Higher Spin Algebras are (infinite-dimensional) Lie algebras, that underlie a theory with Higher Spin spectrum. In different space-time dimensions Higher Spin algebras have different properties. I will present a review and some new results on Higher Spin algebras in different dimensions.

In M-theory, the only AdS_7 supersymmetric solutions are AdS_7 x S^4 and its orbifolds. We find and classify new supersymmetric solutions of the type AdS_7 x M_3 in type II supergravity. While in IIB none exist, in IIA with Romans mass (which does not lift to M-theory) there are many new ones. We use a pure spinor approach reminiscent of generalized complex geometry. Without the need for any Ansatz, the system determines uniquely the form of the metric and fluxes, up to solving a system of ODEs. Namely, the metric on M_3 is that of an S^2 fibered over an interval; this is consistent with the Sp(1) R-symmetry of the holographically dual (1,0) theory. By including D8 brane sources, one can numerically obtain regular solutions, where topologically M_3 = S^3.

I will be discussing the relation between scale and conformal symmetry in unitary Lorentz invariant QFTs in four dimensions.

Over the last five years, the gauge/gravity correspondence has been applied to describe quantum critical systems at finite density. The simplest model to consider is Einstein-Maxwell gravity, and the ground state of the system is described by Reissner-Nordstrom black hole where all the charge is carried by the black hole. However, it turns out that this solution is unstable to the formation of both fermionic and bosonic matter, corresponding in the dual field theory to the creation of a Fermi surface and the onset of superconductivity, respectively. We consider Einstein-Maxwell system coupled to a perfect fluid of charged fermions and a charged scalar field. In addition to the black hole, electron star and holographic superconductor solutions, we find new asymptotically AdS 4 solutions, dual to 2+1 CFTs at zero temperature and finite chemical potential, which contain both scalar hair and an electron star. We compute the free energy and show that these new solutions are thermodynamically favoured when they exist. Moreover, we find evidence for a continuous phase transition between the holographic superconductor and the new solutions.

Compactifications of string theory on Calabi-Yau threefolds lead to supersymmetric four-dimensional vacua with unstable moduli. In order to stabilise these moduli, one may introduce background fluxes in the compact 6-manifold. However, such fluxes backreact on the internal geometry so that the Calabi-Yau condition is broken and a weaker condition of reduced structure group is imposed instead. In contrast to the vast number of Calabi-Yau manifolds, few manifolds with the relevant structure group have been constructed, and this lack of examples has left important properties of flux compactifications in obscurity. In this talk, I will report on recent progress in the construction of SU(3) structures on 6-dimensional smooth compact toric varieties (SCTVs). I will review the topological criterium for the existence of an SU(3) structure on a 6-manifold, which can be fulfilled on an infinite class of SCTVs. Since in string vacua the torsion of the SU(3) structure are constrained, I will then present a method to explicitly construct the SU(3) structure and compute its torsion. I will discuss when parametric choices can be made to tune the torsion classes, and illustrate the construction with several examples.

Instantons in higher-dimensional gauge theories appear, for example, in the context of string compactification. The instanton condition on the compact part of spacetime ensures supersymmetry preservation. My aim is to better understand instantons on special holonomy manifolds. I introduce higher-dimensional instantons and show how the instanton condition can be rewritten as a set of differential equations and algebraic conditions. These equations can be solved under certain simplifying assumptions. The algebraic conditions can be interpreted as relations of a certain quiver gauge theory. I describe the construction of these quivers and show that the instanton conditions match the quiver relations.

I will review the concept of Borel transform and resurgence behavior for the perturbative expansion of generic physical observables presenting particular examples coming from quantum mechanics and supersymmetric localized QFT. I will then discuss more in details the physical role of non-perturbative saddle points of path integrals in theories without instantons, using the example of the asymptotically free two-dimensional principal chiral model (PCM). Standard topological arguments based on homotopy considerations suggest no role for non-perturbative saddles in such theories. However, resurgence theory, unifying perturbative and non-perturbative physics, predicts the existence of several types of non-perturbative saddles associated with features of the large-order structure of perturbation theory. These points are illustrated in the PCM, where we found new non-perturbative fractionalized saddle point field configurations, and give a quantum interpretation of previously discovered `uniton’ unstable classical solutions.

Over the last decade, we have witnessed remarkable progress in our understanding of Quantum Field Theories. New insights have emerged from a multitude of fronts, ranging from the Gauge/Gravity Correspondence to Integrability. In this seminar I will discuss Bipartite Field Theories (BFTs), a new class of QFTs embodying many of these new approaches. BFTs are 4d, N=1 quiver gauge theories with Lagrangians defined by bipartite graphs on Riemann surfaces. Remarkably, they underlie a wide spectrum of interesting physical systems, including: D-branes probing Calabi-Yau manifolds, their mirror configurations, integrable systems in (0+1) dimensions and scattering amplitudes in N=4 SYM. I will introduce new techniques for studying these gauge theories. I will explain how their dynamics is captured graphically and the interesting emergence of concepts such as Calabi-Yau manifolds, the Grassmannian and cluster algebras in the classification of IR fixed points. Finally, I will introduce a new framework for analyzing general systems of D3 and D7-branes over toric Calabi-Yau 3-folds. These ideas can be exploited for embedding BFTs in String Theory but have a much wider range of applicability.

We consider holography for Lifshitz spacetimes with dynamical exponent z=1+epsilon^2, where epsilon is small. Тhe holographically dual field theory is a specific deformation of the relativistic CFT, corresponding to the z=1 theory. We set up the holographic dictionary for Einstein-Proca models and explain how renormalization turns the relativistic conformal invariance into non-relativistic Lifshitz invariance with dynamical exponent z=1+epsilon^2. Using only QFT arguments we show that a particular class of deformations of CFTs generically leads to Lifshitz scaling invariance and we provide examples of such deformations. An analytic Lifshitz black brane up to second order in ε is constructed. Relation to some top-down construction will be discussed. Based on 1304.7776 and 1306.3344.

In this talk, we focus on strongly-coupled, translation-invariant holographic phases at finite density. We show that they can be classified according to the scaling behavior of the metric, the electric potential and the electric flux, introducing to new scaling exponents (cohesion and conduction). Solutions fall into two classes, depending on whether they break relativistic symmetry or not. We show that the dimensions of IR operators are governed by the new scaling exponents, as well as the low-frequency scaling of the optical conductivity. We show that thermodynamically stable phases are always gapless. Finally, we examine a refinement of the holographic entanglement entropy sensitive to the IR behaviour of the electric flux, and show that the minimal surface thus obtained can be different from the Ryu-Takayanagi proposal depending on the cohesion exponent.

I will discuss some interesting connections between quantum and classical integrable systems (e.g. Hitchin system vs. quantum spin chains) in the context of 3 dimensional quiver gauge theories. Gauge theory dualities (Mirror symmetry, Seiberg duality) have direct counterparts on the integrability side.

We discuss thermodynamics of N M2- and M5-branes by using the method proposed by Smilga and Wiseman, which explains the black Dp-brane thermodynamics from the maximally supersymmetric U(N) Yang-Mills theories. As result we obtain the consistent results with the predictions from the eleven-dimensional supergravity by very simple calculations: The free energy of M2-branes is evaluated by using ABJM theory as F ~ N^{3/2}k^{1/2}T^3, and that of M5-branes is estimated by assuming some natural properties of 6d conformal field theory as F ~ N^3T^6.

We discuss tree level scattering of any number of massless open superstring states on a worldsheet of disk topology. The entire state dependence of the tree amplitude can be expressed in terms of gauge theory subamplitudes from the point particle limit. The string corrections entering through momentum dependent integrals over the disk boundary can be disentangled from the YM seeds and analyzed separately. Their power series expansion in the string length and momenta involves multiple zeta values (MZVs). We review some mathematical background on MZVs and the network of relations between them. The explicit form of any tree level string correction to YM theory is derived from the generating function of MZVs -- the Drinfeld associator. It interpolates between the worldsheet integrals in N-point and (N-1)-point scattering and leads to a recursive formula for the momentum expansion of any disk amplitude. Our results apply for any number of spacetime dimensions or supersymmetries and chosen helicity configurations.

Although at first sight it may seem an odd idea, I will argue that it is actually quite natural to investigate the properties of General Relativity and its black holes in the limit in which the number of spacetime dimensions grows to infinity. The theory simplifies dramatically: it reduces to a theory of non-interacting particles, of finite radius but vanishingly small cross sections, which do not emit nor absorb radiation of any finite frequency. This leads to efficient calculational approaches in an expansion around this limit, as well as to intriguing connections to low-dimensional string-theory black holes.

From the supersymmetric partition functions of gauge theories on S5 and CP2 x S1, we calculate the superconformal index of the 6d (2,0) SCFT on S5 x S1 and explore its physics.

I will review the basic techniques and ideas involved in the study of dualities in D=3,N=4 supersymmetric gauge theories using partition functions on a sphere and demonstrate how these partition functions can provide an extremely non-trivial check of mirror symmetry for a large class of quiver gauge theories. In particular, I will focus on theories whose Type IIB descriptions involve D3 branes ending on orbifold/orientifold 5-planes. In addition to providing a convincing check, this procedure allows one to extract important information about the duality like the "mirror map" rather trivially.

We explain how to define, in matrix (gauge) field theory, the notion of the effective action of k ``probe'' branes in the presence of N ``background'' branes on which the field theory lives. The analysis of the large N planar diagram expansion which computes the effective action yields a simple and generic mechanism explaining the emergence of holographic space dimensions. The resulting effective action matches the non-abelian D-brane action in the closed string background dual to the field theory. We shall discuss briefly a few explicit examples, and also provide a general introduction to the notion of emerging space.

We present a collection of results supporting holographic dualities between 3D higher spin gravity and 2D higher spin-symmetric minimal model CFTs at large central charge, focusing on those that are not fixed by higher spin symmetry alone. This includes the first bulk-boundary matching of correlators in 3D higher spin gravity whose functional form is not fixed by conformal invariance, namely, 4-point functions of certain scalar primary operators. In the bulk, this involves the study of propagating scalars in higher spin gravitational backgrounds, which will also help clarify what constitutes a higher spin black hole.