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

TL;DR

This paper introduces nonlinear, formal pullback mappings between function spaces on supermanifolds, based on canonical relations, which induce $L_{}$-morphisms for homotopy Poisson structures, extending classical notions.

Contribution

It develops a formal category framework for nonlinear pullbacks using generating functions, enabling new $L_{}$-morphisms in homotopy Poisson geometry.

Findings

Nonlinear pullbacks are constructed via canonical relations and generating functions.

These pullbacks induce $L_{}$-morphisms for homotopy Poisson structures.

The approach extends classical pullbacks to a formal, nonlinear setting.

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.