the theory and the conformal dimensions of the fields to be real. Such restrictions could be dictated by particular applications. From the point of view of conformal field theory, the new theories are however well-defined, and their correlators manifestly satisfy crossing symmetry, a very stringent constraint. It is therefore possible to study issues like the solution of these theories on Riemann surfaces with boundaries.

I am very grateful to Vladimir Dobrev for pointing out that the notion of a subsingular vector is relevant here, and for patiently explaning some of the literature on this topic to me.

Acknowledgments

I am grateful to Volker Schomerus, Thomas Quella and Vladimir Fateev for useful discussions and comments on the manuscript, and to DESY, Hamburg for hospitality while part of this work was done. In addition, I thank the theoretical physics groups at King's College London and Sogang University in Seoul for listening and reacting to informal talks on this subject.


Matthieu Perreira
email: mperreir@univ-lr.fr

Da Silva

Vincent Courboulay

Armelle Prigent

Pascal Estraillier

Real-time face tracking for attention aware adaptive games

This paper presents a real time face tracking and head pose estimation system which is included in an attention aware game framework. This fast tracking system enables the detection of the player's attentional state using a simple attention model. This state is then used to adapt the game unfolding in order to enhance user's experience (in the case of adventure game) and improve the game attentional attractiveness (in the case of pedagogical game).

Introduction

Usually, computer software has a very low understanding of what the user is actually doing in front of the screen. It can only collect information from its mouse and keyboard inputs. It has no idea of whether the user is focused on what it is displaying, it doesn't even know if the user is in front of the screen. One of the first steps to making computer software more context aware is to make it attention aware. In [START_REF] Roda | Attention Aware Systems: Theories, Applications, and Research Agenda[END_REF] attention aware systems are defined as Systems capable of supporting human attentional processes.

From a functional point of view, it means that these systems are thought as being able to provide information to the user according to his estimated attentional state. This definition shares some objectives with the definition of adaptive systems [START_REF] Weibelzahl | Evaluation of Adaptive Systems[END_REF] which are system that are able to adapt their behaviour according to the context. An attention aware system must be an adaptive system. According to [START_REF] Roda | Attention Aware Systems: Theories, Applications, and Research Agenda[END_REF], one of the best ways to estimate user's attention is using gaze direction. Consequently, building good adaptive attention aware system requires estimating reliably user's gaze, but in an unconstrained environment this is a very difficult task. To overcome this problem we propose, in a first step, to use head pose as an estimator for gaze direction. Head pose provides less accurate but more robust gaze estimation. In this paper, we present a real time vision based face tracking and head pose estimation system coupled with a simple inattention model that will allow an adaptive game to detect attentional shifts and adapt its unfolding accordingly. This model is implemented in two different types of systems (see 1):

-A pedagogical game in which information about the user's attention is used to adapt the game unfolding in order to refocus the player's attention. It is used as a tool for pedo-psychiatrists working with children with autism in the pedo-psychiatric hospital of La Rochelle in order to improve children's attention. -An adventure game in which user's attentional state helps modifying the game scenario in order to make the game more immersive and fun to play.

In the next section we introduce the attention aware adaptive game framework which is using our head pose based gaze estimation algorithm.

An attention aware adaptive game framework

Adaptive game is a subcategory of adaptive systems that can take good advantage of being attention aware. They are designed to react and adapt their unfolding according to the players (explicit) actions. Thus, being able to see if the player is facing the screen and is attentive allows the game to adapt its actions to the situation.

Attention

Attention is historically defined as follows [START_REF] James | [END_REF]:

Everyone knows what attention is. It is the taking possession by the mind in clear and vivid form, of one out of what seem several simultaneously possible objects or trains of thought...It implies withdrawal from some things in order to deal effectively with others. Thus, attention is the cognitive process of selectively concentrating on one thing while ignoring other things. In spite of this single definition, it exists several types of attention [START_REF] Ould Mohamed | Attention analysis in interactive software for children with autism[END_REF]: awakening, selective attention, maintained attention, shared attention, internal or external absent-mindedness and vigilance. For an interactive task, we are mainly interested in selective and maintained attention. The analysis of the first one allows knowing whether people are involved in the activity. The second one enables us to assess the success of the application. It has been proven that the same functional brain area were activated for attention and eye movements [START_REF] Corbetta | A common network of functional areas for attention and eye movements[END_REF]. Consequently, the best attention marker we can measure is undoubtedly eyes and gaze behaviour. A major indicator concerning another type of attention, vigilance, named PERCLOS [START_REF] Dinges | Evaluation of techniques for ocular measurement as an index of fatigue and the basis for alertness management[END_REF] is also using such markers. In continuity of such studies, we based our markers on gaze behaviour, approximated by head pose, to determine selective and maintained attention. A weak hypothesis is that a person involved in an interesting task focuses his/her eyes on the salient aspect of the application (screen, avatar, car, enemy, text...) and directs his/her face to the output device of the interactive application (screen). Nevertheless, if a person does not watch the screen, it does not necessarily mean that he/she is inattentive; he/she can be speaking with someone else about the content of the screen [START_REF] Kaplan | The challenges of joint attention[END_REF]. In order to treat these different aspects, we have decided to adopt the following solutions: if the user does not watch the screen during a time t, we conclude to inattention. In the following subsection, we present how t is determined. If inattention is detected, we inform the application.

A simple model of human inattention. The goal of this model is to define the delay after which the application tries to refocus the user on the activity. Actually, we easily understand that in this case, an interactive application does not have to react the same way if people play chess, role player game or a car race. Until now, this aspect of the game was only directed by the time during which nothing was done on the paddle or the keyboard. We based our model of what could be named inattention on two parameters:

1. the type of application; 2. the time spent using the application.

The last parameter depends itself on two factors:

1. a natural tiredness after a long time 2. a disinterest more frequent during the very first moments spent using the application than once attention is focused, this time corresponds to the delay of immersion.

Once the parameters are defined, we propose the following model in order to define the time after which the application try to refocus the player who does not look at the screen.

Potential of attention.

As we mentioned, potential of attention depends mainly on two parameters, tiredness and involvement. We have decided to model arousal (the opposite of tiredness), or potential of attention, by a sigmoid curve parameterized by a couple of real number β 1 and β 2 . β 2 represents the delay after which the first signs of fatigue will appear and β 1 is correlated to the speed of apparition of tiredness (Figure 2).

P arousal = exp -β1t+β2 1 + exp -β1t+β2 , (1)
where β 1 and β 2 are two real parameters.

For the second parameter, we have once again modelled involvement, or interest probability, by a sigmoid. We started from the fact that activity is a priori fairly interesting, but if the person is involved after a certain time ruled by α 2 , we can consider that interest is appearing at a speed correlated to α 1 (Figure 2).

P interest = 1 1 + exp -α1t+α2 .
(

) 2
For our global model of potential of attention, we couple both previous models in order to obtain: Delay of inattention. Once this model is defined, we are able to determine the time after which the software has to react (if the person still does not look at the screen). Here, it is an arbitrary threshold γ guided by experience, which characterizes each application. The more the application requires attention, the higher this coefficient is. The model we have adopted is an exponential function.

P attention = P interest * P arousal , (3)
D game = exp γ(t) * Pattention , (4)
γ is a function of time because we have estimated that it can exist several tempo in an application (intensive, stress, reflection, action ...). As a conclusion, we can summarize our model of inattention by the two following steps (Figure 3):

depending on the time elapsed from the beginning of the application, we estimate the potential of attention P attention (t); depending on this potential and the application, we estimate the delay D γ(t) (P attention (t)) after which the software has to refocus the inattentive player.

Please note that this model was validated on only five children and still need to be validated on more people. Nevertheless, the first tests are quite promising.

An adaptive framework

As we already mentioned, our objective is to implement a system that interacts in a smart way with the person using the application. It consists in establishing a multi-mode and multimedia dialogue between the person and the system. Consequently, the conception of the platform, represented Figure 4, was guided by the following constraints:

it must track gaze behaviour, determined as the best markers of attention;

it must allow to focus user attention and be able to refocus it if necessary; it must not perturb the user by inopportune interaction.

The system can be divided into three sub-parts:

the system of observation and behaviour analysis: monitors the player's attention; the decision system: adapts the execution of games; the action system: runs the game. This architecture needs a fast and robust gaze estimation system in order to respond to its environment in an adaptive and dynamic manner. To achieve this goal, we have built a fast tracking system dedicated to adaptive systems. This system is described in the following section.

3 Face tracking for adaptive games

Constraints

As the system is designed to be used by a wide range of applications and users (from educational games for children with autism to adventure games for "common gamers"), some constraints have emerged:

non invasive material; low cost; single user; recordable information; standard computer; unconstrained environment.

Our system is based on a low cost ieee-1394 camera connected to a standard computer. Despite its low cost, this camera captures video frames of size 640x480 at 30 frames per second which are suitable characteristics for both accurate face features localization and efficient face features tracking. The choice of a grayscale camera instead of a more common colour camera is driven by the fact that most of the aimed applications are performed in an indoor environment. In such an environment, the amount of light available is often quite low, as grayscale cameras usually have more sensitivity and have a better image quality (as they don't use Bayer filters), they are the best choice. Another advantage of grayscale cameras is that they can be used with infra-red light and optics that don't have infra-red coating in order to improve the tracking performance by the use of a non invasive more frontal and uniform lightning.

Architecture

The tracking algorithm we have developed is built upon four modules which interoperate together in order to provide a fast and robust face tracking system (see Figure 5). The algorithm contains two branches: one for face detection and a second for face tracking. At run time, the choice between the two branches is made according to a confidence threshold, evaluated in the face checking and localization module. Pre-processing. Before face or radial symmetry detection, the input image must be pre-processed in order to improve face and face feature detection performance. The main pre-processing steps are image rescaling and lightning correction, also called contrast normalization. The input image is rescaled so that low resolution data is available for radial symmetry detection algorithms. Contrast normalisation consists in adapting each pixel intensity according to the local mean intensity of its surrounding pixel. As this task is performed by humans retina, several complex models have been developed to mimic this processing [START_REF] Beaudot | The neural information in the vertebra retina: a melting pot of ideas for artifical vision[END_REF]. As our system needs to be real-time, we have chosen to approximate this retinal processing by a very simple model which consists in the following steps:

1. For each image pixel, build a weighted mean M x,y of its surrounding pixels.

In our implementation we used first order integral filtering (Bartlett Filter) in order to achieve fast filtering. Note that first order integral filters are an extension of the commonly used zero order integral filters (box filters). For more information about generalized integral images see [9] 2. Calculate the normalized pixel intensity:

I x,y = S x,y (M x,y + A)
with S the source image, I the normalized image, and A a normalization factor.

Figure 6 shows the result of our simple contrast normalization algorithm on a side lit scene.

Radial symmetry detection for face shape and eye detection. Once the image is pre-processed, we use a set of radial symmetry detector in order to localize a face region candidate that will be further checked by the face checking and localization module. Once again our real-time constraint guided the choice of the algorithms we used. Face ovoid shape is detected in low resolution version of the input image (typically 160x120) using an optimized version of the Hough transform (Fig. whereas eyes are detected using an optimized version of the Loy and Zelinsky transform [START_REF] Loy | Fast radial symmetry for detecting points of interest[END_REF](Fig. 6.d). In order to speed up both transforms, the following improvements have been made: only pixels with a gradient magnitude above a pre-defined threshold are processed, the algorithms vote in only one accumulator for all radius and accumulators are smoothed only once at the end of the processing. Using these two symmetry map and a set of face geometry based rules we define the face candidate area.

Face checking and localization. This module serves two purposes:

-When called from the face detection branch, it checks if the face candidate area really contains a face and outputs the precise localization of this face. -In the case of face tracking, it only finds the new position of the face.

In both cases, the module outputs a confidence value which reflects the similarity between the interface face model and the real face image. Face checking and localization is based on the segmentation of the face candidate image into several blobs. The source frame is first filtered by a DoG filter; the result image is then adaptively thresholded. The resulting connected components (blobs) are then matched on a simple 2D face model in order to check and localize the face.

Head pose estimation.

Similarly to previous research we use triangle geometry for modelling a generic face model. For example, [START_REF] Kaminski | Head orientation and gaze detection from a single image[END_REF] resolves equations derived from triangle geometry to compute head pose. Since we favour speed against accuracy, we use a faster and simpler direct approximation method based on side length ratios.

Performance and robustness

Processing time. We measured the processing time of he algorithm on a laptop PC equipped with a 1,83GHz Intel Core Duo processor. We obtained the following mean processing times:

-Face detection: 30 milliseconds (first detection or detection after tracking failure).

-Face tracking: 16 milliseconds Processing time include all processing steps, from pre-processing to head pose estimation. Since image capture is done at 30fps and the algorithm is using only one of the two processor cores, our system uses 50% of processor time for face detection and 25% of processor time for face tracking. Consequently, the algorithm is fast enough to enable running a game in parallel on a standard middle-end computer.

Robustness. As can be seen from the tracking example shown on figure 7, the algorithm can handle a broad range of head orientation and distance. The contrast normalization step also allows the algorithm to run under different illumination conditions. However, this algorithm is designed to be fast, as a consequence tracking performances still need to be improved under some lightning conditions (backlit scene, directional lighting casting hard shadows, etc.). A future version of the algorithm may use a colour camera and skin colour detection algorithms (as in [START_REF] Schwerdt | Robust face tracking using color[END_REF] or [START_REF] Séguier | A very fast adaptive face detection system[END_REF]) to improve face detection and tracking robustness. This modification would however prevent us from using infra-red light to improve the algorithm performances under poor lightning conditions. Another possibility would be to add an infra-red lightning and adapt the current algorithm to this new lightning in order improve the robustness of the system. 4 Discussion and future work

We described a complete framework for attention aware adaptive games. This framework uses a fast face tracking and head pose estimation algorithm coupled with a simple model of human inattention in order to generate feedback information about the attentional state of the player. This system is currently implemented in an adventure and a pedagogical game.

In order to improve this framework, we are currently working at building a more complex and realistic attentional model which would not only react to attentional shifts but could also predict them. We are also working at using colour information to enhance the accuracy and robustness of our face tracking system as it can provide performance improvements without impacting too much processing time.

Fig. 1 .

 1 Fig. 1. Left: A screen capture of the adventure game prototype. Right: a screen capture of the pedagogical game together with a preview of the head tracking system.

Fig. 2 .

 2 Fig. 2. Left: model of tiredness evolution. Middle: model of interest probability. Right: model of potential of attention. Abscissa represents time in minutes. Each curve is designed for a single person involved in an unique activity. (α1 = 1, α2 = 3, β1 = 0.3, β2 = 12)

Fig. 3 .

 3 Fig. 3. Curves used to determine the delay of software interaction.

Fig. 4 .

 4 Fig. 4. General architecture.

Fig. 5 .

 5 Fig. 5. Architecture of the face tracking and pose estimation system.

Fig. 6 .

 6 Fig. 6. a) Tracking result of a side lit scene. b) Source image after lightning correction. c) Result of face ovoid detection (Hough Transform). d) Result of eyes detection (Loy and Zelinsky Transform).

Fig. 7 .

 7 Fig. 7. Left: Result of the tracking algorithm for different face distances and orientation.

Acknowledgement

This research is partly funded by the French Poitou-Charentes county and the Orange Foundation. The authors would also like to thank doctor Mr. D. Lambert Head of Department of Child Psychiatry of La Rochelle hospital (France) and his team, in particular: Mr. V. Gabet for their useful advices regarding the rehabilitation methods dedicated to children with autism.