Strings on AdS_3 x S^3 and the Plane Wave Limit

Strings on AdS_3 x S^3 and the Plane Wave Limit Oswaldo Zapata The AdS_{d+1}/CFT_d holographic correspondence suggests a relationship between string theory, the only known consistent formulation of quantum gravity, and a superconformal field theory where gravity is absent. In this correspondence the strings live in a ten-dimensional space-time given by a direct product of an anti de-Sitter space (AdS), a space with negative cosmological constant, and a sphere (S). On the other hand the conformal field theory is only defined on the boundary of such an AdS space. In the last years this conjecture has passed a large number of tests, especially in the AdS_5/CFT_4 case, and it continues to be intensively studied, in part due to the weak/strong coupling connection that links the two sides of the correspondence. Another realization of the correspondence is given by superstrings living on a AdS_3 x S ^3 x M_4 bulk space and a two-dimensional superconformal boundary field theory that is expected to be a non-linear sigma-model on a symmetric orbifold. The bosonic case is relatively well known (WZW model) and it provides by itself useful insight into some basic aspects of holography, nevertheless it is not completely satisfactory and new issues should come from the study of the superstring. So far we have made some progress in understanding and extending the bosonic results to the full fermionic string ( M. Bianchi, and O. Zapata, to appear soon ). An interesting limit of AdS x S is the plane wave background (pp-wave), which corresponds to the local background seen by an observer moving at the speed of light in the AdS x S space. This process of zooming-in around a null geodesic is known as Penrose limit. In the line of what we said above we expect string theory in this pp-wave background to be described by a special kind of WZW model. In particular, for the Penrose limit of AdS_3 x S ^3 the worldsheet current model is a generalization of the Nappi-Witten (NW) model. Using powerful CFT techniques we have been able to give explicit expressions for the string amplitudes (two, three and four insertions) in the relevant pp-wave background. Moreover, we have found a relationship between string amplitudes in the pp-wave and the correlators coming from the original formulation in the AdS space. Preliminary results suggest that in the boundary theory the correlation functions should mimic what has been done in the bulk side ( M. Bianchi, G. D'Appollonio, E. Kiritsis and O. Zapata, [arXiv:~hep-th/0402004], published in JHEP 0404:074,2004). We are convinced that following this approach we will be able to show how holography precisely works after the Penrose limit has been taken on both sides of the correspondence, something that has so far remained unclear. Then, by studying a simpler case (AdS_3/CFT_2) we hope to shed light on the understanding of pp-holography in the more realistic and complicated AdS_5/CFT_4 duality. A general overview of the PP-wave/CFT holographic correspondence and some preliminary results were presented in the `Tenth Marcel Grossmann Meeting', 20-26 July 2003, CBPF-ICRA, Rio de Janeiro, Brazil [arXiv: hep-th/0312316].