Supersymmetry and Spencer cohomology

Bosonic backgrounds in supergravity theories are classical solutions in which all fermionic fields vanish. Such a solution is said to be supersymmetric if it is preserved by a supersymmetry transformation of the theory. Apart from their intrinsic interest as the simplest solutions in supergravity theories, supersymmetric backgrounds arise in a number of other guises: as string theory vacua, as particular types of black holes and as settings with curvature on which one can place (rigid) supersymmetric field theories. They also often carry interesting geometric structures, such as (hyper-)Kaehler structures.

In this talk, I will give a (very) brief overview of supersymmetry and supergravity theories (familiarity with these is not necessary to understand the main points of the talk) before describing supersymmetric backgrounds and their supersymmetry algebras. I will discuss the structure of these algebras and an algebraic tool we use to study them, called Spencer cohomology, and if there is time, I will say a bit about how Spencer cohomology gives us a way of thinking about supersymmetry in curved spacetimes that is more general than the notion which arises from supergravity.