Vera Serganova

The goal of this lecture is to show interplay between supersymmetry and tensor categories. The main idea of supersymmetry is to equip all objects with parity ( $$\mathbf{Z}_2$$-grading) and modify usual identities by so called sign rule. Original motivation comes from physics and topology, for example, a complex of differential forms on a manifold is a supermanifold and De Rham differential can be realized as a vector field on this super manifold. One way to approach supersymmetry is via rigid symmetric tensor categories.

After elementary introduction to supersymmetry and tensor categories, I will formulate theorem of Deligne that any rigid symmetric tensor category satisfying certain finiteness conditions is in fact the category of representations of a supergroup.

Then I illustrate how both theories enrich each other on two examples:

1. Decomposition of tensors in superspace;
2. Construction of universal symmetric tensor categories and proof of a conjecture of Deligne using results of representation theory of supergroups.