In recent years, due to the method of supersymmetric localization, many exact results have been achieved in the study of supersymmetric gauge theories on compact spaces of various dimension and topology, leading to the discovery of surpraising structures. An important example is provided by the correspondence introduced by Alday, Gaiotto and Tachikawa, relating the partition functions of a large class of supersymmetric gauge theories on S4 and S2 to correlators in Liouville CFT. In this talk, I will explain how this picture can be lifted to higher dimensional gauge theories via the correspondence of partition functions on S5, S4xS1, S3 and S2xS1 to correlators in theories whose underlying symmetry is given by a quantum deformation of the Virasoro algebra. In particular, I will discuss how 3-point functions can be derived by the bootstrap approach and used to define this novel class of q-deformed CFTs. I will also discuss some aspects related to integrable structures in these models, such as reflection coefficients, as well as possible generalisation.

It will be shown how supersymmetry and superalgebras arise in lattice models both in one and higher dimensions. Examples using correlated fermions and bosons will be primarily used. Some of the examples will have associated superalgebras and others will have supersymmetry more commonly used in field theory. Some aspects of continuum limits will also be touched upon.

I will show that the non-stringy holographic description of hadrons, namely the one based on fields that reside on the bulk and flavor branes, fails to reproduce the hadronic spectra. I will briefly review the stringy duals of Wilson loops and determine the sufficient conditions of confining backgrounds. I will show that the latter are also the requirements for holographic stringy mesons to admit a Regge-like spectrum. I will determine a map between holographic stringy mesons and strings with massive endpoints in 4d flat space-time. Models of classical rotating strings with massive endpoints will be written down. I will discuss the (still unsolved) quantization of such systems. Fits of the models to mesonic data will be presented. A universal model that describes mesons of u,d,s,c quarks will be developed. Mesons with b quarks will also be fitted. I will describe the construction of stringy baryons in holography. I will show that the best description of baryons is in terms of a single string with a quark and a di-quark on its ends. I will present fits of such a model to the baryonic data. Finally I will describe the stringy decays of holographic hadrons.

In this talk we will try to motivate why the study of the Regge limit in a correlation function of a conformal field theory is interesting. We will review the results obtained in the past few years and present some on going work in phi^3 theory using skeleton expansion.

I will show how to construct the familiar heterotic NS5 brane as a topological soliton in a supercritical version of heterotic string theory. Closed string tachyon condensation removes the extra dimensions, leaving the NS5 in 10d, in a process highly reminiscent of the K-theoretical description of type II D-branes, but linking non-trivial gauge bundles and geometry. The construction requires a modification of the anomalous Bianchi identity for H_3 in supercritical heterotic string theory. I will give various proofs for the existence of this modification.

We study geometric aspects and holomorphic properties of N = (0, 2) Heterotic Sigma Models over symmetric spaces, and derive a number of exact relations for the β functions in terms of the analogy to the NSVZ β function in four-dimensional Yang-Mills. We will show that in spite of the heterotic nature, the isometries are preserved at the quantum level. We will also discuss the N = (0, 2) supercurrent multiplet (the so-called hypercurrent) and its anomalies, as well as the “Konishi anomaly.” This gives us another method for finding two-loop β functions. The talk is based on joint work with J. Chen, M. Shifman and A. Vainshtein.

We systematically analyze a broad class of dual heterotic and F-theory models that give four-dimensional supergravity theories, and compare the geometric constraints on the two sides of the duality. In this talk I will show that F-theory gives new insight into the conditions under which heterotic vector bundles can be constructed. We show that in many cases the F-theory geometry imposes a constraint on the extent to which the gauge group can be enhanced, corresponding to limits on the way in which the heterotic bundle can decompose. We explicitly construct all dual F-theory/heterotic pairs in the class under consideration where the common twofold base surface is toric, and give both toric and non-toric examples of the general results. Finally, we provide evidence for important new aspects of G-flux in four-dimensional compactifications.

Motivated by the search for new backgrounds with solvable string theories, the talk classifies the D-brane geometries leading to integrable geodesics. This analysis gives severe restrictions on the potential candidates for integrable string theories. It is demonstrated that the Hamilton-Jacobi equation for massless geodesics can only separate in elliptic or spherical coordinates, all known integrable backgrounds are covered by this separation, and new examples are constructed. The Killing and Killing-Yano tensors associated with such separation and their transformations under string dualities are also discussed.

I will discuss the Maldacena-Lewkowycz derivation of the Ryu-Takayangi prescrition for holographic entanglement entropy, and extend it to a more general class of theories of gravity in the bulk. This analysis results in a new euclidean entropy functional, that generalises Wald's entropy. In lorentzian signature, this functional is a natural extension of Wald's black hole entropy to time dependent situations.

We give a survey on recent results on the physics of black holes in four-dimensional anti-de Sitter space. In particular, we present a general recipe for constructing rotating black holes in N=2 matter-coupled gauged supergravity. Several physical aspects of these solutions are discussed, like their thermodynamics and dual hydrodynamical interpretation. It turns out that there exist new exotic black holes whose horizon is a noncompact manifold with finite volume. Finally, we show how to construct multi-centered black hole solutions in anti-de Sitter space.

I will consider supersymmetric field theories on compact manifolds M and obtain constraints on the dependence of their partition functions Z_M on the geometry of M. For N=1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. I will show how to describe the theory in terms of twisted variables that make easy to analyze the dependence of Z_M on the parameters entering the Lagrangian. I will also show that Z_M is "almost" topological: Z_M is independent of the Hermitian metric and depends holomorphically on the complex structure moduli.

We determine the bosonic part of the superstring sigma model Lagrangian on eta-deformed AdS_5 x S^5, and use it to compute the perturbative world-sheet scattering matrix of bosonic particles of the model. We then compare it with the large string tension limit of the q-deformed S-matrix and find exact agreement. We comment on the structure of the RR sector of the type IIB SUGRA background.

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with N=2 superconformal symmetry.

I will discuss the question of whether four dimensional scale invariant unitary quantum field theories are actually conformally invariant. I will present a complete analysis of possible scale anomalies in correlation functions of the trace of the stress-energy tensor in such theories and discuss recent attempts to show that scale invariance implies conformal invariance.

A brief review will recapitulate some of the essential features of T-duality symmetry in string theory. A proposal will be presented to show how a scattering amplitude of some string excitations could be related to another amplitude by T-duality transformation for a class of stringy states. This is considered in a simple scenario. Some illustrative examples will be presented as application of the proposal.