"Nonlinear pullbacks" of functions and $L_{\infty }$-morphisms for homotopy Poisson structures

Abstract

We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening" of the usual category of supermanifolds; it is close to the category of symplectic micromanifolds and their micromorphisms considered recently by A. Weinstein and A. Cattaneo--B. Dherin--Weinstein.) There are two parallel settings, for even and odd functions. As an application, we show how such nonlinear pullbacks give $L_{\infty }$-morphisms for algebras of functions on homotopy Schouten or homotopy Poisson manifolds.

25 pages. LaTeX2e. Exposition in this version has been substantially reworked

References in corpus (3)

• Symplectic Microgeometry III: Monoids
• Thick morphisms of supermanifolds and oscillatory integral operators
• Quantum microformal morphisms of supermanifolds: an explicit formula and further properties

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• Reducible gauge symmetry versus unfree gauge symmetry in Hamiltonian formalism
• L-infinity bialgebroids and homotopy Poisson structures on supermanifolds
• Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology
• Symplectic Microgeometry IV: Quantization
• Thick morphisms of supermanifolds, quantum mechanics, and spinor representation
• Lagrangian Relations and Quantum $L_{\infty }$ Algebras
• A Graphical Calculus for Classical and Quantum Microformal Morphisms