We develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an indexfree notation where tensors are encoded by polynomials in auxiliary polarization vectors. The efficiency of the formalism is demonstrated by computing the tensor structures allowed in n-point conformal correlation functions of tensors operators. Constraints due to tensor conservation also take a simple form in this formalism. Finally, we obtain a perfect match between the number of independent tensor structures of conformal correlators in d dimensions and the number of independent structures in scattering amplitudes of spinning particles in (d + 1)-dimensional Minkowski space.

We generalize Regge theory to correlation functions in conformal field theories. This is done by exploring the analogy between Mellin amplitudes in AdS/CFT and S-matrix elements. In the process, we develop the conformal partial wave expansion in Mellin space, elucidating the analytic structure of the partial amplitudes. We apply the new formalism to the case of four point correlation functions between protected scalar operators in N=4 Super Yang Mills, in cases where the Regge limit is controlled by the leading twist operators associated to the pomeron-graviton Regge trajectory. At weak coupling, we are able to predict to arbitrary high order in the 't Hooft coupling the behaviour near J=1 of the OPE coefficients C_{OOJ} between the external scalars and the spin J leading twist operators. At strong coupling, we use recent results for the anomalous dimension of the leading twist operators to improve current knowledge of the AdS graviton Regge trajectory - in particular, determining the next and next to next leading order corrections to the intercept. Finally, by taking the flat space limit and considering the Virasoro-Shapiro S-matrix element, we compute the strong coupling limit of the OPE coefficient C_{LLJ} between two Lagrangians and the leading twist operators of spin J.Comment: 27 + 24 pages, 7 figures; v2 Typos corrected, references added; v3 Typo corrected; v4 Typos corrected, references adde

For conformal field theories in arbitrary dimensions, we introduce a method to derive the conformal blocks corresponding to the exchange of a traceless symmetric tensor appearing in four point functions of operators with spin. Using the embedding space formalism, we show that one can express all such conformal blocks in terms of simple differential operators acting on the basic scalar conformal blocks. This method gives all conformal blocks for conformal field theories in three dimensions. We demonstrate how this formalism can be applied in a few simple examples.

We develop the embedding formalism to describe symmetric traceless tensors in Anti-de Sitter space. We use this formalism to construct the bulk-to-bulk propagator of massive spin J fields and check that it has the expected short distance and massless limits. We also find a split representation for the bulk-to-bulk propagator, by writing it as an integral over the boundary of the product of two bulk-to-boundary propagators. We exemplify the use of this representation with the computation of the conformal partial wave decomposition of Witten diagrams. In particular, we determine the Mellin amplitude associated to AdS graviton exchange between minimally coupled scalars of general dimension, including the regular part of the amplitude.

We consider new cosmological solutions with a collapsing, an intermediate and an expanding phase. The boundary between the expanding (collapsing) phase and the intermediate phase is seen by comoving observers as a cosmological past (future) horizon. The solutions are naturally embedded in string and M-theory. In the particular case of a two-dimensional cosmology, space-time is flat with an identification under boost and translation transformations. We consider the corresponding string theory orbifold and calculate the modular invariant one-loop partition function. In this case there is a strong parallel with the BTZ black hole. The higher dimensional cosmologies have a time-like curvature singularity in the intermediate region. In some cases the string coupling can be made small throughout all of space-time but string corrections become important at the singularity. This happens where string winding modes become light which could resolve the singularity. The new proposed space-time causal structure could have implications for cosmology, independently of string theory.

Abstract:We introduce the impact parameter representation for conformal field theory correlators of the form A ∼ O 1 O 2 O 1 O 2 . This representation is appropriate in the eikonal kinematical regime, and approximates the conformal partial wave decomposition in the limit of large spin and dimension of the exchanged primary. Using recent results on the two-point function O 1 O 1 shock in the presence of a shock wave in Anti-de Sitter, and its relation to the discontinuity of the four-point amplitude A across a kinematical branch cut, we find the high spin and dimension conformal partial wave decomposition of all tree-level Anti-de Sitter Witten diagrams. We show 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 compute the anomalous dimensions of the high spin O 1 O 2 composites. Finally, we conjecture a formula re-summing crossed-ladder Witten diagrams to all orders in the gravitational coupling.

We initiate a program to generalize the standard eikonal approximation to compute amplitudes in Anti-de Sitter spacetimes. Inspired by the shock wave derivation of the eikonal amplitude in flat space, we study the two-point function E ∼ O 1 O 1 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 relate the shock two-point function E to the discontinuity across a kinematical branch cut of the conformal field theory four-point function A ∼ O 1 O 2 O 1 O 2 , where O 2 creates the shock geometry in Anti-de Sitter. Finally, we extend the above results by computing E in the presence of shock waves along the horizon of Schwarzschild BTZ black holes. This work gives new tools for the study of Planckian physics in Anti-de Sitter spacetimes.

We derive an eikonal approximation to high energy interactions in Anti-de Sitter spacetime, by generalizing a position space derivation of the eikonal amplitude in flat space. We are able to resum, in terms of a generalized phase shift, ladder and cross ladder graphs associated to the exchange of a spin j field, to all orders in the coupling constant. Using the AdS/CFT correspondence, the resulting amplitude determines the behavior of the dual conformal field theory four-point function O 1 O 2 O 1 O 2 for small values of the cross ratios, in a Lorentzian regime. Finally we show that the phase shift is dominated by graviton exchange and computes, in the dual CFT, the anomalous dimension of the double trace primary operators O 1 ∂ · · · ∂O 2 of large dimension and spin, corresponding to the relative motion of the two interacting particles. The results are valid at strong t'Hooft coupling and are exact in the 1/N expansion.