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# Eikonal Approximation in AdS/CFT correspondence

The AdS/CFT correspondence opens the fascinating possibility of having a non–perturbative formulation of string theory, and therefore of a quantum theory of gravity, based on non–gravitational gauge theories deﬁned over ﬂat spacetime. On the other hand, it gives hope of understanding strong coupling phenomena of gauge theories, such as conﬁnement and the existence of a mass gap in QCD, in terms of a gravitational theory in a curved background with one higher dimension.

Virtually all computations done so far in the literature neglect string loop eﬀects in AdS, considering only the planar limit of the gauge theory. Clearly the strong coupling regime of strings (or gravity) is presently beyond our reach, however, in certain kinematical regimes, the non-perturbative behavior of string scattering amplitudes is known. In particular, in ﬂat space, the limit of small scattering angle is controlled by the well known eikonal approximation. Recently, we initiated a programme to generalize the eikonal approximation to Anti–de Sitter space. Our main goal is to explore the AdS/CFT correspondence beyond the planar limit, obtaining information about the strongly coupled gauge theory at ﬁnite N . Inspired by the shock wave derivation of the eikonal amplitude in ﬂat space, we derived the two-point function < O 1 O1 >_shock in the presence of a shock wave in Anti-de Sitter, where O 1 is a scalar primary operator in the dual conformal ﬁeld theory. At tree level in the gravitational coupling, we related the shock two–point function to the discontinuity across a kinematical branch cut of the conformal ﬁeld theory four-point function A ∼< O1 O2 O1 O2 >, where O2 creates the shock geometry in Anti-de Sitter. We then used this result to ﬁnd the high spin and dimension conformal partial wave decomposition of all tree level Anti–de Sitter Witten diagrams, showing that, as in ﬂat space, the eikonal kinematical regime is dominated by the T–channel exchange of the massless particle with highest spin (graviton dominance). We also computed the anomalous dimensions of the high spin O 1 O2 composites.

The application of eikonal methods in the AdS/CFT correspondence is a research programme with many questions:

- The crucial step of re-summing the contributions from all crossed–ladder Witten graphs and obtain ﬁnite N results is still unclear, though we have conjectured a possible solution.

- So far we have only considered the zero α' limit. However , in ﬂat space, it is known that the leading string eﬀects simply Reggeize the gravitational interaction, lowering its eﬀective spin from 2 to 2 + α' t, with t the square of the transfered momentum.

- It would also be very important to test our results against calculations performed directly in the CFT duals at ﬁnite N . This will probably require the generalization of our methods to AdS × compact space and superspaces, as well as the inclusion of string eﬀects.

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