We discuss quasi-classical quantization of AdS4 x CP3 superstring using the algebraic curve techinque. We exemplify this procedure on some configurations so called giant magnons. It turns out that there are two types of configurations and they are quantizied in slightly different ways. Also we discuss testing AdS/CFT correspondence in this case.

I will explain two algorithms that compute numerically the Weil-Petersson metric on the moduli space of polarized Calabi-Yau manifolds. I will show some explicit examples. Finally, I will outline some applications of these techniques in the physics of Calabi-Yau compactifications of string theory.

We introduce the Wilson loop/amplitude duality, which states that certain gluon amplitudes in N=4 SYM are equivalent to light-like polygonal Wilson loops. We will illustrate the duality's power by computing the one loop Wilson loop analytically (matching with the known amplitudes) and then the two loop Wilson loop numerically for any number of edges. If the duality continues to hold we can thus compute two loop MHV amplitudes for any number of incoming particles. In the second part of the talk we discuss the recently conjectured new symmetry dual superconformal symmetry of the complete S-matrix of N=4 SYM. We prove its presence at tree-level and find new constraints it puts on the one loop amplitude.

In this talk, I will present several results concerning a class of supersymmetric Wilson loop operators in N=4 SYM. These operators can be defined for any loop on a three-sphere and are in general 1/16-BPS. On the string theory side of the AdS/CFT duality, these supersymmetric Wilson loops are described by string worldsheets which are pseudo-holomorphic with respect to a novel almost complex structure defined on a AdS4xS2 subspace of AdS5xS5. A notable subclass of the general 1/16-BPS operators, which will be the main focus of this talk, is obtained by restricting the loop to lie on a S2 in space-time. In this case supersymmetry is doubled, and we propose a conjecture that the expectation value of these operators can be computed exactly in terms of the analogous observables in bosonic 2d Yang-Mills on S2, or equivalently by a gaussian matrix model. Several evidences for this conjecture, both on the gauge theory and on the string theory side, will be presented.

We review a duality between N=4 super Yang-Mills and a string theory in AdS5xS5 background and integrability arising in both theories. New results conserning finite length operators/short strings will be presented.

We first quickly review some basic facts about quantum theories in 1+5 dimensions with 2,0 superconformal symmetry. They arise as decoupled subsectors of string or M theories on space times containing certain defects like branes or singularities. By compactification, they are also related to Yang-Mills theories in lower dimensions. We then describe, in somewhat more detail, recent work on these theories considered on a spatial five torus. In particular, one may determine the spectrum of vacuum states, and also learn much about the spectrum of BPS states. Hopefully, this will eventually be helpful for giving a definition of these mysterious theories.

We attempt to indirectly deduce the form of quantum geometry on the M5-brane in the presence of a constant C-field 3-form potential. We first show how the noncommutative geometry on the D3-brane in a constant 2-form B-field can be deduced by analysing open strings ending on the D3-brane. We show that the string boundary condition is a modified Nahm equation, and that these modifications can be understood in terms of the noncommutative geometry. Similarly, the boundary condition for M2-branes ending on the M5-brane is given by the Basu-Harvey equation, modified by the C-field. By an analogy, we interpret this modification in terms of a proposed quantum geometry on the M5-brane. This geometry is specified by a 3-bracket, unlike the usual 2-bracket for noncommutative geometry.

In some recent work with Kronheimer we have carried over a version of Juhasz' Heegaard Floer Homology for sutured three manifolds to Instanton Floer Homology. This leads to amongst over things a streamlined proofs of property P for all knots and that symplectic manifolds have non-vanishing Donaldson invariants.

From dimer models, as introduced in string theory, we can produce a class of Calabi Yau algebras which are candidates for non-commutative crepant resolutions of Gorenstein 3 fold affine toric singularities. In this talk I will introduce, via examples, dimers and their corresponding toric varieties. I will then talk about the consistency condition that underlies the Calabi Yau property.

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.

N=4 super Yang-Mills theory has a remarkable feature of being integrable. This discovery has made it possible to achieve strong evidence for the AdS/CFT duality. But what about its siblings: marginal deformations of N=4 SYM that preserve conformal symmetry. In this case much less is known about the conjectured supergravity dual. In particular, the Leigh-Strassler deformations are of this kind, which are only integrable for some special values of the deformation parameters, or in some restricted subsectors. In order to gain more insight into the theory, one would like to understand the symmetries of the theory better. We know that the symmetries of integrable systems can be described by Hopf algebras. Here we would like to show that a Hopf algebra structure actually emerges for the generic Leigh-Strassler deformation.

In recent years there have been quantum leaps in the development of calculational technology for field theory amplitudes, partly captured by the buzzwords in the title of this talk. Interestingly, most of these developments are rooted within field theory ideas of the sixties which centered around the 'analytic S-matrix'. This line of thought led to the birth of string theory with Veneziano's amplitude. It therefore seems a natural question if these recent field theory ideas can be found within the frame-work of string theory amplitudes. On the basis of three examples, I will argue that this is not only natural but profitable.

We construct new exact solutions of pure gravity in five dimensions representing black rings in Taub-NUT spaces, both at zero and at non-zero temperature. We use these solutions to study properties of the D0-D6 system in the supergravity approximation. In particular we compute the interaction energy between D0 and D6 branes, analyze equilibrium configurations and their stability.

Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by writing the quaternionic geometry on the hypermultiplet moduli space in terms of twistor geometry. Using this twistor space approach, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions.

In this talk we will consider dynamics with infinitely many time derivatives, such equations follow directly from string field theory and have many interesting properties. First we will review results for the case of Minkowski background and then consider coupling to nontrivial background, in particular, to Friedmann-Robertson-Walker metric. New methods for solving corresponding nonlocal Friedmann equations will be presented and resulting solutions in the view of cosmological applications will be discussed.

A brane tiling is a bicolored graph on an oriented real 2torus, which conjecturally describes both the derived category of coherent sheaves on a 2dimensional toric Fano stack and the derived category of the directed Fukaya category of the mirror. When the toric Fano stack is the projective plane, the corresponding brane tiling divides the torus into three hexagons. In the talk, based on a joint work in progress with Masahiro Futaki, I will describe the analogue of brane tiling for the projective space, which divides the real 3torus into four truncated octahedra, and explain how it helps to study a torus-equivariant version of homological mirror symmetry.

We will discuss the missing pieces in the understanding of the effective field theory description of string creation, the S-dual of the Hanany-Witten effect, both in the open and closed string picture. We explain the origin of a crucial bare Chern-Simons term, that used to be added in by hand for consistency. Finally we summarize the remaining unsettled issues, concerning the need to modify the DBI action and the interpretation of the bare term in M-theory.