On odd parameters in geometry

TL;DR

This paper proves that certain simple complex Lie superalgebras do not admit deformations with odd parameters and demonstrates the existence of non-split supermanifolds of superdimension m|1, highlighting new phenomena in supergeometry.

Contribution

It confirms the conjecture that no simple Lie superalgebras have odd-parameter deformations and provides examples of non-split supermanifolds of superdimension m|1.

Findings

No simple Lie superalgebras have odd-parameter deformations.

Existence of non-split supermanifolds of superdimension m|1.

Obstructions to splitness relate to odd parameters.

Abstract

1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic. 2) Any supermanifold which is a ringed space of the form (a manifold $M$, the sheaf of sections of the exterior algebra of a vector bundle over $M$) is called split. Gaw\c{e}dzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a split supermanifold. So far, researchers considered, mostly, even obstructions to splitness. This lead them to the conclusion that any supermanifolds of superdimension $m|1$ is split. I'll show that there are non-split supermanifolds of superdimension $m|1$; for example, certain $1|1$-dimensional superstrings, the obstructions to their splitness correspond to odd parameters.