THEORY OF CONNECTIONS ON GRADED PRINCIPAL BUNDLES

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. We first review the basic elements of this theory establishing at the same time supple mentary properties of graded Lie groups and their actions. Particular emphasis is given in introducing and studying free actions in the grad ed context. Next, we investigate the geometry of graded principal bund les; we prove that they have several properties analogous to those of ordinary principal bundles. In particular, we show that the sheaf of v ertical derivations on a graded principal bundle coincides with the gr aded distribution induced by the action of the structure graded Lie gr oup. This result leads to a natural definition of the graded connectio n in terms of graded distributions; its relation with Lie superalgebra -valued graded differential forms is also exhibited. Finally, we defin e the curvature for the graded connection and we prove that the curvat ure controls the involutivity of the horizontal graded distribution co rresponding to the graded connection.