Microformal geometry and homotopy algebras

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal, or “thick,” morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a “nonlinear algebra homomorphism” and the corresponding extension of the classical “algebraic–functional” duality. There is a parallel fermionic version. The obtained formalism provides a general construction of $L_\infty $-morphisms for functions on homotopy Poisson ($P_\infty $) or homotopy Schouten ($S_\infty $) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to $L_\infty $-algebroids, we show that an $L_\infty $-morphism of $L_\infty $-algebroids induces an $L_\infty $-morphism of the “homotopy Lie–Poisson” brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular $L_\infty $-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation to the classical version is like that of the Schrödinger equation to the Hamilton–Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits as $\hbar \to 0$ of certain “quantum pullbacks,” which are defined as special form Fourier integral operators.