In this talk, I will discuss how the two geometric notions "fan" and "monoid" can be very fruitful for the understanding of the monopole formula for 3d N=4 gauge theories. After a brief reminder of the monopole formula, I will introduce the matter fan and reorganise the monopole formula accordingly. I then discuss the resulting benefits such as: (1) Explicit expressions for the Hilbert series built from well-studied constituents. (2) Proof that the order of the pole at t=1 and t → ∞ equals the complex or quaternionic dimension of the Coulomb branch. (3) Identification of a sufficient set of chiral ring generators.

I am going to talk about how to derive an exact partition function of supersymmetric field theories on a supersymmetry-preserving quotient of Euclidean AdS_3 by localization. This is one of first examples about localization on non-compact spaces. I will describe the situation by comparing it to a situation in compact spaces. Then, I will explain how to perform a localization procedure on the manifold, and finally, I will discuss the features of the partition function we derived.

I review the theory of non-linear realisations and Kac-Moody algebras, I explain how to construct the non-linear realisation based on the Kac-Moody algebra $E_{11}$ and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a spacetime with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of spacetime, lead to precisely the equations of motion of eleven dimensional supergravity theory. By taking different group decompositions of $E_{11}$ we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the non-linear realisation is a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the $E_{11}$ conjecture given many years ago.

Weyl semi-metals are an exciting new class of 3D materials with exotic transport properties. They are characterised by point like singularities in the Brillouin zone at which conduction and valence bands touch. Around these points the electronic quasi-particle excitations can be described by either left- or right-handed Weyl spinors. This makes high energy physics to be the low energy electronics of these materials. Holographic models might be useful in two ways: first some Weyl semi-metals might be strongly interacting and second holography is our best tool to understand and discover new exotic transport properties related to the chiral nature of the charge carriers, especially to chiral anomalies. I will review the construction of a holographic model supporting a topological quantum phase transition between a topological Weyl semi-metal phase and a trivial phase and use it to predict the existence of a new dissipation less form of viscosity (Hall viscosity) in the quantum critical region.

In a generic CFT the spectrum of operators carrying a large U(1) charge can be analyzed semiclassically in an expansion in inverse powers of the charge. The key is the operator state correspondence by which such operators are associated with a finite density superfluid phase for the theory quantized on the cylinder. The dynamics is dominated by the corresponding Goldstone hydrodynamic mode and the derivative expansion coincides with the inverse charge expansion. I will illustrate this situation by first considering simple quantum mechanical analogues and then will systematize the approach by employing the coset construction for non-linearly realized space-time symmetries. Focussing on CFT3 I will illustrate that the three point function coefficients turn out to satisfy universal scaling laws and correlations as the charge and spin are varied.

We numerically construct asymptotically global AdS_5 x S^5 black holes that are localised on the S^5. These are solutions to type IIB supergravity with S^8 horizon topology that dominate the microcanonical ensemble at small energies. At higher energies, there is a first-order phase transition to AdS_5-Schwarzschild x S^5. By the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, this transition is dual to spontaneously breaking the SO(6) R-symmetry of N=4 supersymmetric Yang-Mills down to SO(5). We extrapolate the location of this phase transition and compute the expectation value of a scalar operator in the low energy phase. In addition, we discuss the construction of localised black holes in type IIA, which are dual (via T-duality) to the low temperature phase of thermal 1+1 dimensional supersymmetric Yang-Mills theory on a circle.

We consider planar hairy black holes in five dimensions with a real scalar field in the Breitenlohner-Freedman window. We show that is possible to derive a universal formula for the speed of sound for any theory. As an example we find all the planar black holes of the single scalar field consistent truncation of type IIB that preserves the SO(3)×SO(3) R-symmetry group of the gauge theory. We find the speed of sound for different values of the induced vacuum expectation value when a double trace deformation is induced in the gauge theory. For any theory, we show that the speed of sound is not bounded by any special value.

The planar scattering amplitudes of 4d N = 4 super Yang-Mills theory and 3d N=6 Chern-Simons theory exhibit a remarkable property known as dual conformal symmetry. Motivated by this, we investigate the consequences of dual conformal symmetry in six dimensions. We find that 6d dual conformal symmetry fixes the structure of the one-loop 4-point amplitude and suggests a Lagrangian with more than two derivatives. Moreover, we obtain a similar structure by generalizing the Alday-Maldacena solution for a string in Anti-de Sitter space to a 2-brane ending on a pillow-shaped surface in the boundary whose seams correspond to a null-polygon, which suggests an all-loop formula for the 4-point amplitude similar to that of N=4 super Yang-Mills theory.

I’ll review old and new aspects of deformed instanton equations derived from D-branes.

The so-called 6d (2,0) conformal field theory in six dimensions, labeled by an ADE Lie algebra, has become of great interest in recent years. Most notably, it gave new insights into lower dimensional supersymmetric field theories, for instance in four dimensions, after compactification. In this talk, I will talk about a deformation of this CFT, the six-dimensional (2,0) little string theory: its origin lies in type IIB string theory, compactified on an ADE singularity. We further compactify the 6d little string on a Riemann surface with punctures. The resulting defects are D-branes that wrap the 2-cycles of the singularity. This construction has many applications, and I will focus on one: I will provide the little string origin of the classification of surface defects of the 6d (2,0) CFT, for ADE Lie algebras. Furthermore, I will give the physical realization of the so-called Bala-Carter labels that classify nilpotent orbits of these Lie algebras.

We numerically construct asymptotically global AdS_5xS^5 black holes that are localised on the S^5. These are solutions to type IIB supergravity with S^8horizon topology that dominate the microcanonical ensemble at small energies. At higher energies, there is a first-order phase transition to AdS_5-Schwarzschild x S^5. By the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, this transition is dual to spontaneously breaking the SO(6) R-symmetry of N=4 supersymmetric Yang-Mills down to SO(5). We extrapolate the location of this phase transition and compute the expectation value of a scalar operator in the low energy phase. In addition, we discuss the construction of localised black holes in type IIA, which are dual (via T-duality) to the low temperature phase of thermal 1+1 dimensional supersymmetric Yang-Mills theory on a circle.

The N-extended massless superparticle in a 4D Minkowski background has a worldline time-reversal invariance that becomes CPT in the quantum theory. This symmetry is anomalous for odd N, which explains why there is no CPT-self-conjugate massless supermultiplet for N=1. In the supertwistor formulation of superparticle mechanics, the anomaly is a close cousin of the parity anomaly of 3D gauge theories. CPT is not anomalous if N is even but if 2N is odd there is a Kramers degeneracy, which explains why the N=2 hypermultiplet is a doubled version of the CPT-self-conjugate supermultiplet of helicities (-1/2,0,0,1/2).

I'll give an introduction to N=2, D=4 SCFTs of "Class-S" and a survey of our programme for classifying such theories. I will put special emphasis on the theories arising from the Exceptional (2,0) SCFTs in D=6.

I will discuss recent advances in addressing the heterotic moduli problem in six and seven dimensions. To map out the reduction to a lower dimensional effective theory, and in a quest to complete our understanding of the generically torsional geometries that appear, it is important to understand the moduli of these geometries. I will briefly review the infinitesimal moduli space in six dimensions, and comment on some recent advances in understanding its geometric properties. I will then explain how a similar story is emerging in seven dimensional compactifications on more exotic manifolds with G2-structure, giving hope that similar tools can be applied to understand moduli in other areas of string theory, with possible applications to phenomenology and AdS/CFT. If time, I will comment on some recent advances is studying finite deformations and obstructions of the moduli space.

Abstract: We are using holography to study quantum phase transitions in a 2+1 D strongly coupled non conformal gauge theory at finite chemical potential and magnetic field B. Our gravity dual is 4D N = 2 Fayet-Iliopoulos gauged supergravity. We found a line of quantum critical points at B= Bc between extremal dyonic black branes and extremal magnetically charged "thermal gas" solutions. We will introduce the two types of solutions, their thermodynamics and their phase diagram. Then we will discuss the physics of the critical point and we will point out qualitative similarities between our field theory and Nambu- Jona- Lasinio model in 2+1 D under the presence of an external magnetic field. We will conclude with some open questions both on the gravity and field theory side and we will discuss future directions.

In this talk I will describe recent progress towards understanding integrable deformations of superstrings in AdS backgrounds. One well-known example of such a deformation is the TsT transformation, which involves a T-duality, a shift introducing the deformation parameter, and a reverse T-duality. After introducing this deformation, I will give two alternative constructions: (i) as a particular non-abelian duality, and (ii) as a particular Yang-Baxter deformation. In the second part of the talk I will outline how this extends to a general relationship between non-abelian duality and Yang-Baxter deformations, and discuss the consequences for the Weyl invariance of the deformed models.

We consider the four-point correlator of the stress tensor multiplet in N = 4 SYM in the limit of large c=N^2, but finite \lambda=g^2 N. For finite values of \lambda single-trace intermediate operators arise at order 1/c and this leads to specific poles in the Mellin representation of the correlator. The sign of the residue at these poles is fixed by unitarity. We consider solutions consistent with crossing symmetry and this pole structure. We show that in a certain regime all solutions result in a negative contribution to the anomalous dimension of twist four operators. This positivity condition can also be proven by assuming the correct Regge behaviour for the Mellin amplitude. The positivity constraints arising from CFT for the Mellin amplitude take a very similar form to the causality constraint for the forward limit of the S-matrix.

At low temperature and high quark chemical potential, QCD exhibits color superconducting phases, namely phases in which the color gauge symmetry is spontaneously broken. I will describe work in progress aimed at constructing string duals of gauge theories that exhibit these types of phases.

I will discuss the coupling of non-relativistic field theories to curved spacetime, and develop a framework for analyzing the possible structure of non-relativistic (Lifshitz) scale anomalies using a cohomological formulation of the Wess-Zumino consistency condition. I will compare between cases with or without Galilean boost symmetry, and between cases with or without an equal time foliation of spacetime. In 2+1 dimensions with a dynamical critical exponent of z=2, the absence of a foliation structure allows for an A-type anomaly in the Galilean case, but also introduces the possibility of an infinite set of B-type anomalies. I will also derive Ward identities for flat space correlation functions in Lifshitz field theories, and develop a method for calculating Lifshitz anomaly coefficients from these correlation functions using split dimensional regularization.

In this talk, I will discuss transport phenomena in two classes of theories with holographic dual. In the first part, I will discuss systems where U(1) current is non-conserved due to anomaly and illustrate how one can show that the anomalous conductivities are non-renormalised in a large class of holographic RG flow. The holographic RG flow we considered is generated by arbitrary dilaton potentials and arbitrary higher derivative terms that do not break global symmetries, incorporating coupling constant corrections to the boundary theory in an expansions around infinite coupling. In the second part, I will focus on systems where translational symmetry is broken by slowly varying scalar fields and show that the bound shear viscosity/entropy density is violate. I will also discuss how to understand the violation in the language of forced fluid dynamics.