On unconventional integrations and cross ratio on supermanifolds

Abstract

The conventional integration theory on supermanifolds had been constructed so as to possess (an analog of) Stokes' formula. In it, the exterior differential d is vital and the integrand is a section of a fiber bundle of finite rank. Other, not so popular, but, nevertheless, known integrations are analogs of Berezin integral associated with infinite dimensional fibers. Here I offer other unconventional integrations that appear thanks to existence of several versions of traces and determinants and do not allow Stokes formula. Such unconventional integrations have no counterpart on manifolds except in characteristic p. Another type of invariants considered are analogs of the cross ratio for ``classical superspaces''. As a digression, homological fields corresponding to simple Lie algebras and superalgebras are described.