## Supergeometry and Its Applications. III

#### Dr Theodore Voronov (University of Manchester)

Frank Adams Room 1,

This is part of a minicourse in the framework of the seminar.

Lecture 3.

Integral calculus: Berezin integral. Superdeterminant (Berezinian) and its properties.

"Super" language in conventional differential geometry: forms and multivector fields as functions on auxiliary supermanifolds \hat M=ΠTM and \check M=ΠT^{*}M. Differential operations such as the exterior differential, Lie derivative, Schouten bracket, and Nijenhuis bracket in this language. Integration of forms as a Berezin integral. Example: integration over fibers.

Some more advanced applications and examples. May include: volumes of supermanifolds, index theorem, L-infinity algebras, Lie algebroids...

Contents of the previous lectures:

Lecture 1 (February 16):

Origins of supergeometry in physics: uniform description of fermions and bosons; supersymmetry. Idea of algebra/geometry duality. Examples of "non-set-theoretic" geometric objects such as "double point" and coordinate superspace. Necessary notions from **Z**_{2}-graded algebra. Two examples of supermanifolds.

Lecture 2 (March 9):

The examples of supermanifolds (the supersphere and the tangent bundle with reversed parity ΠTM, for an ordinary manifold M) carefully worked out. Differential calculus in the super case. Definition of a supermanifold in the language of ringed spaces. Discussion of "points" of a supermanifold.