We will discuss the properties of geometries dual to BPS states of N=4 supersymmetric Yang-Mills theory. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS5 x S5 ground state corresponds to excising the interior of an S5, while the 1/8 BPS excitations correspond to deformations (including topology change) of the S5 and/or the excision of additional droplets from the base.

The study of supersymmetric solutions in supergravity have provided invaluable insight into quantum gravity via AdS/CFT correspondence. In this talk I present a new class of intriguing Kahler geometry which appears in the study of AdS solutions from D3 and M2-branes wrapped on 2-cycles. Solving the master equation, which has higher-order couplings in Ricci tensor, we obtain new AdS solutions.

I discuss some aspects of the full moduli space of N=1 supersymmetric gauge theories, focusing on those arising on D-branes at Calabi-Yau singularities. Of particola importance is the full moduli space for one brane, called the master space. I discuss its geometrical properties and its relation with the uncovering of hidden symmetries and the counting of BPS states of the gauge theory.

The homogeneous and symmetric space sine-Gordon models (HSG- and SSSG-models, for short) are two groups of two-dimensional integrable quantum field theories which belong to a larger class of models: the non-Abelian affine Toda field theories. In this talk I intend to review the main results known up to date about these two classes of models, paying special attention to my own contributions to the subject. These contributions have focused on the one hand, on the development of the bootstrap program for the HSG-models (TBA-analysis, computation of form factors and correlation functions etc) and, on the other hand, on the study of the quantum integrability and spectrum of a subset of the SSSG-models, known as split models.

The global symmetry groups that result from compactifying eleven-dimensional supergravity on an n-dimensional torus play a central role in our understanding of U-dualities in string and M-theory. The mechanism leading to the En Lie algebra upon compactification on Tn is well-known for n less than 9, but the situation for n=9, corresponding to the compactification to 2 dimensions, has been described much less clearly in the literature. We give an elementary, and completely explicit, description of the infinite-dimensional symmetries of all symmetric-space coset models in 2-dimensional gravitational backgrounds, including symmetries of both Kac-Moody and Virasoro type.

We explain the relation between the holomorphic anomaly equations in topological string theory and the loop equations in matrix models. We explain a strategy for their solutions using modular forms, which applies to open and closed topological string theory on (non) compact Calabi-Yau manifolds. (A description how to find Lecture Theatre 1 is given on http://brahms.mth.kcl.ac.uk/cgi-bin/main.pl?action=triangle)

Anomalous dimensions of long operators in SYM are described by the asymptotic Bethe ansatz. There exist, in addition, finite size effects due to wrapping interactions. In this talk I would like to argue that within the AdS/CFT correspondence these additional wrapping interactions are described by finite size corrections in the relevant integrable quantum field theory and analyze the finite size corrections to the giant magnon.

I will discuss the targetspace manifestation of the most general (2,2) sigma models with H flux, these include chiral, twisted chiral and semi chiral superfields. I will describe the appearance of a generalized Kahler potential and also outline the use of the deformation space of generalized complex structures in supergravity.

The question of a modification of the running gauge coupling of (non-) abelian gauge theories by an incorporation of the quantum gravity contribution has recently attracted considerable interest. In this talk we present an involved diagrammatical calculation in the full Einstein-Yang-Mills system both in cut-off and dimensional regularization at one-loop order. It is found that all gravitational quadratic divergencies cancel in cut-off regularization and are trivially absent in dimensional regularization so that there is no alteration to asymptotic freedom at high energies. This settles the previously open question of a potential regularization scheme dependence of the one-loop beta-function traditionally computed in the background field approach. Furthermore, we show that the remaining logarithmic divergencies give rise to an effective Einstein-Yang-Mills Lagrangian with a counterterm of dimension six.

Heavy-ion collision experiments manage to probe the quark-gluon plasma of QCD. The plasma created is strongly coupled and is difficult to analyze using conventional means. In this talk, we shall use the AdS/CFT correspondence to study properties of particles moving through a quark-gluon plasma. Such probe particles provide a model for understanding several unanswered puzzles exhibited by the experiment.

Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS5 x S5 superstring world-sheet theory in terms of physical degrees of freedom we investigate a Pohlmeyer-reduced version of the corresponding supercoset sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting gauge-fixed equations can be obtained from a Lagrangian of a non-abelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. The final form of the Pohlmeyer-reduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. Its small-fluctuation spectrum is that of 8 bosonic and 8 fermionic degrees of freedom with equal masses. We show that in the special case of the AdS2 x S2 superstring model the reduced theory is supersymmetric: it is equivalent to the (2,2) supersymmetric extension of the sine-Gordon model.

I will talk about a new method of performing a strong coupling expansion for many superconformal field theories in four dimensions, in particular those that are relavant for the AdS/CFT correspondence. I will first explain the methods for the case of N=4 SYM, as well as what calculations can be done (both analytically and numerically) and I will show how they compare with the dual AdS geometry. I will then explain what generalizations are required for other setups and what new field theory calculations can be done with these methods that were not available before.

The chiral symmetry of QCD shows up in the linear Weyl--Wigner mode at short Euclidean distances or at high temperatures. On the other hand, low-lying hadronic states exhibit the nonlinear Nambu--Goldstone mode. An interesting question was raised as to whether the linear realization of the chiral symmetry is asymptotically restored for highly excited states. We address it in a number of ways. On the phenomenological side we argue that to the extent the meson Regge trajectories are observed to be linear and equidistant, the Weyl--Wigner mode is not realized. This picture is supported by quasiclassical arguments implying that the quark spin interactions in high excitations are weak, the trajectories are linear, and there is no chiral symmetry restoration. Then we use the string/gauge duality. In the top-down Sakai--Sugimoto construction the nonlinear realization of the chiral symmetry is built in. In the bottom-up AdS/QCD construction by Erlich et al., and Karch et al. the situation is more ambiguous. However, in this approach linearity and equidistance of the Regge trajectories can be naturally implemented, with the chiral symmetry in the Nambu--Goldstone mode. Asymptotic chiral symmetry restoration might be possible if a nonlinearity (convergence) of the Regge trajectories in an intermediate window of n,J, beyond the explored domain, takes place. This would signal the failure of the quasiclassical picture.

I present new analytic solutions of type IIB supergravity with fully backreacting D7-branes describing the addition of an arbitrary number of flavors to the Klebanov-Tseytlin and Klebanov-Strassler theories. I provide a detailed analysis of the field theory and of the duality cascade which describes its RG flow, Seiberg duality is understood as a large gauge transformation in supergravity. Moreover the string background suggests that the UV behavior is a duality wall.

The lectures will start by reviewing closed topological theories, before moving to more recent work in the open/closed theory on spaces with boundary. From the work of Witten, the topology of the so-called Deligne-Mumford moduli spaces of Riemann surfaces with nodes plays a fundamental role in 2-dimensional topological gravity (known to mathematicians as Gromov-Witten theory). For example, by the work of Kontsevich and Manin, it is seen to underly the Witten-Dijkgraaf-Verlinde-Verlinde equation, and hence is intimately related to the theory of Frobenius manifolds and of integrable systems. Most work on these moduli spaces has been focussed on the case of closed topological field theory. In these lectures, I will explore the moduli spaces, analogous to Deligne-Mumford moduli spaces, which play the corresponding role in the open theory. In this case, the world sheet (Riemann surface) has a boundary, and as a result, the moduli spaces are no longer complex orbifolds, but rather real orbifolds with corners. These moduli spaces may be viewed as an explanation of the way that algebraic structures, such as A-infinity categories, cyclic homology, and the Cardy condition, enter topological field theory in two dimensions. This theory should also have applications to understanding the foundations of string theory.

We consider N=1 supersymmetric quantum chromodynamics (SQCD) with the gauge group U(Nc) and Nc+N quark flavors. Nc flavors are massless, the corresponding squark fields develop (small) vacuum expectation values (VEVs) on the Higgs branch. Extra N flavors are endowed with small (and equal) mass terms. We study this theory through its Seiberg's dual: U(N) gauge theory with Nc +N flavors of dual quark fields plus a gauge-singlet mesonic field M. The original theory is referred to as quark theory while the dual one is termed monopole theory. The suggested mild deformation of Seiberg's procedure changes the dynamical regime of the monopole theory from infrared free to asymptotically free at large distances. We show that, upon condensation of the dual quarks, the dual theory supports non-Abelian flux tubes (strings). Seiberg's duality is extended beyond purely massless states to include light states on both sides. Being interpreted in terms of the quark theory, the monopole-theory flux tubes are supposed to carry chromoelectric fields. The string junctions - confined monopole-theory monopoles - can be viewed as constituent quarks of the original quark theory. We interpret closed strings as glueballs of the original quark theory. Moreover, there are string configurations formed by two junctions connected by a pair of different non-Abelian strings. These can be considered as constituent quark mesons of the quark theory.

We will discuss the nonabelian strings found recently in SUSY QCD and non-SUSY gauge theories with scalars. Their properties will be considered in some details. In particular, their rich worldsheet structure involving localized monopoles will be explained. The dependence on the SUSY breaking parameters will be analysed.