Final Report Summary - PURESP (Covariant quantization of the superstring) An important open question in string theory concerns the study of string propagation in backgrounds with nonzero fluxes for the so-called Ramond-Ramond (RR) fields. It is a problem with manifold aspects and a great variety of related questions. It is also a long-standing problem, since the presence of the background RR fields renders the use of the standard formalism for the string quantisation very difficult, so that one has to resort to a different formulation. Nevertheless, a suitable one was not known until around the year 2000, when Berkovits proposed his pure-spinor formalism. By now a consensus has formed that this is indeed the solution to the long-standing problem; however many questions still remain, regarding the foundational aspects of the formalism and the associated practical applications. In the course of this project, the researcher focussed mostly on the second aspect, i.e. the practical usefulness of the formalism, in that it allowed computations of scattering amplitudes that preserved manifestly all the underlying symmetries of the systems, namely Poincaré invariance and supersymmetry. As a result, an efficient computation of the low-energy effective action that described the dynamics of the massless modes of the string in a given background could be performed. In particular, even in the simplest case of flat space, the effective action including all the couplings to the RR fields was not known in a completely explicit form prior to the work of Policastro and Tsimpis in 2006. In the course of the project, the researcher, again in collaboration with D. Tsimpis, built on these previous results to clarify the structure of the quartic action under the SL(2,Z) duality symmetry of strings of the IIB type. In particular, they were able to assess, and disprove, a conjecture related to the form of the quartic RR couplings. The obtained results could be useful in many situations where RR fields were present, e.g. for the computation of the corrections to the entropy of charged black holes. Their usefulness would be increased by an extension of the project to compute the effective action to higher-order; however this was left for future work. A parallel line of research that was pursued in the project was related to the holographic correspondence, one of the main and most fertile topics of the last decade. The correspondence related the string propagating in anti-de Sitter space, which was a background supported by RR fluxes, with a four-dimensional conventional field theory living in Minkowski space. The correspondence was very successful in describing the properties of the theory at very high temperature, in the plasma phase. Indeed, a relatively simple string computation allowed for the extraction of the transport properties of strongly coupled plasma, such as the shear viscosity which was first computed by Policastro, Son and Starinets in 2002. It was since then understood that this quantity attained a universal value for all theories that admitted a holographic description, i.e. to a large class of different theories. The result was compared, and found in remarkable agreement, with the measured value of the viscosity in the real-world quantum chromodynamics (QCD) plasma produced in heavy ion collisions. In the course of the project, Policastro continued this research by considering the supersymmetric sector of the theory. In the N=4 maximally supersymmetric theory, described by the simplest version of the correspondence, there were fluctuations of the density of supersymmetry charge which had hydrodynamic wave-like modes. The holographic theory allowed for the computation of their speed and attenuation rate. In this case, as opposed to the shear viscosity, no direct comparisons to experiments could be made since supersymmetry was not experimentally observed. Nevertheless, the result was theoretically interesting given that that it could predict another universal hydrodynamic quantity. The verification of this property would need further work, which extended beyond the scope of this project.