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Deformation Quantization of Odd Dimensional anti-de Sitter Spaces as Contact Manifolds

We quantize odd dimensional anti-de Sitter spaces by applying the method of deforming contact manifolds proposed by Rajeev. The construction in the present paper consists of the identification of the odd dimensional anti-de Sitter space as a hypersurface of contact type and the subsequent use of 'symplectization' principle. We also show that this construction generalizes to any odd dimensional hypersurface which can be represented as a nonzero level set of a homogenous function.


Introduction
Recently Rajeev, in his work on quantization of thermodynamics [1], proposed a method of quantizing contact manifolds. Basic idea behind this method is the symplectization principle of contact manifolds. This principle allows one to set up a 1-1 correspondence between the functions on a contact manifold and a subset F of functions on a symplectic manifold. Then one uses the usual deformation of the symplectic manifold to deform F and map the result back to the contact manifold. The first order correction to the resulting star product is given by the Legendre bracket which is the analog of Poisson bracket in contact geometry. In [1] this method was applied to quantize odd dimensional spheres.
Considering the central role of AdS/CFT correspondence [2,3,4] in Mtheory and the important relations between noncommutative geometry [5] and string theory [6] we find it useful to apply Rajeev's method to quantize odd dimensional anti-de Sitter spaces.
As usual we will regard AdS 2n+1 as a hypersurface in R 2n+2 defined by the equation We will regard R 2n+2 as a symplectic manifold with the canonical symplectic form. Our symplectization will rely on the fact that AdS 2n+1 is a hypersurface of contact type. Then we will show how one can lift functions on AdS 2n+1 to a subset of homogenous functions on R 2n+2 . Following the general strategy of [1] we will use the star product on R 2n+2 to induce one on AdS 2n+1 . The homogeneity of the function defining Ads 2n+1 is an essential ingredient of the constructions presented in this paper. In fact we will show that our results can be applied to any odd dimensional hypersurface which can be represented as a nonzero level set of a homogenous function.

Contact Geometry
In this section we will summarize basic results from contact geometry that will be useful in our constructions. Detailed accounts of the subject can be found in [7,8].
Let M be a manifold of dimension 2n+1 and ξ a 2n dimensional subbundle of T M. We will assume that there exist a global 1-form α such that kerα = ξ and dα is nondegenerate on ξ. Then ξ is called a contact structure and α is called the contact 1-form. The assumptions on α are equivalent to kerα = ξ and α ∧ (dα) n = 0. Since dα is a 2-form on an odd dimensional manifold, it must be degenerate. However, by assumption dα is nondegenerate on a 2n dimensional space ξ. Therefore there exist a unique direction along which dα is degenerate. This gives us the definition of the Reeb vector field as the unique vector field Y such that A vector field X satisfying £ X α = g X α where g X ∈ C ∞ (M) is called a contact vector field. In particular the Reeb vector field Y is a contact vector field with g Y = 0. There is a one-to-one correspondence between the set of all contact vector fields and C ∞ (M). Given a contact vector field X the corresponding function H is defined as

Conversely given a function H the corresponding contact vector field is
The analog of the Poisson brackets in contact geometry is the Legendre bracket defined by Legendre bracket is bilinear, antisymmetric and satisfies Jacobi identity. However the Leibnitz rule is not satisfied. Instead it satisfies: which is sometimes called the generalized Leibnitz rule. Since the Leibnitz rule is not satisfied it follows that the constant function is not in the center of the algebra. In fact, upon quantization [1] the constant function does not become a constant multiple of the identity. This is the novel feature of quantization on contact manifolds. Let Q be a hypersurface in a symplectic manifold (N, ω). If there exist a 1-form α in Q such that dα = ω| Q then α is a contact form. In this case Q is called a hypersurface of contact type. A Liouville vector field in a symplectic manifold is defined as a vector field satisfying £ T ω = ω. An equivalent condition for Q to be a hypersurface of contact type is the existence of a Liouville vector field T which is transverse to Q.

Contact Geometry of AdS 2n+1
Consider the 2n + 1 dimensional anti-de Sitter space AdS 2n+1 . This is a hypersurface in R 2n+2 defined by the equation It will be useful to call the even coordinates p and odd coordinates q. Then We will think of R 2n+2 as a symplectic manifold with the canonical symplectic form Clearly ω = dα 0 with Consider the vector field T in R 2n+2 given by It follows that T is a Liouville vector field on R 2n+2 . Notice that ι T ω = α 0 . Moreover we have This is nothing but Euler's homogenous function theorem applied to R, a homogenous function of degree 2. Consequently we see that on AdS 2n+1 where R = −1: So T is transversal to AdS 2n+1 . Thus odd dimensional AdS spaces are hypersurfaces of contact type. The contact form on AdS 2n+1 is Let us consider the Hamiltonian vector field corresponding to the function R Explicitly we have Clearly X R is tangent to AdS 2n+1 . For any vector field V tangent to AdS 2n+1 we have Thus on AdS 2n+1 we have which, when restricted to AdS 2n+1 , gives Thus X R is the negative of the Reeb vector field on AdS 2n+1 .

Deformation Quantization of AdS 2n+1
Now let us try to lift functions on AdS 2n+1 to R 2n+2 . We will consider the functions that can be expanded in the eigenfunctions of the Laplacian on AdS 2n+1 [9,10,11,12]. Thus we can expand in e.g.
Here F is a hypergeometric function as given in [11], Y l,{m} are the spherical harmonic on S 2n−1 . The coordinates of a point on AdS 2n+1 in the ambient space are given in terms of the angular variables as Here n i=1 Ω 2 i = 1, 0 ≤ ρ < π 2 , 0 ≤ τ < 2π and we denoted the coordinates p 2 , q 2 , . . . by x 1 , x 2 , . . ., respectively. Expressing Φ's in terms of the original variables and continuing the result to R 2n+2 gives us a function on R 2n+2 . The effect of this on the coordinates is the substitution: The resulting function is defined only on ( Geometrically this can be understood as follows. The transverse vector field T maps a level surface of R onto another, infinitesimally close level set. In such infinitesimal steps one covers the region in question which is thence diffeomorphic to the trivial bundle AdS 2n+1 × R + T . Notice that the resulting function is homogenous of degree 0. In particular it is annihilated by the vector field T . Given a function F on AdS 2n+1 we denote its extension described above by F 1 and define the desired lift by This lift is homogenous of degree 2. Notice that 1 = −R. Now we can check that the Poisson bracket of two such lifts gives us a function whose restriction to AdS 2n+1 gives the Legendre bracket of the original functions. Let F be a function on AdS 2n+1 . We will denote the corresponding contact vector field by X F . The Hamiltonian vector field corresponding to the lift F will be denoted by X F . In order to compare the Poisson bracket with the Legendre bracket we must relate Hamiltonian vectors field to contact vector fields. We will show that the latter are the restrictions of the former on AdS 2n+1 . According to the general theory Let {V a } be a basis for ξ and recall that dα = ω| AdS . Expanding Let us also express X e F in the basis {T, X H , V a } as Then Now and On the other hand we have Contracting this with V b we get Comparing this with (30) we see that A a = Z a . Similarly contraction with T gives and finally contracting by X R we get Substituting these into (35) we get the desired result Now, following Rajeev [1] we can define the star product on AdS 2n+1 as This provides the deformation of AdS 2n+1 as a contact manifold.

Generalizations and Conclusion
Our results can be generalized as follows. Let Q be a hypersurface defined as a nonzero level set of a homogenous function on an even dimensional space. If Q is defined by an equation of the form R = a = 0 where R is a homogenous function of degree r the transverse Liouville vector field can be chosen as This turns Q into a hypersurface of contact type. Again thanks to the homogeneity of R the Hamiltonian vector field X R gives the Reeb field on the hypersurface. The crucial point in the problem is the construction of the lift. This can be accomplished if one can find a suitable basis for the functions on the hypersurface which can be lifted to homogenous functions of degree r in the ambient space. In particular this construction works for odd dimensional spheres [1], for de Sitter spaces and, as we have shown explicitly, for anti-de Sitter spaces.