Breakthroughs in numerical relativity (around 2005) have allowed a tremendous progress in solving the two body problem in general relativity. It is now possible to follow the inspiral, merger and ringdown phases of a binary black hole spacetime, extract accurate wave forms for the gravitational radiation emitted and learn new phenomena, like black hole kicks. It has also been possible to study high energy black hole scattering, and obtain the ratio of energy converted into gravitational radiation and the cross section for black hole formation. In this talk I shall describe the very first steps to apply the same type of techniques to high energy physics scenarios suggested by string theory: namely TeV gravity, the AdS/CFT duality and the study of the non-linear evolution of higher dimensional unstable black hole spacetimes.

I will discuss the structure of 5d supergravity dual to the universal gauge sector of $N=1$ CFT's. This model follows from a consistent truncation of type IIB string theory on 5-dimensional squashed Sasaki-Einstein manifolds and allows for interesting applications in the context of the gauge/gravity correspondence.

The free energy of four-dimensional BPS black holes is given by the generalized Hesse potential. We construct new variables for the Hesse potential, and we exhibit the relation of this Hesse potential with the functions computed on the topological string side.

In tree level Yang-Mills theory and Einstein gravity on-shell recursion relations have been proven which allow one to express any amplitude in terms of certain sums over three point amplitudes. These relations look remarkably like CFT bootstrap equations and are closely related to crossing symmetry. In this talk it will be shown how these recursion relations can be proven for string amplitudes on the disc and sphere in a flat background. A completely CFT based argument will be presented. This suggests that the techniques explored may extend to a much more general setting in string theory. Some simple examples of this will be discussed.

Quaternion-Kahler geometry appears in the hypermultiplet sector of $N=2$ supergravity, and is obtained e.g. by compactifying type II string theory on Calabi-Yau manifolds. In this talk, we discuss the status of finding the instanton corrections to the quaternionic hypermultiplet moduli space using twistor space techniques.

Surface operators are two-dimensional defects in a gauge theory. Certain (half-BPS) surface operators in a four-dimensional supersymmetric $N=2$ gauge theory admit an alternative description as a two-dimensional supersymmetric $N=(2,2)$ gauge theory that is coupled to the four-dimensional gauge theory. The two-dimensional gauge theory itself localizes onto the moduli space of vortex equations. In this seminar we study a duality between vortex counting in the two-dimensional gauge theory and open BPS invariants associated to Lagrangian $3$-manifolds. We employ this duality to compute equivariant vortex partition functions and to learn more about surface operators. For example, we find that the geometric transition yields an alternative description of these surface operators as degenerate insertions in a certain two-dimensional CFT.

We discuss the main features of a phase space noncommutative model of quantum mechanics and its application for quantum cosmology and black holes in the context of the Kantowski-Sachs minisuperspace geometry. References: hep-th/0505064; hep-th/0509207; gr-qc/0606131; hep-th/0611257; arXiv:0712.4122; arXiv:0907.1818; arXiv:0912.4027

Manifolds with $G$-structures are suitable as internal manifolds for supersymmetric flux compactifications, once the backreaction of fluxes on the internal geometry is taken into account. After a general introduction to flux compactifications, I will focus on supersymmetric type IIA compactifications on $SU(3)$ structure manifolds, where necessary and sufficient conditions on the internal geometry have been found. As an illustration, I will discuss the Nilsson-Pope example, where the internal manifold is a $\mathbb{CP}3$. Noting that $\mathbb{CP}3$ is a toric variety, I will outline a generalization to other toric varieties, which could also be interesting for string compactifications.

Toric geometry has been an indispensable tool in constructing Calabi-Yau manifolds that arise in string compactifications without background fluxes. In this talk, I will show that toric varieties can also be equipped with $G$-structures, and are thereby also candidates for internal manifolds of flux compactifications. After a brief introduction to toric geometry, I will review the (partial) classification of three-dimensional smooth complex toric varieties (SCTV) by Oda. I will then propose a method for constructing $SU(3)$ structure on SCTVs. For some examples, I will discuss the torsion classes of the $SU(3)$ structure, and check if the conditions for supersymmetric string theory vacua are fulfilled.

Linearized ten dimensional super Yang Mills equations as well as supergravity can be conveniently formulated as cohomology equations of a (BRST) differential. To this end, superspace is extended with commuting spinorial coordinates that obey the so-called pure spinor constraint. The BRST differential acts on holomorphic functions defined on this ”pure spinor superspace”. For the analysis of the cohomology it would be extremely useful to identify a reduced ”physical pure spinor superspace” parametrized by coordinates that are in the Kernel of the BRST differential. One way of defining such BRST invariant coordinates is to solve the pure spinor constraint explicitly in local coordinate patches. An alternative way is to introduce the complex conjugate of the pure spinor. One thus either looses globality or holomorphicity and has to take care of this loss. This approach has certain similarities with the so-called harmonic superspace in four dimensions. I may or may not have the time to point them out. Note that this is an ongoing project with Antonio Grassi from Alessandria which unfortunately is still missing a conclusion. Comments different from ”that’s all trivial” and ”there is a no-go-theorem” are thus very welcome.

Studying the volume of moduli space of Riemann surfaces is a central problem in string theory. In particular, the computation of Gromov-Witten invariants of Toric Calabi-Yau threefolds is a fascinating subject where many different aspects of physics and mathematics beautifully coexist. Following methods developed in the context of random matrix theory, it has been recently conjectured that this problem can be solved simply by induction on the Euler characteristic of the surfaces enumerated. In this talk, I will briefly present this conjecture and explain its proof in the simplest cases by showing the equivalence between this recursion and the topological vertex formalism which is known to compute generating functions of these invariants. For this purpose, I will explain how the main character of this topological recursion, the spectral curve, arises from mirror symmetry. I will then show the use of promoting the considered generating functions to differential forms on the spectral curve. Indeed, expanding these forms at different points of the spectral curve allows to go from the topological vertex formalism to the topological recursion.

One dimensional quantum mechanical systems known as spin chains provide the connecting link between gauge theories and string theories. More precisely, the common spectral problem of the gauge and string theory entering the AdS/CFT correspondence appears to be described by an integrable spin chain. The prevailing oppinion is that integrability has to break down when one takes into account non-planar corrections in the gauge theory and correspondingly string interactions in the string theory. We will try to quantify this statement.

In this talk, I first give a basic introduction to AdS/CFT and then explore the formalism of AdS/CFT in the long wavelength regime to show the relation between bulk configurations obeying Einstein's equations and holographic fluid configurations obeying the Navier-Stokes equation. As a concrete example, we will explore black branes in $\operatorname{AdS}_5$ and examine their long wavelength dynamics in thecontext of the fluid-gravity correspondence.

In this talk, we will build upon the basic formalism of the fluid gravity correspondence to discuss various simple solutions of fluid mechanics and discuss their holographic bulk interpretation.

I'll recall the definition of the Gelfand-Zeitlin (extended eigenvalue) map $\gamma$ for Hermitian and upper-triangular $n$ by $n$ matrices. It defines a completely integrable Hamiltonian system with respect to the standard Poisson structures on these spaces. Using the coordinate system defined by a certain planar network $N$, we define a "tropical" analogue $\gamma_{trop}$ of the Gelfand-Zeiltin map. This is a piece-wise linear transformation of $\mathbb{R}^{n(n+1)/2}$ with interesting combinatorial properties described in terms of multiple paths on $N$. We establish a relation between fibers of $\gamma$ and $\gamma_{trop}$ and show that $\gamma_{trop}$ defines an open dense Darboux chart. (Joint work with I. Davydenkova, M. Podkopaeva and A. Szenes)