Characteristic classes of supergauge fields

Semidensities on Odd Symplectic Supermanifolds

“…In this case there exist analogies of differential forms considered as integration object. (See [8,9], [1,2] and for a more detailed analysis [25,26].) For example if ΠT * M is (r + s.r + s)-dimensional supermanifold associated with cotangent bundle of (r.s)-dimensional supermanifold M then considering analogue of formula (3.4a) we obtain relations between semidensities in ΠT * M and so called pseudodifferential forms: functions on supermanifold ΠT M .…”

“…The properties of the integral invariant do not change drastically. In particular one can prove that the integrand in (5.1) (the density of the weight σ = 1 and of the rank k = 1) is locally total derivative and all invariant densities on surfaces are exhausted by (5.1) as well as in the case of usual symplectic structure [18,1,2].…”

Batalin–Vilkovisky formalism and integration theory on manifolds

“…In the 4-th Section we consider the densities [7,8,9,10] (the general covariant objects which can be integrated over supersurfaces in the superspace). Following [8] and [10] we consider a special class of densities -pseudodifferential forms on which the exterior derivative can be defined correctly.…”

“…In the 4-th Section we consider the densities [7,8,9,10] (the general covariant objects which can be integrated over supersurfaces in the superspace). Following [8] and [10] we consider a special class of densities -pseudodifferential forms on which the exterior derivative can be defined correctly. Using Baranov-Schwarz (BS) transformations [8] we rise these forms to integration objects on the enlarged space and formulate the condition of closure of these forms in terms of the ∆-operator.…”

Odd Invariant Semidensity and Divergence-like Operators on an Odd Symplectic Superspace

“…In usual mathematics densities are the natural generalization of differential forms (if k = 1). In supermathematics even in the case of k = 1 this object is of more importance, since differential forms in supermathematics are no more integration objects [9,10,11].…”