Asymptotic commutativity of quantized spaces: the case of $\mathbb{CP}^{p,q}$

We present a procedure for quantizing complex projective spaces $\mathbb{CP}^{p,q}$, $q\ge 1$, as well as construct relevant star products on these spaces. The quantization is made unique with the demand that it preserves the full isometry algebra of the metric. Although the isometry algebra, namely $su(p+1,q)$, is preserved by the quantization, the Killing vectors generating these isometries pick up quantum corrections. The quantization procedure is an extension of one applied recently to Euclidean $AdS_2$, where it was found that all quantum corrections to the Killing vectors vanish in the asymptotic limit, in addition to the result that the star product trivializes to pointwise product in the limit. In other words, the space is asymptotically anti-de Sitter making it a possible candidate for the $AdS/CFT$ correspondence principle. In this article, we find indications that the results for quantized Euclidean $AdS_2$ can be extended to quantized $\mathbb{CP}^{p,q}$, i.e., noncommutativity is restricted to a limited neighborhood of some origin, and these quantum spaces approach $\mathbb{CP}^{p,q}$ in the asymptotic limit.


Introduction
The AdS/CF T correspondence principle posits strong/weak duality between the quantum gravity in the bulk of an asymptotically anti-de Sitter (AdS) space and a conformal field theory (CFT) on the boundary of this space. [1,2] For obvious reasons, however, most practical applications of the correspondence principle utilize classical gravity in the bulk. Even though a fully consistent quantum theory of gravity remains out of reach, there are model independent indications that any theory of quantum gravity will require a quantization of spacetime [3][4][5]. The quantization of AdS, or more generally asymptotically AdS, spacetimes has been examined in two dimensions, [6][7][8][9][10] and four dimensions. [11] Its application to the correspondence principle has received only some initial work in two dimensions. [12,13] While in this article we do not directly address the quantization of general AdS spaces of dimension larger than two, we do present a procedure for quantizing another set of non-trivial non-compact geometries generalizing the two dimensional case, namely indefinite complex projective spaces in arbitrary dimensions, CP p,q , q ≥ 1. We also introduce relevant star products for these spaces. CP p,q is a noncompact version of CP n . The simplest example of an indefinite complex projective space is CP 0,1 , which is equivalent to two dimensional anti-de Sitter space, or more precisely Euclidean anti-de Sitter space, EAdS 2 . Another example is CP 1,2 , which is an S 2 bundle over AdS 4 . [11] While the noncommutative generalization of the compact CP n has received some attention [14], the same cannot be said about the non-compact case, or other non-trivial non-compact spaces. Hasebe has done a study of quantized, or 'fuzzy', hyperboloids, [15] while Steinacker and Sperling have applied such spaces, more specifically the fuzzy four-hyperboloid, or noncommutative AdS 4 , to quantum cosmology. The quantization in [11] is made unique with the demand that it preserves the full isometry algebra of the metric of the four-hyperboloid. An isometry preserving quantization and star product can also be constructed for a general CP p,q , as we demonstrate here. Although the isometry algebra, namely su(p + 1, q), is preserved by the quantization, the isometry generators, i.e., the Killing vectors, can pick up quantum corrections.
As stated above, the simplest example of an indefinite complex projective space is CP 0,1 , or EAdS 2 . Its isometry preserving quantization, which we denote by ncEAdS 2 , has been examined previously. [6-10, 12, 13] Among the results found in this case is the fact that the star product (when expressed in a suitable set of coordinates) approaches the point-wise product in the asymptotic limit (which corresponds to the boundary limit of anti-de Sitter space). [12] It was also argued that the quantum corrections to the Killing vectors vanish in this limit. Thus ncEAdS 2 asymptotically approaches commutative anti-de Sitter space. In other words, the quantum features of ncEAdS 2 occur, for all practical purposes, in a limited neigh-borhood of some origin. Since ncEAdS 2 is an asymptotically anti-deSitter space it can then be of relevance with regard to the AdS/CF T correspondence principle, which posits that for every asymptotically anti-de Sitter space there is a strong/weak duality correspondence between a bulk theory and a conformal field theory living on the conformal boundary. According to the correspondence principle, the isometries of anti-de Sitter space are mapped to conformal symmetries of the CF T on the AdS boundary. It is then reasonable to speculate that it has a conformal dual, barring known difficulties of the correspondence principle for two dimensional anti-de Sitter space (see for example, [16,17]). This was pursued in [12,13] where correlation functions were computed on the boundary.
As we argue in this article, the quantization procedure for EAdS 2 can be extended to any CP p,q , q ≥ 1. We can ask whether analogous conclusions can be reached regarding their asymptotic behavior. The question therefore is whether there is a quantized version of CP p,q which asymptotically becomes commutative. In other words: 1) Does the star product between two functions with support "near the boundary" reduce the commutative one, and 2) do the noncommutative corrections to the Killing vectors vanish in the boundary limit? Of course, "the boundary" refers here to the asymptotic CP p,q region, rather than a sharp edge of the manifold. The results obtained here do indeed support the affirmative answer to these questions. For the examples we consider we find that, in the asymptotic limit, the relevant star product trivializes to the commutative product and noncommutative corrections to the Killing vectors vanish.
In Section 2 we review the quantization of Euclidean AdS 2 . We parametrize the manifold in terms of two different sets of coordinates (which differ from those used in [12,13]), specifically, local affine coordinates and canonical coordinates. The former have the advantage that they can be applied to any complex projective space. The canonical coordinates, on the other hand, are useful for the purpose of quantization, and satisfy three requirements: The first, is of course, the requirement that they obey the canonical Poisson brackets. The second, which is surprisingly non-trivial to ensure, is that they cover the entire complex plane. Dropping this condition would necessitate a careful treatment of the boundary of the domain in the quantum theory [18,19]. The boundary is never a sharp one, the domain of definition is always an open set, but when the coordinates are such that the boundary is at the finite value of these coordinates the quantization scheme we are using cannot be applied. The third requirement is that the geometric measure is identical, up to a factor, to the integration measure of standard coherent states in the resulting quantum theory. In this regard, the quantum theory, and corresponding coherent states, naturally follow from canonical quantization of the canonical Poisson brackets. We quantize the space with the introduction of a noncommutative star product of the Wick-Voros type, constructed from coherent states. We show that the product asymptotically goes to the point-wise product after re-expressing it in terms of local affine coordinates. A crucial point concerns the symmetries, implemented by the analogues of the Killing vectors, which as stated above, preserve the full isometry algebra, here su(1, 1). We perform a perturbative expansion (with respect to the quantization parameter) for the symmetry generators and compute the leading order corrections to the Killing vectors. In agreement with results in [12,13], these corrections are seen to vanish in the asymptotic limit. The Wick-Voros product lends itself naturally to a matrix approximation, and considering finite matrices is tantamount to the imposition of a cutoff geometry [20,21], which provides both an ultraviolet and an infrared cutoff. We do not do the finite matrix approximation here.
We review CP p,q in Section 3, along with its parametrization in terms of local affine coordinates and canonical coordinates. The quantization procedure outlined above for EAdS 2 naturally extends to CP p,q . We do not have a universal expression for the Darboux map from local affine coordinates that is valid for all p and q, and instead present the map for specific examples. The examples are the two 4-(real)-dimensional indefinite complex projective spaces, CP 1,1 and CP 0,2 , in Section 4 and 5, respectively, along with their higher dimensional analogues given in Section 6. Like with EAdS 2 , the canonical coordinates obey the canonical Poisson brackets, cover all of C p,q , and the resulting geometric measure is proportional to the integration measure of standard coherent states in the quantum theory. We carry out the quantization explicitly for the examples in Section 4 and 5, and show, like with ncEAdS 2 , that upon taking the asymptotic limit, the star product trivializes to the commutative product and quantum corrections to the Killing vectors vanish. These quantum spaces are thus asymptotically CP 1,1 and CP 0,2 , respectively. Some concluding remarks are given in Section 6.
2 Quantization of Euclidean AdS 2

Euclidean AdS 2
To define AdS 2 , or its Euclidean counterpart, EAdS 2 , it is convenient to first introduce a three-dimensional Minkowski background R 2,1 , which we shall coordinatize with x α , α = 1, 2, 3, using the metric diag(+, +, −). The spaces AdS 2 , or EAdS 2 , results from constraining the SO(2, 1) invariant x 2 1 + x 2 2 − x 2 3 to be a constant, associated with the scale. The AdS 2 surface corresponds to a positive constant, while EAdS 2 corresponds to a negative constant. We shall restrict our attention in this section to the Euclidean case, as this has been of traditional interest for the AdS/CF T correspondence. Therefore we take where for convenience we fixed the scale to be one. The surface identified by this relation is a two sheeted hyperboloid. The reason why it is called Euclidean AdS 2 is that the induced metric has a Euclidean signature. ¶ Later we shall restrict to a single component of the hyperboloid H 2 . This space is maximally isotropic, and the three Killing vectors, which we denote by K α , α = 1, 2, 3, form a basis for an so(2, 1) algebra Because H 2 could be thought of as a co-adjoint orbit, a natural Lie-Poisson structure exists on it. It is easily defined by setting the Poisson brackets of the embedding coordinates to satisfy the so(2, 1) algebra With such a choice, one can then use Lie-Poisson structure to implement the action of the Killing vectors on arbitrary functions f on H 2 . Specifically, if one defines K α acting on f by then from the Jacobi identity, one recovers so(2, 1) algebra of the Killing vectors (2.2).

Local coordinates
A number of coordinatizations have been introduced to EAdS 2 . A popular choice has been Fefferman-Graham coordinates [22] because of its convenience in the AdS/CF T correspondence principle. Here, we shall instead work with two other sets of coordinates, local affine coordinates and canonical coordinates. The former has the advantage that it can be applied to any non-compact projective space, while the latter provides a useful step for quantization. Although the local affine coordinates for EAdS 2 are not defined on the entire complex plane, it is expedient, for the purpose of quantization, that the canonical coordinates span all of C. We shall make this requirement below. Note that the canonical coordinates we use here differ from those used in [12,13], because the latter are not very useful for the higher dimensional generalizations. Both sets of coordinates are, of course, related by a canonical transformation. ¶ For example, in the so-called global coordinates the induced metric takes the form:

Local affine coordinates
We denote the local affine coordinate of H 2 by ζ, and its complex conjugate ζ * . The map from the (ζ, ζ * ) to the embedding coordinates (x 1 , x 2 , x 3 ) corresponds to the non-compact analogue of a stereographic projection of S 2 . It is By imposing the condition |ζ| > 1, we restrict to the 'upper' hyperboloid, x 3 ≥ 1. |ζ| → ∞ maps the point (x 1 , x 2 , x 3 ) = (0, 0, 1) on the hyperboloid, while |ζ| → 1 corresponds to the asymptotic limit. Starting with the Lorentz metric on R 2,1 , and using (2.5), we obtain the following induced metric on H 2 This is the Fubini-Study metric, and as was indicated above, it has Euclidean signature. The metric tensor g ζ,ζ * = 2 (|ζ| 2 −1) 2 can be expressed in terms of the Kähler potential g ζ,ζ * = ∂ 2 ∂ζ∂ζ * V , V = −2 ln(|ζ| 2 − 1). The geometric measure resulting from this metric is (2.7) Using (2.5), the so(2, 1) Poisson brackets algebra of the embedding coordinates (2.3) results from the following fundamental Poisson bracket on H 2 , (2.8) Then from (2.4) we get explicit expressions for the Killing vectors in terms of the local affine coordinates

Canonical coordinates
We next apply a Darboux transformation from the local affine coordinates to canonical coordinates (y, y * ), satisfying {y, y * } = −i (2.10) As stated above, for the purpose of quantization it is necessary to have y span all of the complex plane, unlike ζ which is defined only outside the unit disc, |ζ| > 1. This fixes (y, y * ) up to canonical transformations. For the natural ansatz y = f (|ζ|)ζ, one obtains the following condition on the function f (x): which has the general solution where C is an arbitrary non-negative constant. From here it follows that |y| 2 = C + 1 |ζ| 2 −1 , and it spans the entire positive real axis (including |y| = 0) only when C = 0. Then for this ansatz, we have Another desirable feature, from the point of view of quantization, is that the geometric measure reduces to a flat measure when expressed in terms of the canonical coordinates. This easily follows from the Jacobian of the transformation, which is ∂(ζ,ζ * ) ∂(y,y * ) ≡ |{ζ, ζ * }| = (|ζ| 2 − 1) 2 . So (2.7) is transformed to dµ geom (y, y * ) = 2 dy ∧ dy * . (2.14) When re-expressed in terms of (y, y * ), the expression (2.5) for the embedding coordinates becomes Therefore the origin of the complex plane spanned by the canonical coordinates is the image of the point (x 1 , x 2 , x 3 ) = (0, 0, 1) on the hyperboloid, while |y| → ∞ corresponds to the asymptotic limit. The Killing vectors (2.9) when expressed in terms of the canonical coordinates become

Quantization
One can now perform canonical quantization by replacing the coordinates (y, y * ) by operators (ŷ,ŷ † ) satisfying commutation relations

17)
k − being the noncommutative parameter, and 1 the identity operator. Equivalently, we have raising and lowering operators, Note that, apart from the commutation relation, it is equally fundamental that the canonical coordinates y, y * were defined on the whole plane (unlike the case with ζ, ζ * ). Otherwise, one would require a delicate treatment of the domain with a boundary. [18,19] The operatorsŷ andŷ † act on the infinite-dimensional harmonic oscillator Hilbert space H spanned by orthonormal states |n , n = 0, 1, 2...
whereâ|0 = 0, and 0|0 = 1. Alternatively, one can introduce standard coherent states {|α ∈ H, α ∈ C} written on C: where α is the eigenvalue ofâ,â|α = α|α . Coherent states form an over-complete set with unit norm. The completeness relation and normalization condition are (2.20) The integration measure for coherent states dµ(α, α * ) is which is, up to a factor, identical to the geometric measure (2.14). Here we have re-introduced the canonical coordinates (y, y * ) using y = √ k − α and y * = √ k − α * . The Wick-Voros star product, ⋆, is constructed from the standard coherent states. Here we briefly review it. For details of the construction see, e.g. [23][24][25]. One first defines symbols A(α, α * ) on the complex plane associated with operator functions A ofâ andâ † using Then given any two functions A and B ofâ andâ † , with symbols A and B, respectively, the symbol of their product is which gives the Wick-Voros star product of the two symbols. It is given explicitly in terms of the canonical coordinates by This expression realizes the fundamental commutation relation, [y, A denotes the star commutator, and gives the desired commutative limit, The star product can be re-expressed in terms of the local affine coordinates using (2.13). One gets The presence of the |ζ| 2 − 1 factor is crucial. As we mentioned earlier, the conformal boundary is obtained in the limit |ζ| → 1, and therefore this shows that the value of the product of two functions asymptotically is not different from the one obtained with the usual commutative multiplication. (Provided the noncommutative corrections to the functions vanish at the conformal boundary.) It also means that the star commutator reduces to ik − times the Poisson bracket in the asymptotic limit. We will characterize noncommutative EAdS 2 in terms of the noncommutative analogues of the embedding coordinates (x 1 , x 2 , x 2 ) [6-10]. We need a set of noncommutativecoordinates, which we call X α , that satisfy the ⋆ analogue of the conditions (2.1) and (2.3): with C > 0, a constant which defines the Euclidean version of noncommutative AdS 2 . In order to recover (2.1) in the commutative limit, we need C = 1 + O(k − ). The X's should be functions of the embedding coordinates (x 1 , x 2 , x 3 ) of the commutative theory, and must reduce to them in the limit (or, in terms of the local coordinates, they should be functions of (ζ, ζ * ) or (y, y * ) and must reduce to (2.5) or (2.15) respectively). Relation (2.29) for the X α 's then defines the so(2, 1) algebra, and C fixes the Casimir. We thereby obtain irreducible representations of so(2, 1).
Given the noncommutative analogues of the embedding coordinates, one can introduce noncommutative analogues of the Killing vectors of EAdS 2 . Denote them by K ⋆ α . They are defined in analogous way to K α , by essentially replacing the Poisson bracket in (2.4) by the star commutator: where f (X) denotes a function on ncEAdS 2 . Like the Killing vectors K α of EAdS 2 , K ⋆ α satisfy the so(2, 1) algebra. Furthermore, from (2.25), we see that K ⋆ α reduce to K α in the commutative limit. On the other hand, the expressions (2.9) for K α do not hold for the noncommutative analogues of the Killing vectors (except for α = 3, and except for the asymptotic limit, as we shall see below). Thus, quantization leads to deformations of the Killing vectors, although the algebra they generate is not deformed. We next write X α in terms of the canonical coordinates y and y * . For this we will need several simple properties of the star product (2.24).
1. The symbol of the operatorŷ †ŷ is |y| 2 . In general, any function F(|y| 2 ) is a symbol of some operator F (ŷ †ŷ ) and vice versa, any operator F (ŷ †ŷ ) has a symbol depending only on |y| 2 : 2. For any function F(y, y * ), we have F(y, y * ) ⋆ y = yF(y, y * ) , y * ⋆ F(y, y * ) = y * F(y, y * ) (2.32) (The ordering on the left hand side of the equations is important.) 3. For any two functions of |y| 2 , F(|y| 2 ) and G(|y| 2 ), we have where the derivative is taken with respect to |y| 2 .
Motivated by (2.15), we look for the noncommutative coordinates X α satisfying (2.29) in the form 34) where S = S(|y| 2 ) is some real function to be determined below. Using the properties of the star product (2.31-2.33), one can easily find where the prime denotes a derivative with respect to |y| 2 . According to (2.29) this should be equal to 4k − X 3 . So we have that X 3 = X 3 (|y| 2 ), and in terms of S is given by Using (2.29) one more time and taking into account that there exists S −1 such that S −1 ⋆ S = 1 (since it exists to zeroth order in k − , and we assume that the expansion in k − is valid) we arrive at the equation for X 3 [y, Using this in (2.36) we arrive at the differential equation for S ⋆ S which is easily solved to give where a is another integration constant. It is clear that one should set a = 0 in order to have non-singular noncommutative corrections for |y| → 0 (and to recover that X 1 , X 2 → 0 in this limit). So, we have The Casimir in (2.28) is now easily computable In general, the constant c should have the form c = 1 + O(k − ). We fix this freedom in quantization by requiring that the symbol X 3 remains undeformed, i.e. by setting c = 1. Then (2.42) looks exactly as in the commutative case (2.15) i.e. S is a symbol of the operator 1 +ŷ †ŷ , which can be formally written using (2.31) as Though we do not have the closed answer for the series (2.45), we can systematically calculate S to any order in k − . Let S n be the functions independent of k − and defined by (2.46) Plugging this into (2.44) and using (2.33) we have after some trivial index relabeling (2.47) From (2.47) we obtain the recursion relations defining S n for any n: For example, for n = 1 we have In general, it is not hard to see from (2.48) that for an arbitrary n, S n will have the following form where P n (x) is some polynomial of degree n, with P 0 = 1. Then we can write our noncommutative coordinates X α in terms of the commutative ones as

51)
x ± = x 1 ± x 2 being the commutative counterparts to X ± . We conclude that X ± → x ± in the asymptotic limit |y| 2 → ∞, Using (2.51) and its asymptotics (2.52) we can easily study the behaviour of the noncommutative Killing vectors, defined by (2.30), near the conformal boundary.
Let us denote by L the sum in (2.51 3 has exactly the same form as its commutative counterpart K 3 in (2.16). Trivial analysis shows that when |y| → ∞, K ⋆ ± behave as where we naturally assumed that K 3 f has the same asymptotic behavior as K ± f . This shows that the noncommutative corrections to K ⋆ α vanish in the asymptotic limit. Of course, the same is true for the case of the local affine coordinates (ζ, ζ * ). In this case the commutative limit for both, the coordinates X α and Killings K ⋆ α , will be recovered as |ζ| 2 → 1.
Thus upon expressing the system in terms of the canonical or local affine coordinates, we see that the noncommutative coordinates X α as well as the so(2, 1) isometry generators of ncEAdS 2 approach the standard EAdS 2 expressions, while the star product approaches the ordinary product, which is seen in local affine coordinates. We can then argue that ncEAdS 2 reduces to EAdS 2 in the asymptotic limit.

CP p,q
The natural generalization of ncEAdS 2 is the quantization of the indefinite complex projective space, denoted by CP p,q , where p and q are positive integers; p can be zero, while q ≥ 1. EAdS 2 corresponds to p = 0, q = 1. In this section we review CP p,q , writing down the Killing vectors and analogues of embedding coordinates in terms of appropriate Fubini-Study coordinates (ζ i , ζ * i ), i = 1, ..., p + q, for these spaces. In order to reproduce the quantization program of the previous section, we will need to find the Darboux transform from the Fubini-Study coordinates to canonical coordinates (y i , y * i ) spanning all of C p+q . As was mentioned for the case of EAdS 2 , if the canonical coordinates do not span the entire C p+q , quantization becomes unmanageable due to the presence of boundaries. We have not found a general expression for the Darboux transformation that applies to all CP p,q spaces. Rather, we can give the transformation for various classes of such spaces, which we shall illustrate in Sections 4 and 5.

Definition
The space CP p,q , q ≥ 1, is defined as the H 2q,2p+1 hyperboloid mod S 1 . It can be constructed starting from a p + q + 1 dimensional complex space C p+1,q , with indefinite metric Say C p+1,q is coordinatized by z a , a = 1, ..., p + q + 1, along with their complex conjugates z a * , where the indices a, b, ... are raised and lowered using the metric η C . To embed H 2q,2p+1 in C p+1,q one imposes the constraint z * a z a = 1 .

(3.2)
To obtain CP p,q one further makes the identification e iχ being an arbitrary phase. The compact complex projective space CP p corresponds to q = 0. We will not be concerned with it in the following. The space CP p,q can be equivalently defined as the coset space SU (p + 1, q)/U (p, q).
The standard metric and Poisson bracket on complex projective spaces are the Fubini-Study metric and the canonical one, respectively. The former is given by while the latter is Using (3.5), it follows that (3.2) is the first class constraint (in the sense of Dirac's Hamiltonian formalism) that generates the phase equivalence (3.3).

Coordinates
Here we are interested in generalizing the two sets of coordinates given previously for EAdS 2 , i.e., local affine coordinates and canonical coordinates. While here we give explicit expressions for the former, we just discuss qualitative features of the latter. We shall postpone giving explicit expressions for the Darboux transformation to sections which follow.

Local affine coordinates
The local affine coordinates (ζ i , ζ * i ), i = 1, ..., p + q, are defined in terms of the coordinates z a by They are invariant under the phase equivalence transformation (3.3). The ζ * i are obtained by taking the complex conjugate of (3.6) and lowering the index using the background metric tensor on the p + q dimensional subspace (3.1). We note that it is the Euclidean metric for the special case of q = 1. From the constraint (3.2), one has As usual, one can replace z p+q+1 in the denominator by another complex coordinate, say z a , which would be valid for z a = 0, thereby defining a local affine coordinates on a different coordinate patch. and it follows that ζ i ζ * i > 1, which further implies that |ζ 1 | 2 + · · · + |ζ p+1 | 2 > 1. Therefore the coordinate patch spanned by (ζ i , ζ * i ) is C p+1,q−1 with the region ζ i ζ * i ≤ 1 removed. For reasons stated below we call the boundary of this region the general asymptotic limit: This is in agreement with the asymptotic limit defined previously for EAdS 2 . While (3.1) is the background metric, the metric on the surface CP p,q is the Fubini-Study metric (3.4). Substituting z i = z p+q+1 ζ i into (3.4) gives the Fubini-Study metric tensor in terms of local affine coordinates where we denote For p = 0, q = 1, g ij (ζ, ζ * ) reduces to the metric tensor (2.6) on EAdS 2 (up to an overall factor). It can be expressed in terms of the Kähler potential (3.11) The geometric measure associated with the metric (3.9) is dµ geom (ζ, ζ * ) = 1 2 p+q Z 2(p+q+1) dζ 1 ∧ · · · ∧ dζ p+q ∧ dζ * 1 ∧ · · · ∧ dζ * p+q , (3.12) which is the generalization of (2.7). To verify (3.12) we only need the identity where v, w ∈ Vec n and 1 n is the n-dimensional identity matrix, which easily follows from the definition of the determinant, detM = 1 n! ǫ i 1 ···in ǫ j 1 ···jn M i 1 j 1 · · · M injn for any M ∈ Mat n . We can write the invariant interval in (3.9) as where The geometric measure is then dµ geom (ζ, ζ * ) = |det G| dζ 1 ∧ · · · ∧ dζ p+q ∧ dζ * 1 ∧ · · · ∧ dζ * p+q . (3.15) In order to recover (3.12), we then use (3.13) to get, From (3.5), the Poisson brackets on the coordinate patch spanned by (ζ i , ζ * i ) are generalizing the Poisson bracket (2.8) for the case of EAdS 2 . The isometry group of CP p,q is SU (p + 1, q). There are then a total of (p + q)(p + q + 2) Killing vectors associated with the metric tensor (3.9). In terms of the local affine coordinates they are given by generalizing (2.9). κ i j , κ i p+q+1 and κ p+q+1 i form a basis for su(p + 1, q) To recover the Killing vectors K 1 , K 2 , K 3 defined previously for EAdS 2 we need K 1 − iK 2 = −2i κ 2 1 and K 3 = 2i κ 1 1 . By generalizing the notion of the real embedding coordinates x i for EAdS 2 (2.5), we can implement the action of the Killing vectors (3.18) using the Poisson bracket (3.17). Call x b a , a, b = 1, ..., p + q + 1, real embedding coordinates for CP p,q (in contrast to the complex embedding coordinates z a ). Their Poisson bracket algebra should correspond to su(p + 1, q) . For this we define x b a in terms of z a 's and then on the coordinate patch spanned by the local affine coordinates (ζ i , ζ * i ). In terms of the complex embedding coordinates we have: Then, as usual, we can write the action of SU (p + 1, q) Killing vectors in terms of these Poisson brackets The appearance of an extra Killing vector due to x p+q+1 p+q+1 is apparent, which could be seen by noticing that not all x b a 's are independent due to the constraint (3.2), which leads to tr x = x a a = 1 , as well as the higher order conditions Since [x b a ] is a finite dimensional matrix, there is a finite number of independent such conditions on x b a . More specifically, there is a maximum number of n = (p + q) 2 independent conditions on the (p + q + 1) × (p + q + 1) on [x b a ] (excluding tr x = 1). So, in particular, from tr x = x a a = 1 follows that κ b a is traceless, i.e. κ p+q+1 p+q+1 is not independent: κ p+q+1 p+q+1 = −κ i i . Now we can trivially repeat this construction on the coordinate patch spanned by the local affine coordinates (ζ i , ζ * i ). Using (3.6) we have It is because the embedding coordinates are in general divergent in the limit (3.8), that we call this the asymptotic limit. (Components of x b a may vanish in the limit in the special cases where ζ i = 0.) The action of the Killing vectors κ j i on functions f on the coordinate patch is written exactly as in (3.22) Upon using (3.17) we can explicitly verify that κ b a has the form (3.18) (though, of course, this should be obvious from the derivation of (3.17) from (3.5)).
For the case of EAdS 2 , the three real embedding coordinates x 1 , x 2 , x 3 of the section 2.1 are recovered from x b a by setting There is only one independent constraint in this case, namely

Canonical coordinates
Following the previous section, the next step is to perform the Darboux transformation. As was mentioned above we have not found a single expression for the Darboux transformation that applies for all CP p,q spaces. The difficulty is due to our restriction that the resulting canonical coordinates (y i , y * i ) are valid for the whole of C p+q , in order that there are no boundaries on our domain in the corresponding quantized theory. As stated above, we shall give the Darboux transformation for various examples in the sections which follow. As in the previous case of EAdS 2 , we find that the Jacobian of the Darboux transformation goes like ∂(ζ, ζ * ) ∂(y, y * ) = Z 2(p+q+1) , (3.28) and hence in terms of the canonical coordinates, the geometric measure is proportional to the flat measure dµ geom (ζ, ζ * ) = 1 2 p+q dy 1 ∧ · · · ∧ dy p+q ∧ dy * 1 ∧ · · · ∧ dy * p+q . (3.29) In order to proceed further, we need to assume a Darboux transformation for CP p,q that takes the local affine coordinates (ζ i , ζ * i ) to coordinates (y i , y * i ) spanning all of C p+q which satisfies the canonical Poisson bracket relations for all i, j = 1, ..., p + q. We do not have a general proof of this existence, nor that (3.28), and hence (3.29) in general hold, but we are able to find such transformations for the examples in Sections 4 and 5.

Quantization
Generalizing the procedure that was adapted for EAdS 2 , we perform canonical quantization, replacing the coordinates (y i , y * i ) by the set of operators (ŷ i ,ŷ † i ) satisfying commutation relations k − once again being the noncommutative parameter. This is the algebra for p + q harmonic oscillators. The lowering and raising operators,â i andâ † i , are obtained by rescalingŷ i andŷ † i , respectivelŷ = 0 for all i, j = 1, ..., p + q.â i andâ † i act on the infinite-dimensional Hilbert space H, now spanned by orthonormal states |n = |n 1 , ..., n p+q = (â † 1 ) n 1 · · · (â † p+q ) n p+q n 1 ! · · · n p+q ! |0 , (3.33) where n i are non-negative integers. The bottom state |0 = |0, ..., 0 is annihilated by anyâ i , and has unit norm 0|0 = 1. It is straightforward to generalize the coherent states (2.19) and Wick-Voros star product (2.24) to C p+q . The former are given by where α i are complex eigenvalues ofâ i ,â i | α = α i | α , and |α| 2 = α * i α i . The completeness relation and normalization condition are now where the integration measure for coherent states dµ( α, α * ) is Upon doing the rescaling back to canonical coordinates, y i = √ k − α i , we see that it agrees, up to a constant factor, with the geometric measure (3.29). Symbols of operators are defined as in (2.22), while the Wick-Voros product of symbols is Then the star commutator gives a realization of the fundamental commutaton relations (3.31), and the requirements (2.25) for the commutative limit are satisfied. The star product can be re-expressed in terms of local affine coordinates. For the examples that follow, as well as the one in section two, we find that the star product reduces to the ordinary product in the asymptotic limit (3.8).
To define the noncommutative version of CP p,q we should construct the noncommutative analogues of the matrix elements x b a . Denoting them by X b a , we demand that they satisfy su(p + 1, q) commutation relations as well as the analogues of the conditions (3.23). The analogues of these conditions fix the Casimirs of the algebra, restricting the allowable representations of su(p + 1, q) of the noncommutative theory. We, of course, demand that X b a → x b a when k − → 0. In Sections 4 and 5 we shall provide perturbative expansions in k − for X b a as functions of local coordinates for the examples CP 1,1 and CP 0,2 , respectively.
Given X b a it is then easy to define noncommtuative analogues κ ⋆ b a of the Killing vectors. Generalizing (2.30) the action of κ ⋆ b a on functions f on noncommutative CP p,q , we have Then κ ⋆ b a are deformations of the Killing vectors κ b a , with the deformation vanishing in the commutative limit k − → 0. In order to extract the leading order corrections to κ b a , we need to obtain [X b a , f ] ⋆ up to second order in k − . Even though κ ⋆ b a are deformations of the Killing vectors, they satisfy the same algebra as κ b a , namely the su(p + 1, q) isometry algebra For the two examples which follow, as well as the one in Section 2, we get that the deformation of the Killing vectors vanishes in the asymptotic limit (3.8).

CP 1,1
In this section and the next one we write down the explicit Darboux transformation from local affine coordinates, and perform the quantization procedure as outlined previously.
There are two complex affine coordinates ζ i , i = 1, 2, along with their complex conjugates. In this case, the background metric on the reduced space is Euclidean, diag(+, +). The condition (3.7) leads to the restriction that the local affine coordinates are defined on a real four dimensional space with a solid three-sphere removed, The quantity Z 2 spans the positive real line, excluding the origin which corresponds to the asymptotic limit, (3.8) or Z 2 → 0. While the background metric for the coordinates is Euclidean, the Fubini-Study metric (3.9) has a Lorentzian signature. The latter solves the sourceless Einstein equations with Λ = 3 [26]. There are eight real embedding coordinates (3.24), x b a , with tr x = 1. Since CP 1,1 has four real dimensions, x b a are subject to four additional independent conditions (3.23).

Darboux map
Here we give the transformation from local affine coordinates to canonical coordinates (y i , y * i ), i = 1, 2, satisfying (3.30). As stated previously, we require the domain of the latter to be all of C 2 , unlike the domain of local affine coordinates. Up to canonical transformations, the Darboux transformation is given by (4.2) Note that the square root is not necessarily real. To see that the coordinates cover the full complex plane once let us express them as: One can see that by fixing ζ 2 , and letting ζ 1 be arbitrary, y 1 covers the complex plane, and of course the same holds exchanging 1 with 2. The asymptotic limit is The Jacobian of the Darboux transformation is ∂(ζ,ζ * ) ∂(y,y * ) = Z 6 in agreement with (3.28), and so we recover the flat measure (3.29).
Substituting the Darboux transformation in the expressions for the embedding coordinates (3.24) gives r being the positive square root of r 2 . We can then check that the constraints (3.23) and the su(2, 1) Poisson bracket algebra (3.21) hold. Substituting (4.5) into (3.25) gives the Killing vectors in terms of canonical coordinates.

Quantization
Quantization proceeds as in Section 3, with the Hilbert space H being that of a two-dimensional harmonic oscillator. The Wick-Voros star product is given in (3.37), and can be re-expressed in terms of local affine coordinates by making the replacement along with the corresponding replacement for ∂ ∂y 2 , obtained by switching the coordinate indices 1 and 2 in (4.6). Since they both contain the over-all factor of Z, it follows that the star product reduces to the ordinary product in the asymptotic limit, Z → 0.
Next we construct the noncommutative analogues X b a of the embedding coordinates (4.5). We take the following ansätse where we assume that R i is a real function of |y i | 2 , and S is a real function of r 2 .
In order to recover (4.5) in the commutative limit, we need that R i → R (0) i = |y i | 2 + 1 2 , and S → S (0) = r when k − → 0. Away from the commutative limit, R i and S can be obtained as a perturbative expansion is k − For this we require that X b a satisfy the su(2, 1) star commutator algebra (3.38). For the leading two corrections we find and where c 1 and c 2 are arbitrary real constants. While tr X = X a a = 1, as in the commutative theory, there are noncommutative corrections to the constraints (3.23). For example, They correspond to the quadratic and cubic Casimir operators for su(2, 1). We note that there is no choice of c 1 and c 2 for which the noncommutative corrections in both trX 2 and trX 3 disappear.
Upon writing the result for the expansion (4.8) in terms of local affine coordinates one gets where for simplicity we set c 1 = c 2 = 0. The zeroth order terms in k − correspond to the commutative result. When substituted into (4.7), and extracting the zeroth order terms, we recover the formulae (3.24) for embedding coordinates. The noncommutative corrections to y i |y i | R i are not valid near ζ i = 0. The noncommutative corrections to y i |y i | R i and S, and hence X b a , contain factors of Z, and so, away from ζ i = 0, these corrections vanish in the asymptotic limit Z → 0. For this we also use the above result that the star product, when expressed in terms of local affine coordinates, reduces to the ordinary product in the asymptotic limit. Finally we can construct the series expansion for the noncommutative analogue κ ⋆ b a of the Killing vector on CP 1,1 using (3.39). The above arguments show that they too reduce to the commutative Killing vectors (3.18) in the asymptotic limit.
Once again there are two complex affine coordinates ζ i , i = 1, 2, along with their complex conjugates. They are defined by ζ i = z i z 3 , z 3 = 0. Unlike the case with CP 1,1 , here the indices i, j, ... are raised and lowered with the Lorentzian metric, diag(+, −). So here (3.7) implies that and so |ζ 1 | > 1. This restriction means that the local affine coordinates are defined on a real four dimensional space with a solid three-hyperboloid removed. The boundary of this region once again corresponds to the asymptotic limit (3.8), Z 2 → 0. While the background metric is Lorentzian, the Fubini-Study metric (3.9) for CP 0,2 has a Euclidean signature. This is opposite the situation with CP 1,1 . As with CP 1,1 , the Fubini-Study metric solves the sourceless Einstein equations with Λ = 3 [26].

Darboux map
We now give the transformation from the local affine coordinates (ζ i , ζ * i ), i = 1, 2, to canonical coordinates (y i , y * i ), satisfying Poisson brackets (3.30). We note that the indices for the former are raised and lowered using the Lorentzian metric, but the latter coordinates are defined on a two-dimensional complex Euclidean space. Because of this fact it is helpful to perform an intermediate step. For this we recognize that local affine coordinates are not unique. Instead of using the coordinates (ζ i , ζ * i ), as defined in (3.6), we can choose to work with the alternative set of coordinates (ξ n , ξ * n ), n = 1, 2, where ξ n = z n+1 z 1 , z 1 = 0. In contrast with (ζ i , ζ * i ), for these coordinates, the indices n, m, ... are raised and lowered with the Euclidean metric, diag(−, −). The transformation between the two sets of local affine coordinates (in the overlapping region) is therefore something like a Wick rotation of the parameter space, although the signature of the Fubini-Study metric, of course, remains Euclidean. The transformation between the two sets of local affine coordinates is given by The two sets of coordinates are valid on different domains and the transformation applies in the overlapping region. From (5.2) and hence (ξ n , ξ * n ) span the interior of a three-sphere of radius one, |ξ 1 | 2 +|ξ 2 | 2 < 1. As usual the boundary corresponds to the asymptotic limit |ξ 1 | 2 + |ξ 2 | 2 → 1. The Fubini-Study metric and Poisson brackets can be re-expressed in terms of the new local affine coordinates (ξ n , ξ * n ). It is now not difficult to find the map from the affine coordinates (ξ n , ξ * n ) to canonical coordinates (y i , y * i ), i = 1, 2, having the desired properties. Up to canonical transformations, it is There are no restrictions on the domain of (y i , y * i ), i.e., they span all of C 2 . To see this note that where we once again define r 2 = |y 1 | 2 + |y 2 | 2 . The right hand side of (5.5) spans the entire positive real line. Moreover, |y 1 | 2 and |y 2 | 2 span the entire positive real line. Just as with the case of CP 1,1 , r 2 → ∞ is the boundary limit. Using (5.2) and (5.4), we can write the Darboux map from the original set of affine coordinates (ζ i , ζ * i ). It is This is an extension of the Darboux map for EAdS 2 (2.13), where ζ and y now correspond to ζ 1 and −iy 2 , respectively. The Jacobian of the transformation is ∂(y,y * ) = Z 6 , so we again recover the flat geometric measure when expressed in terms of canonical coordinates. Writing the embedding coordinates (3.24) in terms of canonical coordinates gives We can then check that the constraints (3.23) and the su(1, 2) Poisson bracket algebra (3.21) hold. Substituting (5.7) into (3.25) gives the Killing vectors in terms of canonical coordinates.

Quantization
Quantization proceeds as in the previous section. The algebra of observables is again that of a two-dimensional harmonic oscillator, which is realized with the Wick-Voros star product (3.37). The star product can again be re-expressed in terms of the original local affine coordinates (ζ i , ζ * i ), now by making the replacement Because of the over-all factor of Z, it follows that the star product reduces to the ordinary product in the asymptotic limit, Z → 0. Next we construct the noncommutative analogues X b a of the embedding coordinates (5.7). We try writing where we assume that S is a real function of r 2 . We need that S → S 0 = √ r 2 + 1 when k − → 0, in order to recover (5.7) in the commutative limit. In order to obtain S away from the commutative limit, we require that X b a satisfy the su(1, 2) star commutator algebra (3.38). We can then get S in a perturbative expansion in k − . So as before we write S = S 0 + k − S 1 + k − 2 S 2 + · · ·. For the leading two corrections we get Once again, while tr X = X a a = 1, as in the commutative theory, there are noncommutative corrections to the constraints (3.23). For example, In comparing the expansion found here with the one found for CP 1,1 , we note that the latter was expressed in terms of undetermined integration constants c 1 and c 2 . Integration constants may appear for CP 0,2 as well upon generalizing the ansatz (5.9). From (5.9), noncommutative corrections to the embedding coordinates only appear for X 2 1 , X 3 1 , X 1 2 and X 1 3 . After writing the leading order terms for these four matrix elements in the original affine coordinates (ζ i , ζ * i ), we get where again this only applies for (a, b) = (1, 2), (1, 3), (2, 1), (3,1). We find that the corrections contain factors of Z 2 , and so they vanish in the asymptotic limit, Z 2 → 0. Finally, we can obtain the leading corrections to the Killing vectors, specifically κ 2 1 , κ 3 1 , κ 1 2 and κ 1 3 , using the definition (3.39) for their noncommutative analogue. Since they involve taking a star product, which reduces to the ordinary product in the commutative limit, we once again see that all noncommutative corrections to the Killing vectors vanish in the asymptotic limit.

Concluding remarks
In this article we have shown how to perform a unique quantization of CP p,q which preserves the full su(p + 1, q) isometry algebra. For the specific examples considered here we found that noncommutativity is effectively restricted to a limited neighborhood of some origin, and that these quantum spaces approach CP p,q in the asymptotic limit. It is likely that this is a universal result that applies for all CP p,q , q ≥ 1 quantized in a isometry preserving manner. Just as a strong-weak duality is postulated to exist between gravity on asymptotically AdS spaces and a CFT on the boundary, it is tempting to speculate that a similar duality could exist between gravity on asymptotically CP p,q spaces and some boundary field theory. Adapting the standard techniques to this case, it should be possible to compute n−point correlation on the boundary, which are expected to be consistent with the su(p + 1, q) algebra, rather than the full conformal algebra. So then if we have that noncommutative CP p,q is asymptotically CP p,q , there could exist a dual SU (p + 1, q) invariant boundary theory.
As was stated in the text, the main reason we do not have an explicit construction for all quantized CP p,q , q ≥ 1, and cannot prove asymptotic commutativity in general, is that we do not have a universal construction of the Darboux map. The Darboux map from local affine coordinates needed to satisfy three requirements, one of which was that the resulting canonical coordinates cover the entire complex plane. We found explicit constructions of the map for all examples in two and four dimensions. Straightforward higher dimensional generalizations of these constructions exist, but they cannot be applied to all cases. There are two types of higher dimensional extensions: 1) CP p,1 and 2) CP 0,q .
More work is required to obtain the Darboux map for other cases, as it appears that a universal formula does not apply. One case, in particular, that is not included in 1) and 2), and may be worth pursuing is CP 1,2 , as it contains Euclidean AdS 4 as a submanifold, and its noncommutative version is of possible interest for quantum cosmology. [11] The noncommutative analogue of Euclidean AdS 4 is constructed from quantized CP 1,2 . Therefore if, as expected, quantized CP 1,2 is asymptotically commutative, it should naturally follow that noncommutative AdS 4 is asymptotically anti-de Sitter, having a dual three-dimensional conformal theory at the boundary.