Calabi-Yau manifolds without flux are perhaps the best-known supergravity backgrounds that leave some supersymmetry unbroken. The supersymmetry conditions on such spaces can be rephrased as the existence and integrability of a particular geometric structure. When fluxes are allowed, the conditions are more complicated and the analogue of the geometric structure is not well understood. In this talk, I will define the analogue of Calabi-Yau geometry for generic D=4, N=2 backgrounds with flux in both type II and eleven-dimensional supergravity. The geometry is characterised by a pair of G-structures in 'exceptional generalised geometry' that interpolate between complex, symplectic and hyper-Kahler geometry. Supersymmetry is then equivalent to integrability of the structures, which appears as moment maps for diffeomorphisms and gauge transformations. Similar structures also appear in D=5 and D=6 backgrounds with eight supercharges. As a simple application, I will discuss the case of AdS5 backgrounds in type IIB, where deformations of these geometric structures give exactly marginal deformations of the dual field theories.

Symmetries are crucial in physics. Here I will develop the concept of symplectic symmetry for Einstein gravity, which is an extension of the concept of asymptotic symmetries. I will then show that the throat of the extremal Kerr black hole admits symplectic symmetries, which leads to an infinite number of conserved charges which form a Virasoro algebra in 4 dimensions and a novel generalized Virasoro algebra in higher dimensions. I will build explicitly the geometries that carry these charges, the so-called "descendants" of the extremal black hole, in the near-horizon region.

Student Triangles are a chance for PhD students to give talks to their fellow PhD students from the London Universities, without the pressure of presenting in front of academics. For this first student triangle of the year we will be holding a gong show with a series of short whiteboard talks. PhD students can sign up for a gong show slot on http://doodle.com/poll/ehgri6kz6f6a8cux

We study the web of correspondences linking the exceptional Lie algebras E8,7,6 and the sporadic simple groups Monster, Baby and the largest Fischer group. We will survey some old observations from the perspective of Moonshine and representation theory and present some new ones from that of congruence groups and enumerative geometry. Based on joint work with John McKay.

5d theories can be at fixed points, in many cases exhibiting rather amusing properties. In this talk we will study several properties of these fixed point theories. In particular, we will describe how to put them supersymmetrically in curved spaces. As a consequence, we will discuss when a deformation to a Yang-Mills theory is possible. The latter typically exhibit a smaller global symmetry that that of the fixed point theory, which, from the YM point of view, appears as an enhanced symmetry. In this enhancement instantons play a pivotal role, and we will discuss several aspects of these.

Counting operators in supersymmetric field theories is very interesting, as the corresponding generating functions encode properties of the field theory. In this lecture we will mostly concentrate on 4d SUSY field theories. After reviewing some basic notions about SUSY field theories leading to the notion of chiral ring, we will introduce a generating function enumerating operators in the ring. This function is the so-called Hilbert series. In the case of SCFTs with a gravity dual, we will show how this function recovers geometric properties of the dual gravity background. The uses of the Hilbert series go beyond this, since, as we will see, being a counting of operators, it is of relevance in the computation of instanton partition functions for pure gauge theories. In the second part of the lecture we will briefly touch upon these issues.

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.

Lecture 3: An example of mixed open/closed amplitudes

Lecture 2: Derivation of the Virasoro-Shapiro amplitude; 4 graviton amplitude in the superstring (no derivation, but discussion of the low and high energy limits)

Lecture 1: Veneziano amplitude: derivation, low and high energy limits

In recent years we have witnessed a revival of the conformal bootstrap approach to CFTs. I will discuss the application of these ideas to six-dimensional conformal field theories with (2,0) supersymmetry, focusing on the universal four-point function of stress tensor multiplets. For these theories the program splits into an analytic and a numerical component. The analytic component yields exact results but in a protected subsector. The numerical component can be used to derive bounds on OPE coefficients and scaling dimensions from the constraints of crossing symmetry and unitarity. The principal numerical result is strong evidence that the A1 theory realizes the minimal allowed central charge (c=25) for any interacting (2,0) theory. This implies that the full stress tensor four-point function of the A1 theory is the unique unitary solution to the crossing symmetry equation at c=25. For this theory, we can estimate the scaling dimensions of the lightest unprotected operators appearing in the stress tensor operator product expansion. We also find rigorous upper bounds for dimensions and OPE coefficients for a general interacting (2,0) theory of central charge c. For large c, our bounds appear to be saturated by the holographic predictions obtained from eleven-dimensional supergravity.

Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant for routing problems, inference and data mining. In real growing networks, topological, structural and geometrical properties emerge spontaneously from their dynamical rules. Here we show that a single two parameter model of emergent network geometry, constructed by gluing triangles, can generate complex network geometries with non-trivial distribution of curvatures, combining exponential growth and small-world properties with finite spectral dimensionality. In one limit, the non-equilibrium dynamical rules of these networks can generate scale-free networks with clustering and communities, in another limit 2 dimensional manifolds with non-trivial modularity. When manifolds of arbitrary dimension are constructed, and energies are assigned to their nodes these networks can be mapped to quantum network states and they follows quantum statistics despite they do not obey equilibrium statistical mechanics.

We start by providing complete insights about the standard ways of finding couplings in effective field theory , and present most of the supersymmetric and non-supersymmetric scattering amplitudes of both type IIA,IIB superstring theory as well as their corrections. We would also like to address the way of obtaining a universal conjecture for DBI, Chern-Simons and new Wess Zumino actions with their corrections. We then move on to provide a comprehensive explanation even for D- brane-anti D-brane systems, where various new techniques will be also illustrated. If time allows , we then mention several issues related to those effective actions and eventually highlight some open problems.

Remarkable formulae for the tree-level S-matrix of gauge and gravity theories were recently discovered by Cachazo-He-Yuan, based on a localisation procedure in the moduli space of Riemann surfaces. These formulae are now known to arise from new string-like chiral models. These naturally produces loop-level amplitudes, also localised in the moduli space of higher genus curves. However, the resolution of the localisation equations is immensely more difficult than at tree-level, and it remained as an open question as to what these formulae computed, and more generally if these string models made sense at the quantum level. In this talk I'll describe a resolution of the localisation equations at one-loop in a particular kinematical regime, and show agreement with known amplitudes. Then I'll describe a much more powerful method, based on an integration by parts which degenerates the torus down to a sphere (plus two new points). The model is then solved like at tree-level. I'll briefly discuss extension to higher loops in conclusion.

The talk reports on recent progress in amplitude computations for broad classes of N=2 Maxwell-Einstein and Yang-Mills-Einstein supergravities, using the framework provided by color/kinematics duality and the double-copy construction. After a review of the main theoretical tools employed in the computations, we discuss the extension of the double-copy construction to a particular infinite family of N=2 Maxwell-Einstein supergravities in four and five dimensions and show that the global symmetries of these theories can easily be gauged, giving the amplitudes of the corresponding Yang-Mills-Einstein supergravities. Finally, we discuss how the construction can be modified to describe spontaneous symmetry breaking and present examples at one loop.

We will show how to obtain half-BPS four-point correlation functions in N=4 SYM at up to 3 loops using symmetries, analytic properties the Wilson loop/correlator duality, planarity and some minimal knowledge of the OPE structure relating four-point correlators of different charges. These can be used to verify recent predictions for OPE coefficients arising from integrability.

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.

In this talk we present recent advances on NLO QCD corrections, performed with BlackHat+SHERPA, to processes with many jets in the final state relevant for the LHC. Furthermore, we will discuss the importance of scattering amplitudes for developing strategies to measure the "out of reach" parameters of the Higgs boson.

I will discuss how to construct rigid supersymmetric gauge theories on Riemannian five-manifolds following a holographic approach. This approach realises the five-manifold as the conformal boundary of a six-dimensional bulk supergravity solution and leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals. The background metric is furnished with a conformal Killing vector, which generates a transversely holomorphic foliation with a transverse Hermitian structure. Finally, I’ll also construct supersymmetric Lagrangians for gauge theories coupled to arbitrary matter on such backgrounds.