We review some results concerning integrable systems on the moduli spaces of holomorphic bundles. The interrelations between bundles of different topological types are given by modifications (singular gauge transformations). The modification provides canonical maps between corresponding integrable systems. It also allows to describe the Backlund transformations. We apply the general scheme to the elliptic Calogero-Moser model and construct a map to an integrable Euler-Arnold top (the elliptic SL(N)-rotator). The models correspond to the bundles of degree zero and one. Explicit formulae represent a bosonization of the coadjoint orbit. The later describes the phase space of the rotator while the Poisson structure of the Calogero model is canonical. The construction can be generalized to more complicated models. For example, field generalization (loop group or 1+1 theories) leads to relation between the field Calogero model and the Landau-Lifshitz equation for XYZ model while the generalization to the isomonodromic deformations leads to relation between Painleve VI equation and Zhukovsky-Volterra gyrostat. In the end we note that the classification of models is described by the center of the structure group which are shown to be characteristic classes of the bundles. This remark allows to generalize the construction to the case of an arbitrary simple Lie group. Three-dimensional generalization of the construction naturally leads to the Bogomolny equations. The modification in this case provides non-trivial charge of the monopole solution. We find the Dirac monopole solution in the case (R)x(Elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it.

We review some results concerning integrable systems on the moduli spaces of holomorphic bundles. The interrelations between bundles of different topological types are given by modifications (singular gauge transformations). The modification provides canonical maps between corresponding integrable systems. It also allows to describe the Backlund transformations. We apply the general scheme to the elliptic Calogero-Moser model and construct a map to an integrable Euler-Arnold top (the elliptic SL(N)-rotator). The models correspond to the bundles of degree zero and one. Explicit formulae represent a bosonization of the coadjoint orbit. The later describes the phase space of the rotator while the Poisson structure of the Calogero model is canonical. The construction can be generalized to more complicated models. For example, field generalization (loop group or 1+1 theories) leads to relation between the field Calogero model and the Landau-Lifshitz equation for XYZ model while the generalization to the isomonodromic deformations leads to relation between Painleve VI equation and Zhukovsky-Volterra gyrostat. In the end we note that the classification of models is described by the center of the structure group which are shown to be characteristic classes of the bundles. This remark allows to generalize the construction to the case of an arbitrary simple Lie group. Three-dimensional generalization of the construction naturally leads to the Bogomolny equations. The modification in this case provides non-trivial charge of the monopole solution. We find the Dirac monopole solution in the case (R)x(Elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it.

There is good evidence that a class of conformal gauge theories at strong gauge coupling and for large rank of the gauge group can be described by classical gravity in five dimensions. In this gauge/gravity duality, the theory of classical gravity always admits a consistent truncation to pure gravity so that there is a sector in the dual gauge theories where the dynamics is universal. This universal sector constitutes phenomena like hydrodynamics, but also other phenomena far away from equilibrium such as decoherence and relaxation. We will use field theoretic tools to understand how the time evolution of all states and the expectation values of all observables in the universal sector get determined by the energy-momentum tensor alone. We will find a way to extrapolate our field-theoretic results to strong coupling to propose a condition on the energy-momentum tensor such that the dual solution in gravity has a smooth future horizon, as a test of our approach. We will also do a first study of how irreversibility emerges at long time scales of observation through gravity.

AdS2-CFT1 correspondence leads to a prescription for computing the degeneracy of a single centered extremal black hole in terms of path integral over string fields living on the near horizon geometry of the extremal black hole. This prescription is called quantum entropy function. In four dimension the near horizon isometry group SL(2,R) x SU(2) for BPS black hole gets enhanced to full SU(1,1,2). Using these enhanced supersymmetries and localization techniques, we will argue that the path integral receives contribution only from special class of field configurations which are invariant under a particular subgroup of SU(1,1,2). We will identify saddle points which are invariant under this subgroup.

Over the last years, gauge/string duality has been extended to include gauge theories with an arbitrary number of flavors. We study the flavoring procedure in the light of calibrated geometry and discuss the special case of a type IIA dual of N=1 super Yang-Mills with flavors. Relating our results to the standard type IIA/M-theory duality, we find that the usual oxidation formulas cannot accommodate for the additional flavor branes. We address and solve this issue by considering M-theory with torsion, which allows us to construct source-modified equations of motion for eleven-dimensional supergravity.

Solvmanifolds, in particular nilmanifolds, commonly known as twisted tori, provide several examples of internal manifolds in flux compactifications towards de Sitter, Minkowski or Anti de Sitter. The relation between these manifolds and the six-dimensional torus, and the string vacua obtained on them, are the main interests of this talk. We will first present some properties of these manifolds. We will give a generic construction of their Maurer-Cartan forms out of the six-dimensional torus, via a transformation called the twist. We will then describe several Minkowski flux backgrounds of type II supergravity obtained on these manifolds. Thanks to the generalized complex geometry approach, we will show that one can obtain those solutions from solutions on the torus, via the twist transformation. The latter then acts as a solution generating technique, being able to relate backgrounds which are not T-duals. Finally, we will apply this twist transformation technique to relate Kahler/non-Kahler solutions of the heterotic string. This talk is based on a work with Enrico Goi, Ruben Minasian and Michela Petrini.

I talk about finite corrections to the large spin asymptotics of N=4 SYM twist operators. These corrections are an all-loop result, not affected by wrapping effects, and agree, after determining the strong coupling expansion, with string theory predictions. The finite order corrections enter the sub-leading poles of the logarithm of multi loop gluon scattering amplitudes and describes the difference between the former and the expectation value of certain Wilson lines. For the complete sub-leading pole structure however a second function is needed. A few comments on the quest for a search of a possible description of the latter in terms of local operators will be given.

We review the derivation of a 6d gauge theory related to the M5 brane from the Bagger-Lambert-Gustavsson (BLG) model with the volume preserving diffeomorphisms gauge group. In particular, the Lagrangian for the 6d self-dual antisymmetric tensor field appears upon converting three of eight scalars in the BLG theory into the components of antisymmetric tensor field in D=6. We develop also a Lorentz-covariant Lagrangian formulation for such an antisymmetric tensor field in the free case. This formulation admits a natural supersymmetrization as well as coupling to 6d gravity.

The Tachyon-Dirac-Born-Infeld (TDBI) action captures some aspects of the dynamics of non-BPS D-branes in type II string theory. We show that it can also be used to study the classical interactions of BPS branes and antibranes. Our analysis sheds light on real time D-Dbar tachyon condensation, on the proposal that the tachyon field can be thought of as an extra spatial dimension whose role is similar to the radial direction in holography, and on Sen's open string completeness conjecture.

Exact solution of 4d nonlinear bosonic higher-spin gauge theory, that provides a higher-spin deformation of the AdS4 Schwarzschild black hole, is found.

TBA

The N=2 boundary superspace formulation of RNS models in the presence of arbitrary NSNS fluxes is developed. This allows for a systematic classification of supersymmetric D-brane configuration which leads e.g. to generalized lagrangian and coisotropic branes.

Recent work on membranes of M-theory has lead to a new type of physical realization of fuzzy 2-spheres in Matrix Brane actions. In standard realizations, the fuzzy two-sphere coordinates transforming in the vector of the spherical rotation symmetry are identified with matrix variables in the adjoint representation of the unitary symmetry of the Matrix brane actions.In these new realizations, spinors of the rotational symmetry are identified with variables in bi-fundamental representations of the unitary symmetry. The vector coordinates of the fuzzy 2-sphere are recovered from bilinears in the spinors by a fuzzy version of the projection map of the Hopf fibration over the two-sphere. The outcome is the emergence of a five dimensional theory from a three dimensional one as expected from the intersection of the fundamental branes of M-Theory in ABJM quotients of eleven dimensional spacetime. The demonstration of the emergent six dimensional theory expected for the brane intersection in M-Theory without a quotient remains an open problem.

In the first part of the talk I will review differential geometric background involving associative submanifolds of 7 manifolds with G2 holonomy and an analogue of the Yang Mills instanton equation for connections on bundles over such a manifold. Then I will describe joint work with Ed Segal in which, assuming these objects have suitable formal properties, we define holomorphic bundles over moduli spaces of Calabi-Yau 3-folds which can be viewed as the complexification of Floer theory. In the last part of the talk I will consider in more detail the problem of establishing the foundations required for the theory. This involves pertubations of the equations and the question of how to count solutions at infinity, which are pairs consisting of a connection and an associative submanifold. The counting problem leads to a nonlinear generalisation of the spectral flow for eigenvalues of Dirac operators.

In this talk, we will review recent work which asks the question, what, if any effects stringy modifications to geometry might have on the predictions of inflation? We couch our discussion within two well motivated frameworks-- non-commutative geometry (likely to be present during brane inflation), and modifications to mode propagation coming from a field theory representation of the stringy uncertainty principle. In the former case, we find that UV/IR mode mixing can imprint potentially observable signatures onto the CMB independent of the scale of inflation. In the latter case, we analytically compute corrections, and comment on the possibility of observing these in the fortunate circumstance that we happen to find ourselves in a weakly coupled corner of moduli space.

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.