Supertime

A supermanifold of values of a dynamical parameter. On manifolds, Time is usually 1-dimensional, a different example is Kadomtsev-Petviashvili hierarchy in which an infinite dimensional manifold is interpreted as Time.

On finite dimensional manifolds, Time is always 1-dimensional as follows from the rectifyability of vector field theorem studied at early courses of differential equations. Shander generalized the theorem on rectifyability of vector fields to nondegenerate fields on supermanifolds and gave the following characterization of such fields, in particular, the ones used in SUSY theories: the nondegenerate (at a point) vector field X can locally be reduced to the form , where x is an even coordinate, if X is even, to the form , where θ is an odd coordinate, if X is odd and X ² = 0, or to the form , if X is odd and X ²≠ 0.

Shander explained that for dynamical systems on supermanifolds supertime runs a (1|1)-dimensional supermanifold with parameters t,τ. Shander gave examples...