This week

Special geometries, such as Calabi-Yau manifolds, play a central role in multiple areas of string theory, as well as gravitational theories more generally. The goal of these lectures is to introduce some of the formalism and tools useful for characterising such geometries, pitched at the level of a starting PhD student. We will start with purely geometrical backgrounds using the general notions of a G-structure and special holonomy and then will go on to describe backgrounds that also have non-trivial fluxes. We will be guided by applications to string phenomenology and the AdS/cft correspondence.

A natural way to extend Einstein's General Relativity in the high-energy regime is to introduce higher-order curvature terms in the gravitational Lagrangian. Indeed, by working in the framework of perturbative QFT one can show that quadratic-curvature gravity in four dimensions is strictly renormalizable. The quadratic-curvature terms are multiplied by dimensionless parameters that are related to the masses of the additional gravitational degrees of freedom and to the interaction couplings. In this talk, after having motivated Renormalizable Quantum Gravity, we will study the limits in which those dimensionless parameters tend to zero or to infinity, and show that different types of decoupling can occur. In particular, it will be shown that the presence of a non-zero cosmological constant affects the decoupling in a non-trivial way in the limit where the coefficient of the Weyl-squared term tends to infinity. We will discuss possible physical implications of this mathematical analysis for the high-energy behavior of the spin-2 massive ghost and for the classical limit of the theory. Several concepts that have been developed in the context of massive gravity will naturally emerge in this talk, sometimes with different relevance.

We study the four point correlator of the stress-energy tensor in N=4 SYM at leading order in inverse powers of the central charge. This corresponds to the Anti-deSitter version of the Virasoro-Shapiro amplitude. At large t'Hooft coupling lambda, we use dispersive sum rules to relate the Wilson coefficients in a 1/lambda expansion to the OPE data of heavy string operators. Assuming that the Wilson coefficients are in the ring of single valued multiple zeta values (as is expected for closed string amplitudes), we solve the dispersion relations to get the first 1/R^2 correction to the flat space amplitude.

In this talk I will discuss how to compute amplitudes on AdS, using the analytic conformal bootstrap and the AdS/CFT correspondence. I will also discuss how to include higher trace operators in a simplified setup.