Noncommutative AdS2/CFT1 duality: The case of massive and interacting scalar fields

Abstract

We continue the study of the nocommutative AdS 2 =CFT 1 correspondence. We extend our previous results obtained for a free massless scalar field to the case of a massive scalar field. Both the free and interacting cases are considered. For both cases it is confirmed that to the leading order in noncommutative corrections the 2-and 3-point correlation functions have the form that is assumed by some (yet unspecified) dual conformal field theory (CFT). We also argue that there does not exist a map which connects the commutative model to its noncommutative counterpart, and therefore the conformal behavior of the noncommutative correlators is a nontrivial result.

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

“…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.…”

“…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].…”

“…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][7][8][9][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 pointwise product in the asymptotic limit (which corresponds to the boundary limit of anti-de Sitter space) [12].…”

“…It is then reasonable to speculate that it has a conformal dual, barring known difficulties of the correspondence principle for twodimensional anti-de Sitter space (see, for example, [16,17]). This was pursued in [12,13] where correlation functions were computed on the boundary.…”

“…II 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.…”