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.

A systematic understanding of the low energy limit of string theory scattering amplitudes is essential for conceptual and practical reasons. In this talk, I shall report on a recent work where this limit has been analyzed using tropical geometry. Our result is that the field theory amplitudes arising in the low energy limit of string theory are written in a very compact form as integrals over a single object, the tropical moduli space. This picture provides a general framework where the different aspects of the low energy limit of string theory scattering amplitudes are systematically encompassed; the Feynman graphs structure and the ultraviolet regulation mechanism. I shall then give examples of application of the formalism, in particular at genus two, and discuss open issues. No knowledge of tropical geometry will be assumed and the topic shall be introduced during the talk.

We construct the first eleven-dimensional supergravity solutions, which are regular, have no smearing and possess only SO(2,4) x SO(3) x U(1)_R isometry. They are dual to four-dimensional field theories with N = 2 superconformal symmetry. We utilise the Toda frame of self-dual four-dimensional Euclidean metrics with SU(2) rotational symmetry. They are obtained by transforming the Atiyah--Hitchin instanton under SL(2,R) and are expressed in terms of theta functions. The absence of any extra U(1) symmetry, even asymptotically, renders inapplicable the electrostatic description of our solution.

We holographically study the entanglement between two CFTs in a thermofield double state at nonzero chemical potential. In the bulk, this entanglement corresponds to entanglement between the two exterior regions of a Reissner-Nordstrom AdS black hole. We will make use of two probes: thermo-mutual information and two-point correlators of scalar operators. In particular, in the zero-temperature limit the entropy density of the black hole remains finite, while neutral correlators and the mutual information of finite regions vanish, implying that these are not good probes of entanglement at zero temperature. However, the correlators of electrically charged scalar operators can be made to remain finite. We comment on the time evolutions of these quantities and other possible applications.

While the emergence of bulk locality in AdS/CFT is not fully understood, progress has been made towards understanding how pieces of the bulk geometry are encoded in subregions of the CFT. Recently, Hubeny and Rangamani have proposed that a modification of the Ryu-Takayanagi entropy called the 'causal holographic information' (so called because extremal surfaces are replaced with causal boundaries) quantifies the minimum information needed to reconstruct certain causally defined bulk regions. I will argue that the boundary dual of the causal holographic information is a coarse-grained entropy which tracks the one-point functions in the associated boundary domain of dependence. The talk will focus on the motivation and evidence for this conjecture as well the prospects for future precision tests.

The deformations of higher-spin symmetries induced by cubic interactions of symmetric massless bosonic fields are analyzed within the metric-like formalism. In particular, we identify a class of couplings which leave the gauge algebra Abelian but deform one (out of three) gauge transformation, and another class of couplings which deform all three gauge transformations in (A)dS but only two in the flat-space limit. The former class is related to higher-spin algebra multiplets (representations of the global algebra). The latter class is what makes (A)dS a distinguished background for higher-spin interactions and includes in particular the gravitational interactions of higher-spin fields, retrospectively accounting for the Fradkin-Vasiliev solution to the Aragone-Deser problem.

Recently the existence of a new quantity which decreases along RG-flows of 4d supersymmetric QFT's with R-symmetry has been conjectured. I will analyze this conjecture from a dual supergravity perspective: using some general properties of domain-wall solutions dual to R-symmetric RG flows, I will define a function interpolating between the correct values at the UV and IR fixed points. I will finally test its monotonicity properties in a class of examples.

Superfields in Superspace provide very economical means of representing supermultiplets. However, there are limitations on the possible actions one may construct in standard superspace when the number of supersymmetries grow. In projective superspace, superspace is enlarged with one CP(1) at each point. I will explain how this improves the situation and exemplify.