Supergeometry and Its Applications. II

Dr Theodore Voronov (University of Manchester)

Frank Adams Room 1,

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

Lecture 2.

Examples and definition of supermanifolds. Language of supergeometry in the conventional world of differentiable manifolds. Forms and multivector fields as functions on auxiliary supermanifolds \hat M=ΠTM and \check M=ΠT*M. Digression into "supercalculus". Superdeterminant (Berezinian). Berezin integral.  Differential operations such as the exterior differential, Lie derivative, Schouten bracket, and Nijenhuis bracket in this language. Integration of forms as Berezin integral. Integration over fibers.

 

Contents of Lecture 1, which was on 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 Z2-graded algebra. Two examples of supermanifolds. [Definition was not yet given.] 

Tentative content of Lecture 3, to follow:

Some more advanced applications and examples (depending on how the previous lectures go). May include: volumes of supermanifolds, index theorem, L-infinity algebras, Lie algebroids...

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