Generalised Geometry in Supergravity theories

I study how ten-dimensional Type II Supergravity theories can be reformulated using an extension of conventional differential geometry known as “Generalised geometry”. I review the dynamics and symmetries of these theories, define the key elements of generalised geometry, including the notion of torsion-free generalised connections, and show how this geometry can be used to give a unified description of the supergravity fields, exhibiting an enlarged local symmetry group. This part will be end showing that the equations of motion for the NSNS sector of Type II Supergravity theories in the framework of Generalised geometry can be reformulated in a similar way of Einstein’s equations of motion for gravity in ordinary geometry. In the second part I investigate the notion of “Leibniz generalised parallelisations”, the analogue of a local group manifold structure in generalised geometry, aiming to characterise completely such geometries, which play a central role in the study of consistent truncations of supergravity. One of the original results we obtained is the solution of the misterious case of consistent truncation on S7 showing that in Generalised geometry all spheres Sd are Leibniz generalised parallelisable. I work out also some explicit examples of manifold that are Leibniz generalised parallelisable (S2 S1, H3, dS3, AdS3) and in particular connecting this results with the consistent truncations of supergravity.