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 $CFT$. We also argue that there does not exist a map which connects the commutative model to its non-commutative counterpart, and therefore the conformal behaviour of the noncommutative correlators is a non-trivial result.
Highlights
In a recent paper [1] aspects of the AdS2=CFT1 correspondence were studied in a noncommutative setting, namely when the geometry on the gravity side of the correspondence is replaced by the noncommutative version of two-dimensional anti–de Sitter space (AdS2)
The motivation for making the AdS2 space noncommutative is to include some quantum gravitational corrections, since there is a general belief [2] that the quasiclassical regime of quantum gravity should appear as a quantum field theory on some noncommutative background
The introduction of noncommutativity on the AdS2 background can be made unique by demanding that it preserves the SOð2; 1Þ isometry group. [3,4,5,6,7,8] In [1], the analogues of the AdS2 Killing vectors generating SOð2; 1Þ were constructed on the noncommutative space, denoted by noncommutative version of AdS2 (ncAdS2)
Summary
In a recent paper [1] aspects of the AdS2=CFT1 correspondence were studied in a noncommutative setting, namely when the geometry on the gravity side of the correspondence is replaced by the noncommutative version of two-dimensional anti–de Sitter space (AdS2). Baring the known difficulties of the correspondence principle for two-dimensional anti–de Sitter space (see e.g., [9,10]), the result that ncAdS2, is asymptotically AdS2 opens up the possibility of a dual conformal field theory on the boundary. We add a cubic term to the action, and from it we obtain an integral expression for the leading order noncommutative correction to the three point correlation function on the conformal boundary. The outline of this article is the following: After briefly reviewing the correspondence principle for free massive scalar field theory on AdS2 in Sec. II, we quantize the background space to get ncAdS2 and derive the leading order noncommutative correction to the two point correlation function on the conformal boundary. In Appendix D we show that the on shell boundary action does not pick up noncommutative corrections
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