We show how "generalised geometry", a class of extensions of conventional differential geometry first introduced by Hitchin, gives a natural way of formulating supergravity theories. The formulation unifies the bosonic fields and symmetries and has a natural action of O(d,d) or the exceptional groups E_d in d-dimensions. By introducing the analogue of the Levi-Civita connection we find that full set of bosonic equations of motion reduce to simply the vanishing of the generalised Ricci tensor. We show that the connection also encodes the supersymmetry variations and fermionic equations of motion. This formalism also gives natural extensions of complex, symplectic and other integrable structures, with implications for describing supersymmetric string theory backgrounds.

I will explain how to put three-dimensional supersymmetric theories on curved three-manifolds preserving some supersymmetry. I will discuss some physical implications of this method, including an exact formula for the two-point function of the energy-momentum tensor in 3d N=2 superconformal theories.

I will describe a new infinite set of AdS/CFT dual pairs which come from M5-branes wrapping Riemann surfaces. These solutions interpolate between and extend beyond a famous pair of solutions by Maldacena and Nunez. Additionally, the dual SCFTs are so-called "non-Lagrangian" theories, which have no weakly coupled UV description yet can (and will) be described explicitly.

I will review the localisation of 2d N = (2, 2) supersymmetric gauge theories with a vector U(1) R-symmetry on the round 2-sphere, leading to a matrix integral expression for the partition function, in the so called "Coulomb branch" representation. For gauge theories which are completely Higgsed in the presence of a Fayet-Iliopoulos term, an alternative localization scheme yields the "Higgs branch" representation of the same partition function as the product of vortex times antivortex partition functions, weighted by semiclassical factors and summed over isolated vacua on the Higgs branch. Finally, I will briefly overview the potential of these exact results for the study of type II string compactifications and supersymmetric gauge theories in various dimensions. Based on arXiv:1206.2356 with F. Benini.

Four-dimensional supersymmetric type II string theory vacua can be described elegantly in terms of pure spinors on the generalized tangent bundle T+T. This leads to constraints on M6 involving Hitchin's "generalized complex geometry". In this talk, we apply the same techniques to any ten-dimensional supersymmetric solution, not necessarily factorized as AdS4xM6 or Mink4xM6. I will describe a system of differential equations in terms of forms describing a "generalized ISpin(7) structure" is equivalent to unbroken supersymmetry, in both IIA and IIB. One of the equations in the system reproduces in one fell swoop all the pure spinors equations for four-dimensional vacua.

Using localization techniques, Pestun has shown how to reduce certain quantum field theory path integrals to finite-dimensional matrix integrals. In this way, the expectation value of certain operators in the gauge theory is equivalent to the expectation value of observables in the associated matrix model. In this talk, Pestun's matrix model is used to compute the expectation value of the circular Wilson loop in N = 2 SuperConformal QCD, in the weak and strong coupling regime. At weak coupling, we define a perturbative scheme that permits to compute the Wilson loop at arbitrary high number of loops. At strong coupling, our computation carries information about the string theory dual of the SuperConformal QCD, that is not known yet. In particular, our results indicate that the string tension in the dual string theory scales as the logarithm of the gauge theory 't Hooft coupling.

The internal dynamics of U-folds (or T-folds) seems to provide interesting generalisations of Einstein gravity. What can we learn from supersymmetry? I describe work in progress, where pure spinor techniques are adapted to the U-duality setting.

TransPlanckian inflationary excursions are very interesting because they are sensitive to Planck-suppressed operators and can generate observable tensor modes. I review one proposed model generating a transPlanckian excursion and argue that consideration of antibrane backreaction renders the assumptions entering the model inconsistent.

I review the recent construction of six-dimensional (1,0) superconformal models with non-abelian gauge couplings for multiple tensor multiplets. A crucial ingredient in the construction is the introduction of higher-rank tensor fields. I discuss various examples and the possible relation to the (2,0) theories underlying the dynamics of multiple M5 branes.

Note: Sunil Mukhi's talk was cancelled!

In this talk we derive a novel and fundamental relation between C*-algebras and Complex Cobordism. We endow the space of complex genera with the structure of an infinite Lie group, with the underlying manifold being isomorphic to the group of automorphisms of the formal disc, and relate it then to operator algebras. The group operation introduced can be understood combinatorially in terms of planar rooted trees. Finally we shall give connections with Conformal Field Theory.

Maximally supersymmetric string theory compactified on d-dimensional torus is invariant under the non-perturbative U-duality groups E_{d+1}(Z). These duality groups are the discrete version over the integers of the real split forms of the simply laced lie groups of rank d. The interactions of the theory are given by automorphic function under E_{d+1}(Z), given in the simplest cases by the residue of maximal parabolic Eisenstein series. The dimensional reduction of the theory to three dimensions gives rise to the duality group E_8 from which all BPS couplings in higher dimensions can be extracted. We will present the construction of the automorphic form relevant for the supersymmetric protected BPS interactions in the low-energy expansion of maximal supersymmetric effective action. We will show that the support of the non-vanishing Fourier coefficients for these automorphic form is given by the expected BPS conditions one can derive from the supersymmetric conditions.

TBA

For arbitrary gauge groups, I will describe how one can check at the one-instanton level that the Nekrasov partition function of pure N=2 super Yang-Mills is equal to the norm of a certain coherent state of the corresponding W-algebra. For non-simply-laced gauge groups, this coherent state is in the twisted sector of a simply-laced W-algebra. I will also talk about the case of gauge theories with hypermultiplets.

Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding self-dual strings, respectively. We study an index which counts BPS instantons with electric charges in Coulomb and symmetric phases. We first prove the existence of unique threshold bound state of U(1) instantons for any instanton number. By studying SU(N) self-dual strings in the Coulomb phase, we find novel momentum-carrying degrees on the worldsheet. The total number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory. We finally propose that our index can be used to study the symmetric phase of this theory, and provide an interpretation as the superconformal index of the sigma model on instanton moduli space.

We show that in order to understand the higher-dimensional origin of all supersymmetric branes in (toroidally) compactified string theory one needs to extend the family of branes with an additional set of so-called ``generalized monopoles''. We will show what these monopoles are and how they lead to interesting ``wrapping rules'' for the branes in string theory.

N = 2 SuperConformal QCD is perhaps the simplest theory outside the N = 4 "universality class" that we can attempt to study holographically. It is moreover continuously connected to the N = 4 class by an interpolating family of N = 2 SCFT. Careful analysis of its chiral spectrum points towards a gravity dual description via a non-critical string background. We then turn to the study of spin chains. The complete one-loop Hamiltonian and some higher-loop results already reveal a rich dynamics. We present past results and current progress.

Recent results (arXiv:1107.1872 and arXiv:1107.5028) will be presented about nonabelian interactions for simple mixed-symmetry gauge fields in AdS background.

Consistent truncations have proved to be powerful tools in the construction of new string theory solutions. Recently, they have been employed in the holographic description of condensed matter systems. In the talk, I will present a rich class of supersymmetric consistent truncations of higher-dimensional supergravity which are based on geometric structures, focusing on the tri-Sasakian case. Then I will discuss some applications, including a general result relating AdS backgrounds to solutions with non-relativistic Lifshitz symmetry.