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.

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.