Microformal geometry and homotopy algebras
Theodore Voronov

TL;DR
This paper introduces microformal morphisms as nonlinear, formal canonical relations extending the category of supermanifolds, enabling new constructions in homotopy algebras, quantum mechanics, and geometric structures.
Contribution
It develops the theory of microformal morphisms, providing a framework for nonlinear pullbacks, $L_{}$-morphisms, and quantum analogs, extending classical dualities and structures in geometry and algebra.
Findings
Constructed a formal category of microformal morphisms with nonlinear pullback operations.
Extended $L_{}$-morphisms to functions on homotopy Poisson and Schouten manifolds.
Developed a quantum version of the formalism, linking it to Schrd6dinger and Hamilton-Jacobi equations.
Abstract
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…
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