Interpretation of higher derived cyclic cocycles for regular foliations

Clarify the precise interpretation of derived cyclic n-cocycles for regular foliations, for every degree n≥1, in terms of classical foliation theory and multilinear functionals on leaf-wise differential forms.

Background

For a regular foliation, the associated foliation Lie algebroid determines a Q-manifold whose functions are leaf-wise pseudo-differential forms and whose homological vector field is the leaf-wise de Rham differential. In this setting, derived cyclic 0-cocycles are identified with generalised closed foliation currents and include classical Ruelle–Sullivan currents.

The paper constructs higher derived cyclic cocycles as generalised closed foliation multiform currents, but does not establish how these objects correspond to known constructions in classical foliation theory. Determining this relationship is therefore an explicitly unresolved interpretive problem.