Asymptotic commutativity of quantized spaces: The case of CPp,q

Abstract

We present a procedure for quantizing complex projective spaces CP p;q , q ≥ 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, suðp þ 1; qÞ, 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 two-dimensional anti-de Sitter space (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 CP p;q , i.e., noncommutativity is restricted to a limited neighborhood of some origin, and these quantum spaces approach CP p;q in the asymptotic limit.

Exact solutions for scalars and spinors on quantized (Euclidean) $AdS_2$ and the correspondence principle

“…Though here we restricted our attention to the case of massless fields, we have extended previous results in two directions. First of all, by considering spinors, we have gone beyond the case of scalar fields examined in [6][7][8]. An essential tool for the study of spinor fields on the quantized EAdS 2 was the map (2.26) to the covariant formulation of the Dirac operator, which we were able to extend to the quantized space [cf.…”

“…3. Natural generalizations of quantized EAdS 2 were obtained in [8]. They correspond to the quantization of CP p,q .…”

“…In an attempt to merge these different approaches, we have studied the AdS/CF T correspondence principle in a non-commutative setting. [6][7][8] By this we mean that the geometry on the gravity side of the correspondence was written on a quantized space-time, which we briefly review below.…”

“…[More precisely, the analysis and quantization was carried out for the Euclidean version of AdS 2 (EAdS 2 ).] Due to rather involved calculations in the approach followed there, the ‡ Although it may not be possible to make the same demand for arbitrary d > 2 dimensional anti-de Sitter spaces, higher-dimensional generalizations of such an isometry preserving quantization procedure do exist for other spaces, namely, indefinite complex projective spaces (CP p,q , q ≥ 1), as was shown in [8].…”