Exceptional generalised geometry (or exceptional field theory) is an extension of generalised complex geometry (or double field theory) "geometrising" all degrees of freedom of 11d or type II supergravity. Concentrating on the case in which the 11d tangent space is split into four and seven dimensional spaces, we will discuss the geometry and gauge structure of these exceptional theories. We will show how closure of the "exceptional Lie algebra" requires adding a tower of four-dimensional p-form fields, known in the 4d gauged supergravity language as the tensor hierarchy. This indicates the existence of an underlying structure compatible with the E_{11} construction.

I will review the known integrable structures of (super-)gravity in the presence of a sufficient number of Killing vectors and how they can be used to generate new solutions. The associated inverse scattering techniques have been widely applied to four-dimensional Einstein gravity, starting with the work of Belinski and Zakharov, but their generalisation to other gravity systems poses some problems. I will discuss progress for the case of STU supergravity by exploiting the underlying group theory.

The microscopic description of the 4 dimensional supersymmetric (or BPS) black holes in flat space has already been well understood via the AdS/CFT correspondence on the black hole horizon. In this talk I address the similar problem of finding the dual description for the static BPS black holes in AdS_4 embeddable in M-theory. The gravity picture of an interpolation between asymptotic AdS_4 and AdS_2xS^2 on the horizon can be understood as a renormalization group (RG) flow between a 3d and a 1d superconformal field theory. I discuss in some detail both the gravity and the field theory side, providing evidence for their match. At the end I present a proposal for the 1d CFT states that make up the black hole entropy.

Holographic entanglement entropy is part of an expanding dialogue has opened between string theorists and physicists in a variety of other fields, eg, condensed matter and nuclear physics. Holographic entanglement entropy also provides an interesting window into the suggestion that quantum entanglement plays an essential role in the emergence of spacetime geometry in theories of quantum gravity. In this lecture, I will review some of the basic aspects of entanglement entropy and holographic entanglement entropy. I will also describe how holographic entanglement entropy leads one to consider associating entanglement entropies with general regions of spacetime in quantum gravity. Finally, I will discuss some recent work to examine this conjecture more precisely in the context of the AdS/CFT correspondence.

Entanglement of quantum states and its measures play an important role in many areas of theoretical physics. Some techniques about how to deal with entanglement in QFT will be discussed. In particular, the strong subadditivity play the crucial role in the analysis of the "c-theorems" in 1+1 and 2+1 dimensions. We will also consider the twist fields and how they are employed to find analytic results for the entanglement entropies of disjoint intervals and the negativity (a measure of entanglement for mixed states) 1+1 CFTs.

I will review what has become the standard model for the origin of structure in the Universe: quantum fluctuations of a scalar field during a period of accelerated expansion ("inflation") in the very early universe. I will discuss some of the latest observational evidence, including recent results from ESA's Planck satellite, and what this might tell us about the physics of inflation.

I'll explain a new way of looking at 4d supergravity --- as a theory of holomorphic maps into Penrose's twistor space. Allowing twistor space to have N fermionic directions, the theory is anomaly free when N=8. Via the Penrose transform, the vertex operators correspond to an N=8 Einstein supergravity multiplet. Conformal symmetry is explicitly broken by the presence of the infinity twistor in the BRST operator. I will show how to compute the complete classical S-matrix from worldsheet correlation functions, and interpret these amplitudes geometrically.

A classic problem in field theory is to compute the critical exponents of the second-order phase transitions in 3d, for example for the Ising model universality class. Traditionally, this problem has been approached via RG-based techniques, such as the Wilson-Fisher epsilon-expansion. Here I will discuss another method to extract the critical exponents, and more, by using conformal field theory.

Chern-Simons theories coupled to vector matter exhibit interesting phenomena. In the planar limit, these theories are conjectured to be holographically dual to generalized theories of gravity, involving high-spin fields. This is a weak-weak holographic duality that is in some aspects very simple, and may serve as a toy model for deepening our understanding of both holography and string theory. On the CFT side, exact calculations performed in the planar limit, along with constraints imposed by a ‘slightly-broken’ high-spin symmetry, have led to many exact results. These have uncovered the details of a 3D bosonization duality, relating theories with bosonic matter to theories with fermionic matter. I will present dynamical evidence for this duality.

We study half-BPS Wilson loops in D = 3 N =4 gauge theories using matrix models obtained from localization techniques. The infrared CFTs of the N=4 theories are subject to 3-dimensional mirror symmetry, which exchanges the Higgs and Coulomb branches of vacua of dual theories. Recently progress have been made in understanding the mapping of BPS Wilson loops under mirror symmetry in abelian theories. Our aim is to understand the operators dual to half-BPS Wilson loops in non-abelian theories. We propose a matrix model for the mirror loops by implementing mirror symmetry directly in the matrix model and we verify the mapping of loop operators by computing explicitly the Wilson loops and mirror loops in non-abelian linear quiver theories. We discuss the possible gauge theory operators that would lead to the matrix model we found. Our results are nicely related to the brane realization of linear quivers in IIB string theory.

We discuss partition functions of 2d conformal field theories. Modular invariance is known to constrain the spectrum of such theories. We investigate these constraints systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We also consider generalized partition functions of N = (2,2) superconformal theories and discuss the application of our results to Calabi-Yau compactifications. This talk is based on 1209.4649 with H.Ooguri and 1307.6562 with D.Friedan.

In this talk I will show that gauge-invariant derivatives of scalar fields grow asymptotically along the event horizon of extremal black holes. This growth originates from conservation laws on such horizons. The resulted conserved quantities can be seen as a natural generalization of the Newman-Penrose constants along null infinity. I will also present recent generalizations by Reall and Lucietti concerning electromagnetic and linearized gravitational perturbations of extremal black holes.

To the full order in fermions, we construct D = 10 type II supersymmetric double field theory. We spell the precise N = 2 supersymmetry transformation rules as for 32 supercharges. In terms of a stringy differential geometry beyond Riemann, the constructed action unifies type IIA and IIB supergravities in a manifestly covariant manner with respect to O(10, 10) T-duality and a ‘pair’ of local Lorentz groups, or Spin(1, 9) × Spin(9, 1), besides the usual general covariance of supergravities or the generalized diffeomorphism. The distinction of IIA and IIB may arise after a diagonal gauge fixing of the Lorentz groups. They are identified as two different types of ‘solutions’ rather than two different theories. References: arXiv:1210.5078 (N=2) arXiv:1206.3478 (bosonic N=2) arXiv:1112.0069 (N=1)

Computation of conformal dimensions in planar N=4 SYM using integrability techniques was a hot topic during the last decade, with more than thousand publications devoted to it. I will tell you about our new results in this domain: Instead of the Y-system used previously, we are now able to encode the conformal dimensions, at any value of the 't Hooft coupling, in much simpler way: through a Riemann-Hilbert problem. This appears to be not only a very beautiful mathematical setup, but also the most efficient approach to explicitly compute the dimensions. For instance, we've analytically computed the so called Konishi anomalous dimension up to 8 loops in perturbation theory.