Non-linear homomorphisms of algebras of functions are induced by thick morphisms

Abstract

In 2014, Voronov introduced the notion of thick morphism of a (super)manifold as a tool for constructing L ∞ L_{\infty } -morphisms of homotopy Poisson algebras. A thick morphism generalises an ordinary smooth map, but it is not itself a map. Nevertheless, it induces a pull-back of C ∞ C^{\infty } functions. These pull-backs which are in general non-linear maps between algebras of functions are “non-linear homomorphisms”. By definition, this means that their differentials are algebra homomorphisms in the usual sense. The following conjecture was formulated: an arbitrary non-linear homomorphism of algebras of smooth functions is generated by some thick morphism. We prove here this conjecture in the class of formal functionals. In this way, we extend the well-known result for smooth maps of manifolds and algebra homomorphisms of C ∞ C^{\infty } functions and, more generally, provide an analog of classical “functional-algebraic duality” in the non-linear setting.