The D3/D7 system provides a simple holographic description of a strongly coupled gauge theory with quarks. By introducing a range of deformations of the AdS5 space a variety of familair phenomena in gauge theory can be investigated quantitatively. These include a running gauge coupling, temperature, chemical potential, and magnetic fields leading to confinement and chiral symmetry breaking, as well as bound mesons and their melting. I'll illustrate these phenomena with examples and derive the phase diagram of a chiral symmetry breaking theory. Finally I shall discuss how hadronization can be modelled in such set ups and propose that rho meson emission should be described by a simple EM radiation problem.

The existence of a modular invariant genus two partition function implies infinitely many relations among the structure constants of a chiral self-dual conformal field theory. All of these relations can be shown to be a consequence of the associativity of the OPE, as well as the modular covariance properties of the torus one-point functions. Using these techniques we prove that for the proposed extremal conformal field theories at c=24k a consistent genus two vacuum amplitude exists for all k, but that this does not actually check the consistency of these theories beyond what is already testable at genus one.

I will review some recent attempts at providing a microscopic description for extremal black holes. First, I will explain a constituent model for extremal non-rotating non-BPS asymptoticaly flat black holes. Second, I will summarise the main claims in the so called extremal BH/CFT correspondence, pointing out how a chiral CFT can emerge as a limit of a non-chiral CFT. Finally, I will use R-charge AdS black holes to derive the existence of emergent IR CFTs, similar to the ones that have been argued to capture some interesting quantum criticality phenomena in some strongly coupled condensed matter systems.

I will introduce a family of supersymmetric Chern-Simons-matter theories in d=2+1 dimensions, labelled by a positive integer n, and argue using a number of different dual descriptions that these describe the worldvolume dynamics of M2-branes at a corresponding family of four-fold hypersurface singularities (the n=1 theory being the ABJM theory). The IR properties of these theories and their deformations are also studied.

After reviewing how small BPS black holes with two charges can be described in perturbative Heterotic strings, we compute the cross section for their pair production at tree level and analyze angular and energy distributions. We consider both toroidal compactifications and FHSV orbifold model, and comment on scenari with large extra dimensions. Finally we briefly discuss how to extend our results to non-BPS, non-extremal and rotating BH's.

Following the work of Aharony, Bergman, Jafferis and Maldacena, a great effort has been made to understand the 3d conformal gauge theories describing M2-branes at conical singularities. Special attention has been directed to N=2 supersymmetric AdS4/CFT3 pairs arising from M2-branes at toric CY 4-folds, where powerful algebraic techniques are available. After reviewing some results concerning brane tilings, the relation between the toric CY3 and CY4 moduli spaces of 4d and 3d toric quiver gauge theories, and M-theory/type IIA duality, I will explain how flavours naturally arise in this framework, and how their matter content and superpotential couplings to bifundamental fields are encoded in geometric data. Quantum effects are crucial in computing the moduli spaces of such flavoured quiver gauge theories. Any quiver Chern-Simons (CS) theory of the kind considered in the AdS4/CFT3 literature so far can be obtained by coupling a 3d gauge theory with the same quiver diagram but no CS terms to chiral flavours, and integrating them out via a real mass term.

The Riemann zeta-function, which encodes information about the primes, is the subject of one of the most important problems in mathematics: the Riemann Hypothesis. In the past few years connections have emerged with random matrix theory/matrix models. I shall review these connections.

In this talk I will first review the recent progress made towards finding instabilities of rotating vacuum black holes in higher dimensions. I will show that singly spinning Myers-Perry black holes exhibit an infinite sequence of zero modes that should connect them to other black hole phases such as black rings and black saturns. In the second part of my talk I will describe the gravitational instabilities of Myers-Perry black holes with all the angular momenta equal. The onset of instability is associated with the appearance of time-independent perturbations which generically break all but one of the rotational symmetries. In nine spacetime dimensions, this is interpreted as evidence for the existence of a new 70-parameter family of black hole solutions with only a single rotational symmetry.

The gauge/gravity duality provides a new set of tools for exploring the physics near 2+1 dimensional quantum critical points. In many highly quantum states of matter, soliton configurations yield clues about the physics of both large and small length scales. I will review a holographic model for a relativistic superfluid. I will then discuss dark soliton and vortex solutions supported by this superfluid.

I will report on some recent progress on defining and computing finite size effects in the entropy of black holes with highly nontrivial agreements between thermodynamics and statistical mechanics. I will describe a number of puzzles and their resolutions along with some exact computations, and then briefly discuss the role of mock modular forms and Borcherds-Kac-Moody superalgebras in this context.

Non-zero mass is compatible with unbroken gauge invariance in three spacetime dimensions (3D). A systematic procedure for the construction of massive gauge theories will be illustrated by new massive gravity, which propagates unitarily two massive spin 2 modes in a Minkowski vacuum. The supergravity extension of this model will be presented along with new results on supersymmetric AdS vacua. The extension to a new N=8 3D supergravity will be discussed, as will be the AdS3CFT2 correspondence and possible connections to string/M-theory.

In this talk, I will discuss work in progress with Bertrand Eynard, in which we derive the BKMP remodelling the B-model conjecture, in the large radius limit. This is the claim that Gromov-Witten invariants of any toric Calabi-Yau 3-fold coincide with the spectral invariants of the mirror curve. Our method consists in explicitly constructing a matrix model which reproduces the topological string partition function obtained via the vertex formalism, and then demonstrating that the spectral curve of this matrix model coincides with the mirror geometry.

This is an introduction to Gerbes aimed at physicists. The approach will be to introduce Cech cohmology and its relation to gauge theories, monopoles and Wilson loops and then give the extension to extend these ideas to the relation between higher order Cech cohomolgy, gerbes and strings.

Leading singularities are invariants of multi-loop scattering amplitudes (the full amplitude at tree level) obtained by generalized unitarity. In this talk we show how to construct multi-loop leading singularities on twistor space for maximally super-symmetric Yang-Mills (and gravity). Building on the tree-level twistor-string representation of scattering amplitudes, they can be represented as integrals over a moduli space of nodal curves in twistor space. We discuss how this might arise from a conjectural twistor-string path integral representation for the full loop amplitude. We also show how the construction relates to the Grassmannian representation of leading singularities conjectured by Arkani-Hamed et. al.. This shows firstly that all leading singularities can be represented in the Grassmannian, and secondly that the complexity is limited, in particular we conjecture that there are no new leading singularities at beyond 3p loops for NpMHV amplitudes.

Tremendous progress in computing perturbative scattering amplitudes in N=4 supersymmetric gauge theory has been made over the past few years. Importantly the planar amplitudes appear to display a dual conformal invariance next to the usual conformal symmetry. Altogether the symmetry enlarges to a Yangian algebra known from the context of integrable models. This infinite-dimensional symmetry might have the power to completely fix the S-matrix by algebraic means. In this talk we review the above developments. We then discuss conformal symmetry for tree and loop scattering amplitudes. It turns out that the free conformal symmetry generators are anomalous which calls for certain deformations to make the symmetries exact. These relate amplitude with different numbers of legs, and thus they contribute substantially to a complete algebraic determination.

In this talk we will review the appearance of an infinite dimensional symmetry algebra of tree level scattering amplitudes of N=4 super Yang-Mills theory, known as the Yangian of psu(2,2 4). It arises through the commutation of standard superconformal and the recently discovered dual superconformal symmetries of scattering amplitudes. At the loop level the conformal and dual conformal symmetry is destroyed by infrared divergencies. The present challenge is whether it may be repaired. We point out a novel IR regularization based on a suitable Higgsing of the theory, which is natural from the dual string perspective. It turns out that in this regularization an extended conformal symmetry of the loop amplitudes arises, which remains free of anomalies.

We study the sub-structure of the heterotic Kahler moduli space due to the presence of non-Abelian internal gauge fields from the perspective of the four-dimensional effective theory. Internal gauge fields can be supersymmetric in some regions of the Kahler moduli space but break supersymmetry in others. In the context of the four-dimensional theory, we investigate what happens when the Kahler moduli are changed from the supersymmetric to the non-supersymmetric region. Our results provide a low-energy description of supersymmetry breaking by internal gauge fields as well as a physical picture for the mathematical notion of bundle stability. Specifically, we find that at the transition between the two regions an additional anomalous U(1) symmetry appears under which some of the states in the low-energy theory acquire charges. We compute the associated D-term contribution to the four-dimensional potential which contains a Kahler-moduli dependent Fayet-Iliopoulos term and contributions from the charged states. We show that this D-term correctly reproduces the expected physics. Several mathematical conclusions concerning vector bundle stability are drawn from our arguments. We also discuss possible physical applications of our results to heterotic model building and moduli stabilization.

Local string models are those where Standard Model degrees of freedom arise on a small region inside a large bulk volume. I study threshold effects on gauge coupling running for such models. The Kaplunovsky-Louis formula for locally supersymmetric gauge theories predicts the unification scale should be the bulk winding mode scale, parametrically large than the string scale where divergences are naively cut off. Analysis of explicit string models on orbifold/orientifold geometries confirms this. The winding mode scale arises from the presence of tadpoles uncancelled in the local model. I briefly discuss phenomenological applications to supersymmetry breaking and gauge coupling unification.

I will present two recent results in the are of Gauge-Strings dualities, applied to field theories with a possible interest in phenomenology.