Representation Theory and Mathematical Physics Seminar: A cohomological approach to the Berezin fibre integral

Seminar | September 27 | 4-5 p.m. | 939 Evans Hall

 Alexander Alldridge, University of Cologne

 Department of Mathematics

The fibrewise Berezin integral is an important tool in mathematical physics, e.g., to derive supersymmetric field equations from variational principles. For a supermanifold, the de Rham complex is underbounded, and differential forms cannot be integrated; the Berezinian sheaf was introduced by Berezin to address this problem. However, the definition is ad hoc, the resulting integral is defined coordinate-independently only for compact supports, and "boundary corrections" appear when changing coordinates for non-compactly supported integrands. So far, there was no systematic and conceptual understanding of these terms.

On the other hand, by the work of Penkov and of Verbovetsky, it has been known for some time that the Berezin sheaf can be obtained as the cohomology of a natural complex of D-modules. However, it was not known how to use this to define the Berezin integral, except in the non-fibrewise case (i.e. over a trivial base) by the work of Rothstein. Unfortunately, his approach uses the trivial base (and the boundedness of the de Rham complex) in an essential way and does not generalise to the case where the base is a supermanifold.

We address these issues and give a definition of the Berezin integral in terms of the cohomology of a complex of D-modules. This is based on the observation that although arbitrary differential forms cannot be integrated on a supermanifold, closed forms can be. Using higher order differential forms, we derive a higher order Stokes's theorem for relative supermanifolds with corners and show how this gives a systematic derivation of the boundary terms.

Joint work with Joachim Hilgert and Tilmann Wurzbacher.