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# Supersymmetry and index theory (Spring 1998)

## I.P.Prokhorenkov

Classical theorem of Atiyah-Singer expresses
the index of the elliptic operator on a compact
manifold M through some topological invariants of M.
We will prove a particular case of this theorem - the
Gauss-Bonnet-Chern theorem. In our proof we will use
ideas of supersymmetry, which came into mathematics from
physics.
In the second part of our course (time permitting)
we will outline the proof of the Atiyah-Singer theorem
for Dirac operators by means of functional integration.

All necessary facts from differential geometry and analysis will
be discussed during lectures. Some knowledge of differential
forms and de Rham cohomology will be useful.

### Outline:

1. Short course in Riemannian Geometry, including
parallel translation and curvature operators.

2. Notion of supersymmetry, Berezin-Patodi formula.

3. Bochner Laplacian and Weitzenbock formula.

4. Elliptic regularity (description of results).

5. Localization of the heat kernel and the completion
of the proof of the Gauss-Bonnet -Chern theorem.

6* Path integrals.

7* Atiyah-Singer formula for Dirac operator.