We give a survey on recent results on the physics of black holes in four-dimensional anti-de Sitter space. In particular, we present a general recipe for constructing rotating black holes in N=2 matter-coupled gauged supergravity. Several physical aspects of these solutions are discussed, like their thermodynamics and dual hydrodynamical interpretation. It turns out that there exist new exotic black holes whose horizon is a noncompact manifold with finite volume. Finally, we show how to construct multi-centered black hole solutions in anti-de Sitter space.

I will consider supersymmetric field theories on compact manifolds M and obtain constraints on the dependence of their partition functions Z_M on the geometry of M. For N=1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. I will show how to describe the theory in terms of twisted variables that make easy to analyze the dependence of Z_M on the parameters entering the Lagrangian. I will also show that Z_M is "almost" topological: Z_M is independent of the Hermitian metric and depends holomorphically on the complex structure moduli.

We determine the bosonic part of the superstring sigma model Lagrangian on eta-deformed AdS_5 x S^5, and use it to compute the perturbative world-sheet scattering matrix of bosonic particles of the model. We then compare it with the large string tension limit of the q-deformed S-matrix and find exact agreement. We comment on the structure of the RR sector of the type IIB SUGRA background.

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with N=2 superconformal symmetry.

I will discuss the question of whether four dimensional scale invariant unitary quantum field theories are actually conformally invariant. I will present a complete analysis of possible scale anomalies in correlation functions of the trace of the stress-energy tensor in such theories and discuss recent attempts to show that scale invariance implies conformal invariance.

A brief review will recapitulate some of the essential features of T-duality symmetry in string theory. A proposal will be presented to show how a scattering amplitude of some string excitations could be related to another amplitude by T-duality transformation for a class of stringy states. This is considered in a simple scenario. Some illustrative examples will be presented as application of the proposal.

This talk addresses a basic question about supersymmetric gauge theory: how does a soliton behave in the vicinity of a heavy charged object? The answer to this question is intuitive and geometrical, and can be expressed in terms of a natural connection over the soliton moduli space. Based on 1309.2644 and 1401.6167 with David Tong.

Certain classes of supersymmetric Yang-Mills (SYM) theories including the well known N=4 SYM, that takes part in the AdS/CFT correspondence, can be formulated on a Euclidean spacetime lattice using the techniques of exact lattice supersymmetry. In this talk I will provide the constructions of such theories and describe how great ideas such as topological field theories, Dirac-Kahler fermions, geometric discretization all come together to create lattice SYM theories that are exact-supersymmetric, gauge-invariant, local and doubler-free on the lattice. Then I move on to discuss the recent lattice constructions of specific classes SYM theories in two dimensions, with matter in various representations.

Higher spin gravity theories in Anti-de Sitter space have been conjectured to be holographically dual to conformal field theories with vector-like matter fields: in the simplest version of the duality, these are just free CFT's of N-component massless scalars or fermions. After reviewing these conjectures, I will discuss recent new tests of the duality based on comparing the bulk and boundary partition functions. In particular, I will show that the sum over one loop free energies of the infinite set of massless higher spin gauge fields in AdS_{d+1} is precisely consistent with the form of the O(N^0) term in the partition function of the CFT on S^d and S^1xS^{d-1}.

This talk will review perturbative and non-perturbative properties of string theory scattering amplitudes with particular emphasis on the constraints imposed by duality with M-theory on their low energy action.

We discuss elliptic genera of supersymmetric two-dimensional conformal field theories with continuous spectra. We argue that they consist of a mock modular term, and a non-holomorphic modular completion. We illustrate the general arguments with a large class of examples, computed using conformal field theory, string theory and localization techniques. We discuss applications, amongst others to black hole physics.

Theories with anomalies exhibit unusual thermodynamic and hydrodynamic behavior. This behavior may be captured entirely by constructing a "thermal" anomaly polynomial which encodes the solution to the Wess-Zumino consistency condition and an additional condition (consistency with the Euclidian vacuum) which the thermodynamic partition function must satisfy. Using the thermodynamic partition function, the role of the anomaly in hydrodynamics can be traced to the coefficients of certain Chern-Simons terms on the base manifold over which the thermal circle is fibered.

In this talk, I will explain how to compute gluon scattering amplitudes in planar N=4 Super-Yang-Mills theory using the flux tube (OPE) picture. The latter, which builds on the duality with light-like Wilson loops, provides a description of scattering amplitudes in terms of excitations evolving on a two-dimensional background. Exploiting the remarkable integrable structures of this background, one can formulate scattering amplitudes, at any values of the 't Hooft coupling, as an expansion around the multi-collinear (a.k.a. OPE) limit. I shall report on recent progress made in this direction and speculate on its possible application to the study of scattering amplitudes in the multi-Regge kinematics.

In this talk I will review recent progress in understanding aspects of 5d gauge theories. Under certain circumstances these can be at fixed points admitting a large N limit, whose gravity dual we will discuss. Such gravity dual suggests a yet more general class of quiver-type fixed point theories which can sometimes be understood in terms of S-duality. To this matter the braneweb construction of the theories will prove very helpful, yielding new insights which we will test using the 5d superconformal index.

I will first review the algebraic structure related to the classical integrability of sigma models at the hamiltonian level. I will then show how this hamiltonian perspective enables to construct an integrable q-deformation of the AdS5 x S5 superstring action.

I will discuss the implications of extended symmetries in higher dimensional CFT's and show that the algebra they generate, the higher-spin algebra, is unique (which extends Maldacena-Zhiboedov results). As an infinite-dimensional extension of conformal symmetries higher-spin algebras can be used to restrict correlation functions and I will present an explicit formula that computes all correlators in unbroken Vasiliev higher-spin theory.

The Kondo effect occurs in metals doped with magnetic impurities: in the ground state the electrons form a screening cloud around each impurity, leading to dramatic changes in the thermodynamic and transport properties of the metal. Although the single-impurity Kondo effect is considered a solved problem, many questions remain, especially about the fate of the Kondo effect in the presence of multiple impurities. In particular, for a sufficiently dense concentration of impurities, a competition between the Kondo effect and inter-impurity interactions can lead to quantum criticality and non-Fermi liquid behavior, which remains poorly understood. In this talk I will present a model of the single-impurity Kondo effect based on holography, also known as gauge-gravity duality or the AdS/CFT correspondence, which may serve as a foundation for a new approach to the multiple-impurity system.

I will review the localisation of 5D supersymmetric Yang-Mills theory on the toric Sasaki-Einstein manifolds. As main examples, the case of sphere and Y^{p,q} spaces will be considered. The large N-limit and factorisation properties of exact partition function will discussed.