Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology

TL;DR

This paper extends the tangent functor to microformal morphisms of supermanifolds, enabling non-linear pullbacks of forms that preserve de Rham differentials and induce transformations in cohomology.

Contribution

It introduces a framework for non-linear pullbacks via microformal morphisms, generalizing traditional maps and canonical relations in supergeometry.

Findings

Microformal morphisms extend tangent functors to supermanifolds.

Non-linear pullbacks of forms preserve de Rham differentials.

These pullbacks can induce non-linear transformations of cohomology.

Abstract

We show how the tangent functor extends from ordinary smooth maps to "microformal morphisms" (also called "thick morphisms") of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on position variables on the source manifold and momentum variables on the target manifold (as formal power expansions), regarded as part of the structure. Microformal morphisms act on functions by non-linear (in general) pullbacks. We obtain here non-linear pullbacks of (pseudo)differential forms and show that they respect the de Rham differentials as "non-linear chain maps" that can induce non-linear transformations of cohomology.