Summary: In these lecture notes we give a simple introduction into double field theory. We show that the presence of momentum and winding excitations in toroidal backgrounds of closed string theory makes it natural to consider double field theories. A tool-kit is developed based on the Courant bracket and generalized Lie derivatives. We construct a background independent action which represents a T-duality covariantization of the Einstein-Hilbert action for gravity coupled to an antisymmetric tensor field and a dilaton.
For the entire collection see [Zbl 1239.81006].