Other

I explain what physicists mean when they say that space and time may be relational and discuss the extent to which different of the contemporary attempts at quantum gravity satisfy this criteria. I then discuss how the current debates between advocates of background dependent and background independent theories may be seen in the light of the historical debate in philosophy between advocates of absolute and relational notions of space and time.

We will see how generalized Calabi-Yau manifolds as defined by Hitchin emerge from supersymmetry equations in type II theories. In the first lecture, we will review the formalism of G-structures, which is central in the context of compactification with fluxes. In the second lecture we will see how (twisted) generalized Calabi-Yau manifolds emerge from supersymmetry equations using SU(3) structure. In the last lecture, we will discuss special features about compactifications on Generalized Calabi-Yau's.

In these lectures, we examine how twisted generalized Calabi-Yau (GCY) manifolds arise in the construction of a general class of topological sigma models with non-trivial three-form flux. The topological sigma model defined on a twisted GCY can be regarded as a simultaneous generalization of the more familiar A-model and B-model. Emphasis will be given to the relation between topological observables of the sigma model and a Lie algebroid cohomology intrinsically associated with the twisted GCY. If time permits, we shall also discuss topological D-branes in this more general setting, and explain how the viewpoint from the Lie algebroid helps to elucidate certain subtleties even for the conventional A-branes and B-branes. The lectures will be physically motivated, although I will try to make the presentation self-contained for both mathematicians and physicists.