Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Theodore Th. Voronov

TL;DR

This work extends the tangent and antitangent functors to thick (microformal) morphisms between supermanifolds, enabling non-linear pullbacks of differential forms. By grounding the construction in canonical diffeomorphisms (Mackenzie–Xu, Tulczyjew) and introducing time-derivative generating functions for tangents, the authors build a coherent framework where even/odd thick morphisms induce tangent/antitangent thick morphisms with parity-flipping behavior. They show that these non-linear pullbacks act compatibly with de Rham differentials to yield non-linear transformations on de Rham cohomology, and they introduce thick morphisms of vector bundles (fiberwise-linear thick morphisms) that preserve fiberwise structure under pullbacks. The results generalize classical linear pullbacks to a rich non-linear setting, with potential implications for Q-manifolds and non-linear cohomological transformations. Overall, the paper provides a comprehensive mechanism to transport thick morphisms through tangent structures and to study their effects on forms and cohomology in both even and odd contexts.

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.

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

TL;DR

This work extends the tangent and antitangent functors to thick (microformal) morphisms between supermanifolds, enabling non-linear pullbacks of differential forms. By grounding the construction in canonical diffeomorphisms (Mackenzie–Xu, Tulczyjew) and introducing time-derivative generating functions for tangents, the authors build a coherent framework where even/odd thick morphisms induce tangent/antitangent thick morphisms with parity-flipping behavior. They show that these non-linear pullbacks act compatibly with de Rham differentials to yield non-linear transformations on de Rham cohomology, and they introduce thick morphisms of vector bundles (fiberwise-linear thick morphisms) that preserve fiberwise structure under pullbacks. The results generalize classical linear pullbacks to a rich non-linear setting, with potential implications for Q-manifolds and non-linear cohomological transformations. Overall, the paper provides a comprehensive mechanism to transport thick morphisms through tangent structures and to study their effects on forms and cohomology in both even and odd contexts.

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.

Paper Structure

This paper contains 15 sections, 33 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology:
  3. Natural diffeomorphisms for cotangent and anticotangent bundles
  4. Mackenzie--Xu isomorphism and its odd analog
  5. Tulczyjew isomorphism
  6. Three odd analogs of the Tulczyjew isomorphism
  7. The tangent and antitangent morphisms for a thick morphism
  8. The tangent functor $T$.
  9. Case of even thick morphisms.
  10. Case of odd thick morphisms.
  11. The antitangent functor $\Pi T$.
  12. Thick morphisms of vector bundles
  13. Thick $Q$-morphisms. Consequences for forms and cohomology
  14. Thick morphisms of $Q$-manifolds
  15. Action of thick morphisms on forms and de Rham cohomology

Key Result

Theorem 1

For a vector bundle $E\to M$, there is a natural diffeomorphism $T^*(E)\cong T^*(E^*)$. It is an antisymplectomorphism of the canonical symplectic structures.

Theorems & Definitions (66)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 2: tv:graded
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • ...and 56 more