We conceive higher bundles with connections entirely Lie algebraically by, essentially, extending the Cartan-Ehresmann definition of a connection from Lie algebras to L-infinity algebras. Using this machinery, we study Chern-Simons n-bundles as obstructions to lifts through String-like extensions of L-infinity algebras.

In this talk I want to describe recent work on geometric (Berry) phases that arise in the quantum theory of certain D-brane states. After briefly reviewing the geometric phase in general theories, I will turn to supersymmetric systems of the type arising in D0-brane configurations in string theory. Interestingly, the resulting Berry phases are related to the four Hopf fibrations. I will show how this fact arises as a consequence of the close relationship between SUSY and the division algebras. The framework of string theory allows to draw some interesting physical consequences from these Berry phases, such as a new interpretation of the Berry phase in terms of gravitational precession and the surprising result that in certain configurations, D0 branes behave as anyons.

These will be introductory lectures surveying GW, MNOP and GV invariants - all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

Recently Romelsberger made some conjectures about an index for N=1 superconformal quantum field theories applied to certain Seiberg dual theories. Recent joint work by me and Hugh Osborn proves and extends these conjectures by employing certain q-series type identities. I will discuss the superconformal index, how it applies for more general N=1 dual theories and, briefly, the proof of these conjectures.

I give an outline of the program of Shadrin and his collaborators, to give explicit formulas for correlators in topological gravity (with descendants), starting with a Batalin-Vilkovisky algebra. In particular, I will try to explain the relationship with Givental's work on Gromov-Witten invariants. No background is assumed.

These will be introductory lectures surveying GW, MNOP and GV invariants -- all different ways of counting curves. For a string theorist this involves seeing the curve as, respectively, the world sheet of a string, a D-brane, or a BPS thingummy. I will describe a 4th way via stable pairs, which in effect means counting D-branes (or stable objects of the derived category, to mathematicians) after a change of stability condition.

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.