Most of my previous research activities lay in the context of the gauge/gravity duality or Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. I've been mainly involved in three research lines:
Holography from CFT
One of the main mysteries in gauge/gravity duality is the emergence of a local gravitational theory living in one higher dimension than the dual gauge theory. The standard AdS/CFT dictionary implies that there are two basic necessary conditions for a CFT to be dual to a weakly coupled local gravitational theory in AdS. The first, is the existence of a large-N expansion providing a notion of single-trace operators in the CFT. The second, is the existence of a large gap in the spectrum of dimensions of single-trace primary operators. The ones with low dimension are dual to the AdS fields present in the bulk lagrangian. In arXiv:0907.0151, we gave strong evidence that these are also sufficient conditions. More precisely, the following conjecture was proposed: any CFT that has a large-N expansion, and in which all single-trace operators of spin greater than two have parametrically large dimensions, has a local bulk dual. One can then think of the bulk dual as providing a consistent CFT for a given operator content (at least to leading order in 1/N). This point of view is useful in applications to condensed matter physics.
Flat space limit of AdS/CFT
An important limitation of the best understood gauge/gravity duality examples is that the gravitational theory lives in a space that is asymptotically AdS. It would be very interesting to have a holographic description of quantum gravity with asymptotically flat boundary conditions. Therefore, it is natural to study the limit of very large radius of AdS. In arXiv:0903.4437, we showed that the 2 to 2 scattering amplitude of the bulk theory in flat space is encoded in a particular kinematic limit of the four point function of the dual CFT. This suggested that it was possible to obtain the full string theory S-matrix from correlation functions of the gauge theory in the large N limit and at finite string coupling, as we showed in arXiv:1002.2641.
The close analogy between the flat space S-matrix and the CFT correlators deserves further exploration. In particular, it suggests a unitarity analysis of CFT correlation functions in the spirit of S-matrix theory, where singularities should have a physical meaning as on-shell intermediate states in AdS. In arXiv:1011.1485 and arXiv:1107.1499, we were able to make this idea precise using the Mellin representation of CFT correlators. I believe there is still much more to be learned from this analogy between CFT correlators and scattering amplitudes.
High energy scattering in AdS/CFT
This was the topic of my Ph.D thesis and is a research program that I am still pursuing in collaboration with Miguel Costa. In a series of papers, methods from high energy scattering in gravity, string theory and gauge theory were brought together and extended in a unified formalism. One basic ingredient of our approach is to focus on the CFT four point function and define the Regge limit in position space. In maximally supersymmetric Yang-Mills (SYM) at large N (tree-level in the bulk theory), high energy scattering is dominated by a Regge pole whose spin varies as one varies the 't Hooft coupling, interpolating from graviton exchange at strong coupling to pomeron exchange at weak coupling. In this formalism, the BFKL impact factors simplified dramatically suggesting the existence of an underlying integrable structure, at least in N =4SYM.
The eikonal approximation was generalized to AdS spacetime and it was shown that the effect of multi-graviton exchanges at high energy amounts to exponentiate the tree-level phase shift in impact parameter space. This tree-level phase shift in AdS was then associated to the leading Regge pole at any value of the 't Hooft coupling. This idea was then explored to describe saturation in Deep Inelastic Scattering (DIS) at small Bjorken-x. Although QCD is a confining theory, there is some evidence that conformal symmetry is a useful approximate symmetry in some regimes. The best known example being the strongly coupled plasma produced at RHIC. In DIS, at large energy and momentum transfer, one also expects approximate conformal invariance and these methods should be relevant. The perspective of this work was to use AdS as a geometric description of the conformal dynamics of QCD in this regime. We extended these techniques to four point functions of operators with spin, in order to obtain predictions for the behavior of the standard structure functions at small Bjorken-x. This formalism was then used to propose a model of saturation in high energy scattering based on the geometric idea of a black (absorbing) disk in AdS. This model predicted that deep inside saturation the transverse and longitudinal structure functions have the same power law scaling, FT ~ FL ~ 1/x^w and that the ratio FL/FT is given by the universal value (1+w)/(3+w), independently of the target.