In this talk, I will discuss a new family of supersymmetric Wilson loop operators in N=4 SYM. These operators can be defined for any loop on a three-sphere in space-time and generically preserve two super-conformal charges. On the string side, via AdS/CFT duality, these loops map to surfaces which are 'pseudo-holomorphic' with respect to a novel almost-complex structure defined on a suitable subspace of AdS5xS5. Of particular interest is the subclass of loops lying on a two-sphere in space-time whose expectation value is conjectured to be computed exactly in terms of the analogous observables in bosonic 2d Yang-Mills on S2. Several evidences for this conjecture, both on the gauge theory and on the string theory side will be discussed.

I will discuss a supergravity solution corresponding to fivebranes wrapped on the S2 of the resolved conifold. By changing a parameter the solution continuously interpolates between the deformed conifold with flux and the resolved conifold with branes. Therefore, it displays a geometric transition, purely in the supergravity context. The solution is a simple example of torsional geometry and may be thought of as a non-Kahler analog of the conifold. I will discuss how one can obtain supersymmetric solutions of type IIB supergravity starting from simpler non-Kahler geometries, by U-dualities or other methods. Applying these transformations to the torsional conifold solution we obtain a solution dual to the baryonic branch of the Klebanov-Strassler theory. Far along the baryonic branch the solution resembles D5 branes wrapping a fuzzy two-sphere in the resolved conifold and this can be matched to a weakly coupled field theory analysis.

A characteristic feature of string theory is the asymptotic nature of the free energy genus expansion indicating the presence of non-perturbative effects due to branes. In certain toy models, such as minimal models or topological strings, the theory admits a matrix model description. In these cases the non-perturbative effects can be interpreted in terms of eigenvalues tunneling in the dual matrix model and the finite N matrix model can be naturally regarded as the non-perturbative completion of the theory. We address the case of the topological string on the Resolved Conifold. We compute the exact non-perturbative contribution to the free energy via Borel analysis and show how it controls the large order behavior of the theory. We interpreted the non-perturbative effects in the dual Chern-Simons matrix models and in the space-time in terms of toric branes.

We consider some novel topological effects, with potentially observable consequences, in classical and quantum gravity. In one scenario, we find that the topology of extra dimensions generically breaks global Lorentz symmetry, leading to distinct experimental signatures in the context of brane worlds. In a different scenario, we show that topological terms in the gravitational action, such as those expected in heterotic string theory, can greatly enhance the instability of four-dimensional de Sitter space by favouring the nucleation of primordial black holes.

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Motivated by the BDS conjecture, we look at 1/N corrections to gluon amplitudes in N=4 SYM.Infrared divergences are analyzed in two formalisms, of Catani and of Sterman/Tejeda-Yeomans. I willanalyze the 1/N expansion of the IR divergences, and find some systematics, at 3-loops, and iterated to any number of loops. Then we observe that up to 2-loop we have an exact relation to graviton scattering in N=8 supergravity, with a simple interpretation in a 't Hooft-like picture, but the higher loop extrapolation is problematic. Finally, I discuss the transcendentality of amplitudes and Catani IR divergent operators

This is the first lecture of a short course on flux compactifications. For more information on the course and the schedule, please visit http://www.strings.ph.qmw.ac.uk/index.htm and follow the link to the Graduate Program in String/Field Theory.

I will discuss the wavefunctions for open strings in warped compactifications, and in particular for D7-branes in warped Calabi-Yau orientifolds, which provide a string theory realizations of the Randall-Sundrum scenario. I will show how certain background fluxes, necessary in the presence of warping, couple to the fermionic wavefunctions and qualitatively change their behavior. Finally, the warped Kahler potential for the D7-brane light modes will be discussed, as well as a proposal to embed it in the full open+closed Kahler potential.

The AdS4/Chern-Simons theory duality, recently conjectured by Aharony, Bergman, Jafferis and Maldacena, provides not only a new testing ground for the gauge theory/gravity correspondence, but also a possible way to access M-theoretic degrees of freedom. In this talk, I shall discuss some of our recent results on the open string sector of the type IIA string theory in AdS4xCP3, with emphasis on D-branes and giant gravitons in particular. Specifically, I shall focus on the so-called dual giant, a D2-brane extended on an S2 in AdS4, its spectrum of small fluctuations and open strings attached to it.

First we review existing results concerning the non-planar spectrum of N=4 SYM. Next, using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2) x SU(2) sector, including all non-planar corrections. This generator is then applied to a series of finite length operators as well as to two different types of BMN operators. As in N=4 SYM, at the planar level the finite length operators are found to exhibit a degeneracy between certain pairs of operators with opposite parity -- a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory.When non-planar corrections are taken into account the degeneracies between parity pairs disappear hinting the absence of higher conserved charges. The analysis of the BMN operators resembles that of N=4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

In Landau-Ginzburg models, B-type boundary conditions are described by matrix factorisations of the superpotential. In cases where such a model corresponds to a rational conformal field theory (e.g. a coset model), we can compare the matrix factorisations to known boundary states. The connection between these complementary descriptions can on the one hand be used to improve the understanding of closed and open string moduli space, on the other hand it could give hints towards the construction of new boundary states.