In essence, supersymmetry is the extension of space-time symmetry (Galileo, Poincare, conformal, etc.) by fermionic generators and it serves as the theoretical scheme which naturally unifies bosons (such as the photon) and fermions (such as the electron).
As a mathematical theory, supersymmetry hasn't been taken seriously for a long period of time. This maybe came about because in its simplest form the ``super'' mathematical objects are the usual ones, only equipped with a Z_2-grading. One might therefore think of this as a simple enough structure which is not worth a whole theory. However, it turned out that this grading is only good enough for algebraic structures like vector spaces, algebras etc. When it comes to, say, manifolds, a supersymmetric refinement is a much more interesting structure.
The purpose of this seminar is to understand the mathematical structure of supersymmetric manifolds, vector bundles, connections, differential forms, and integration.
This is a ``learning seminar'' and participants will alternately give talks. In the first meeting on Thursday Sept. 25, we will give a brief introduction and then survey the contents of the various lectures, so that participants can pick their favorite topic.
We will go through the Notes on supersymmetry by P. Deligne and J. Morgan, which follow lectures given by J. Bernstein in 1996 at the Institute for Advanced Study. We will supply copies of these Notes. They start with lectures on graded algebra, then introduce super manifolds and super Lie groups and finally study the differential geometry of super manifolds. (Another potentially helpful reference is Dan Freed's Lectures on supersymmetry.)
Sean discovered an excellent-looking reference by Varadarajan. Chapter 3 is on vector spaces and algebras, chapter 6 on Lie groups.
Provisional schedule:
October 2: Sean Raleigh Super vector spaces
October 9: Jana Comstock The Berezinian
October 16: Gregg Musiker Supermanifolds
October 23: Henning Hohnhold Super Lie groups
October 30: Li Yu Super vector bundles and connections
November 6: Peter Teichner Integration on supermanifolds
November 13: Justin Roberts The de Rham complex in
supergeometry
December 4: Jeff Rabin Super Riemann surfaces
justin@math.ucsd.edu
