Manchester Geometry Seminar 2012/2013

This is the second talk in the series devoted to integration and analogs of differential forms on supermanifolds.

In the first talk, I recalled the notion of the Berezin integral over a supermanifold and introduced the most straightforward ("naive") super analog of differential forms. Unlike ordinary manifolds, on a supermanifold the complex of differential forms is not bounded from the right --- no "top degree" forms --- and forms cannot be integrated over the supermanifold. A partial remedy is given by the complex of so-called `integral forms'. It is bounded from the right, with the Berezin volume forms playing the role of forms of top degree. It can be shown that integral forms are suitable for integration over submanifolds of odd codimension zero, such as $P^{k|m}\subset M^{n|m}$. Differential forms, on the other hand, can be integrated over purely even submanifolds, i.e., of odd dimension zero, such as $P^{k|0}\subset M^{n|m}$. (I will recall all that briefly.)

The problem is to find suitable objects of integration in the "intermediate" dimensions. Solution is given by the notion of an $r|s$-form defined as a first-order Lagrangian (function of velocities) satisfying certain conditions. This will be the main subject of the present talk. The construction of $r|s$-forms exhibits non-trivial features. It is based on the language of calculus of variations. (In particular, I will show that the familiar de Rham differential on ordinary manifolds also allows a re-formulation in variational terms.) At the same time, the defining equations for forms contain, in the odd-odd sector, the equation known in classical integral geometry as the John equation. This link is very surprising and has not been understood fully yet.