I will review recent progress in understanding the 3d N=2 superconformal field theories which describe the low energy dynamics of M2-branes probing toric CY_4 cones in M-theory, based on a KK reduction of M-theory to type IIA string theory. A key role in field theory is played by 't Hooft monopole operators, disorder operators which encode the geometric details of the reduction.

After briefly reviewing the notion of geometries characterised by SU(3) structures, I will discuss a particular class known (in String Theory) as non-Kahler. I will explain how this arises both in Type II and Heterotic Supergravities. Two explicit constructions of these geometries will be discussed. One is a one-parameter solution corresponding to fivebranes wrapped on the two-sphere of the resolved conifold, that can be thought of as a non-Kahler analog of the conifold. The other is a general construction of one-parameter non-Kahler deformations of Calabi-Yau manifolds with a U(1) isometry, where the non-Abelian Yang-Mills field of the Heterotic is non-trivial. The presentation will be loosely based on the two papers: arXiv:0906.0591 and arXiv:1010.4031.

The Y-system, originating from integrability, is a tool describing the exact spectrum in 4D N=4 Super-Yang-Mills theory in a planar limit. We describe the construction and discuss predictions and tests for weak, strong and intermediate couplings.

Quantized complex curves play a central role in both topological string theory and Chern-Simons theory with complexified gauge group. In both cases these quantum curves yield operators that annihilate partition functions. However, in both cases, the actual quantization of these curves has only been understood indirectly (via matrix models on one hand, via recursion relations for the Jones polynomial on the other). I will discuss an intrinsic, geometric quantization scheme that should produce such quantum Riemann surfaces directly. N.B. Such quantum curves also show up in conformal field theory, as the operators that annihilate correlators with degenerate insertions.

Though defects in a general sense are ubiquitous and much-studied within statistical mechanics models it is only recently that they have been considered within integrable field theory. At first sight, defects could be considered disastrous since the property of integrability might be lost. However, it turns out that not only is it possible to have 'integrable defects' but they have a range of interesting properties and cast some new light on traditional features. Several examples will be described, together with their properties in classical and quantum versions of the models.

Condensed matter systems at quantum critical points can be described by strongly coupled field theories exhibiting anisotropic scale invariance. Lifshitz geometries been proposed to be holographic duals to these theories. I will discuss top down constructions of Lifshitz geometries in Type IIB and D=11 dimensional supergravities.

I will give an overview of recent developments in F-theory GUT model building. First I will discuss the basic ideas of F-theory model building, exemplifying this by the construction of supersymmetric SU(5) GUTs. Then I will outline how global string theoretic consistency requirements impact the SUSY phenomenology. The main focus of this talk will be on the phenomenological implications of these models.

I will review the matrix model which calculates the partition function of ABJM theory on S3 as well as the expectation value of Wilson loop operators. I will then explain how this matrix model is solved and present the results for these quantities at all values of the couplings. At strong coupling these calculations reproduce the results of supergravity on Ads4 x CP3 and in particular the N to the 3/2 scaling of the free energy of the theory.

We discuss relations between two a priori unrelated classes of objects: (i) W-algebras, which are certain symmetry algebras of two-dimensional conformal field theories, and (ii) four-dimensional N=2 gauge theories in the presence of surface operators (certain two-dimensional defects). In particular, we relate the classifications of W-algebras and surface operators.

Vertex operator algebra (VOA) is the algebraic setup formalising conformal field theory. It develops in a mathematically complete way the idea of constructing quantum field theory using the algebra of symmetry currents and their modules. On the other hand, conformal loop ensembles (CLE) are measures on random loop configurations that are known, in certain cases, to describe the continuous limit of statistical models at critical points. There is a one-parameter family of such measures, supposed to correspond to all central charges between 0 and 1. These two constructions enjoy complete mathematical rigour, and give the opportunity to understand with more precision the relation between the statistical interpretation of QFT, and its algebraic description. I will describe some of my recent works in this direction: I will explain how to construct the Virasoro VOA (the stress-energy tensor and its descendents) in terms of random objects in CLE. No prior knowledge of either VOA or CLE is needed as I will review both subjects.

Effective quiver gauge theories arising from a stack of D3-branes on certain Calabi-Yau singularities can be studied from many points of view. In this talk we adopt a first principle approach via open topological string theory, which means that we construct the natural A- infinity-structure of open string amplitudes in the associated D-brane category. This precisely reproduces the results of the method of brane tilings, without having to resort to any effective field theory computations. In particular, one obtains a general and simple formula for effective superpotentials.

In 2009 there has been progress in understanding and classifying the set of four dimensional field theories with N=2 SUSY. These models arise as M5 branes wrapped over a Riemann surface. We review this construction and describe a five dimensional point of view, using (p,q)-webs of 5branes in Type IIB string theory. This point of view makes many properties of the theories explicit. We will also touch on the AGT correspondence, that associates a 2-dimensional CFT, similar to the Liouville CFT, to the protected sector of four dimensional N=2 models.

We discuss the effect of vacuum polarization on the propagation of photons in curved spacetime in QED. A compact formula is presented for the full frequency dependence of the refractive index for any background in terms of the Van Vleck-Morette matrix for its Penrose limit. This shows explicitly how the superluminal propagation found in the low-energy effective action is reconciled with causality. The geometry of null geodesic congruences is found to imply a novel analytic structure for the refractive index and Green functions of QED in curved spacetime, which preserves their causal nature but violates familiar axioms of S-matrix theory and dispersion relations. The Kramers-Kronig dispersion relation and the optical theorem for QFT in curved spacetime are discussed critically. The significance of the Penrose limit for black hole spacetimes and their relation to homogeneous plane waves is explained and unexpected features of light propagation in a number of spacetimes are described.

We will discuss how by integrating out a global U(1)B gauge field, the U(n)xU(n) ABJM models at level k are equivalent to SU(n)xSU(n) N=6 Chern-Simons theories with a Zk identification on the fields and a modified flux quantisation condition, but only when n and k are relatively prime. As a consequence, the ABJM model for two M2-branes in R8 can be identified with the N=8 SU(2)xSU(2) theory at k=1. We will also argue that the original N=8 SO(4)-theory of Bagger and Lambert, without modified flux quantisation, is equivalent to the U(2)xU(2) ABJM model at k=2 and hence describes the IR fixed point of a maximally supersymmetric three-dimensional O(4) gauge theory obtained in M-theory by an R8/Z2 orbifold without torsion.