Matrix models are toy models for quantum field theories. They can be extremely complicated but can also be solved in a variety of ways. In my talk I will discuss general properties of matrix models and their solutions and focus on particular matrix models that arise in the study of 4d SUSY field theories. Those matrix models describe the index of the field theory, counting the number of states of the theory (with + sign for a boson and - for a fermion) and have been known for over 10 years. Though they look very complicated I will show how some simple tricks allow in certain cases to solve those matrix models exactly in terms of elementary functions. My talk will focus on the matrix model calculation and no specialized knowledge of SUSY field theories or indices would be required to follow it.

Entanglement entropy is a very important quantity in holography, nevertheless it is only a measure of the total amount of entanglement between two complementary subsystems in a global pure state. A more detailed knowledge about the pattern of entanglement is essential for a deeper understanding of the relation between bulk geometry and quantum information in the boundary theory. I will discuss how negativities and other measures can be used in this context focusing in particular on qubit systems for which a classification of possible entanglement structures is available.

In this talk, I will summarize recent progress in the description of thermoelectric transport using gauge/gravity duality. I will first review thermoelectric transport in hydrodynamics, where momentum conservation implies infinite zero-frequency conductivities. By a change of basis of the conserved currents, a universal, finite conductivity can be extracted. It can be computed holographically. I will discuss its low-temperature scaling in terms of critical exponents characterizing time and space anisotropy and anomalous dimensions for the free energy and conserved current. When momentum is almost conserved, the zero-frequency delta functions broaden into Drude-like peaks. A holographic computation precisely identifies the redistribution of the low-frequency spectral weight between two contributions originating from the non-conservation of momentum and intrinsic dissipation respectively. It also sheds some light on how to construct effective theories of thermoelectric transport when momentum is not conserved.

Gauge/gravity duality can be used to study the transport properties of strongly interacting systems with no quasi-particles. I will give an overview of some holographic toy models of states like this, in which momentum is not conserved and thus the transport of energy and charge is non-trivial. Specifically I will discuss the thermoelectric transport properties of these toy models and their possible relations to the phenomenology of the strange metals.

Describing work in collaboration with Louise Dolan, I will discuss the scattering equations, originally introduced in 1972 by Fairlie and Roberts searching for new dual models, rediscovered by Gross and Mende in 1988, discussing the high energy behaviour of string theory, and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing remarkable formulae for tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons). We reformulate the scattering equations for N particles as a system of N -3 homogeneous polynomial equations in N - 2 complex variables, which are linear in each variable separately. The linearity of the equations enables their explicit solution in terms of the roots of a single-variable polynomial of degree (N-3)!, which can itself be explicitly constructed in terms of the Mandelstam variables formed from the momenta. The possible extension to one loop and the special case of four-dimensional space-time will also be briefly discussed.

The recent focus on entanglement entropy in holography has many motivations, ranging from the applied (e.g. AdS/CMT) to the foundational (emergence of gravity). For all of these programs It is important to find examples, where the quantities of interest can be directly calculated in strongly-coupled field theories and, moreover, the dual geometry constructed at strong coupling. In this talk I will describe joint work with Crossley and Dyer on using localization methods to obtain entanglement and (super-) Renyi entropies of the N=4 SYM theory with gauge group SU(N) in 4D at all values of the ’t Hooft coupling \lambda and number of colors N. Since obtaining quantities like entanglement and Renyi entropies involves working on singular spaces, which typically break the supersymmetry, we focus on a supersymmetric generalization, the so-called super-Renyi entropy where the supersymmetry breaking effects of the singularities are suitably compensated. I will also discuss dual gravity solutions as five-dimensional BPS black holes with hyperbolic horizon. I will conclude with a description of Wilson loops, that is the contribution to the entanglement and Renyi entropies due to adding fundamental matter to the theory.

While a lot of work has been done on understanding thermalization in 2d CFTs, several confusing aspects remain, in particular regarding integrable aspects of 2d CFTs. In this talk I will try to summarize some of these confusions and how these connect to AdS/CFT and black hole formation. There appears to be a significant difference between thermalization in low c CFTs versus thermalization in large c CFTs with gravitational duals.

Certain materials, such as the cuprate superconductors and heavy fermion materials exhibit fascinating, yet hard to explain transport properties. Holography provides a consistent framework to examine linear response and in particular transport of strongly coupled matter. I will discuss momentum dissipation in holography and show that DC transport is fixed via a Stokes flow of an "auxiliary fluid" residing on the horizon of black holes.

I will discuss knot-like defects in CS theory with complex gauge group SL(N), in the context of its connection with 3d N=2 theory (the so-called 3d-3d correspondence). I am hoping to discuss this problem from a number of different perspectives, including cluster algebras, state-integral models, 3d N=2 non-Abelian gauge theories, 5d N=2 SYM, and holographic dual, and discuss the consistency checks as well as new predictions/implications. This talk is mostly based on my recent papers with D. Gang, N. Kim and M. Romo.

Dual superconformal symmetry is a remarkable, hidden feature of N=4 SYM in 4 dimensions. Via AdS/CFT, such a symmetry corresponds to the invariance of the AdS(5) x S(5) superstring under specific combinations of bosonic and fermionic T-dualities. We show that AdS(d) x S(d) x S(d) superstrings with D(2,1;\alpha) isometry supergroup are T-self-dual if additional T-dualities along complexified S(d) directions are performed. This implies that CFTs dual to AdS(d) x S(d) x S(d) x T(10-3d) superstrings enjoy a new type of dual superconformal symmetry.

The aim of this talk is to extract the quadratic corrections to Fronsdal equations from Vasiliev's equations, and discuss their structure. The issue of locality in higher-spin theories will be addressed. Implications of this analysis in relation to Giombi-Yin checks of higher-spin/vector model duality will be discussed. Some of the previous puzzles will be resolved, while others will be elucidated.

We would like to talk about all kinds of supersymmetric and non-supersymmetric scattering amplitudes of type IIA,IIB superstring theory as well as their corrections. In fact we want to address how to get to a universal conjecture for DBI, Chern-Simons and more importantly new Wess Zumino actions with their all order $\alpha'$ corrections. Indeed we try to provide a comprehensive explanation even for D- brane-anti D-brane systems, where various new techniques will be also introduced. If time allows , we then mention several issues related to those effective actions and eventually highlight some comments about higher point functions of the Mixed closed string RR , scalar fields in different pictures of the string theory amplitude.

We study the general formulation of gauged supergravity in seven dimensions with sixteen supercharges keeping duality covariance by means of the embedding tensor formalism. We first classify all inequivalent duality orbits of consistent deformations. Secondly, we analyse the complete set of critical points in a systematic way. Interestingly, we find the first examples of stable de Sitter solutions within a theory with such a large amount of supersymmetry.

We introduce connection, torsion and curvature for Courant algebroids. We discuss in detail torsionless connections compatible with a generalized metric and the related Einstein-Hilbert actions.

We derive the localization formula for N=4 supersymmetric quiver quantum mechanics in the Higgs and Coulomb branch. The partition function (index) is exactly evaluated and it is shown that the path integral is localized at fixed points, which are given by solutions to the BRST equations combined with D-term and F-term conditions. We give various examples of the quiver theory and classifications of their fixed points. The indexes completely agree with the mathematical wall crossing formulae for the quiver moduli spaces. We also discuss a gravitational picture of the localization in the Coulomb branch.

We consider bound states of strings which arise in 6d (1,0) SCFTs that are realized in F-theory in terms of linear chains of spheres with negative self-intersections 1,2, and 4. These include the strings associated to N small E8 instantons, as well as the ones associated to M5 branes probing A and D type singularities in M-theory or D5 branes probing ADE singularities in Type IIB string theory. We find that these bound states of strings admit (0,4) supersymmetric quiver descriptions and show how one can compute their elliptic genera.

Two dimensional Warped Conformal Field Theories (WCFTs) may represent the simplest examples of field theories without Lorentz invariance that can be described holographically. As such they constitute a natural window into holography in non AdS space-times, including the near horizon geometry of generic extremal black holes. I’ll explain that WCFTs posses a type of boost symmetry. Using this insight, I’ll discuss how to couple these theories to background geometry. This geometry is not Riemannian. We call it Warped Geometry and it turns out to be a variant of a Newton-Cartan structure with additional scaling symmetries. With this formalism the equivalent of Weyl invariance in these theories will be discussed as well as examples of WCFTs. Lastly I’ll present a systematic description of the holographic duals of WCFTs. I’ll argue that the minimal setup is not Einstein gravity but an SL(2,R) x U(1) Chern-Simons Theory, which we call Lower Spin Gravity.

I will argue that the holographic correspondence can be reconstructed in the large N and strongly interacting limit as a specific generalisation of Wilsonian RG flow, which can be defined via three simple principles. I will also give an explicit example for such a reconstruction, where dual non-linear classical gravity equations in the long wavelength approximation will emerge from specific kinds of coarse graining of hydrodynamics in the field theory.

In this talk I will review some of the main results of my research in this area, which stated in 2007 in collaboration with John L. Cardy and Benjamin Doyon. I will emphasise how a special type of field we have named branch point twist field has become an essential tool for performing computations of the entanglement entropy in non-critical systems. I will show how the relationship between correlators of twist fields and entanglement entropy allows us to recover well-known results for critical systems but also to predict new results for theories with a finite correlation length. Time permitting, I will mention some more recent results extending our understanding to non-unitary critical and non-critical systems.