On unconventional integrations and cross ratio on supermanifolds

Abstract

(A) The conventional integration theory on supermanifolds had been constructed in order to have (an analog of) the Stokes formula in which a~sub-supermanifold is of codimension $1=(1|0)$. I review other integrations and formulate related open problems: (1) On the $1|1$-dimensional superstring associated with the trivial bundle, in presence of a~contact structure there is a special integration useful in describing super versions of elliptic functions. It is needed to construct a~particular spinor representation of the Neveu-Schwarz superalgebra. (2) Versions of the Stokes formula with ``over-supermanifold" of codimension $(0|-1)$ due to Shander and Palamodov should be developed further. (3) Apply Shander's integration with odd parameters over chains to inverse problems. (4) Establish existence of conjectural integrations (apparently, not leading to any analog of the Stokes formula) related to various (super)traces on various Lie superalgebras and the corresponding (super)determinants. (B) I offer analogs of the cross ratio for ``classical superspaces'', including infinite-dimensional versions. For four solutions of a~super matrix Riccati equation, these invariants are first integrals.