Microformal geometry and homotopy algebras

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 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 with the classical version is like that of the Schrödinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hslash \to 0$ of certain "quantum pullbacks", which are defined as special form Fourier integral operators.

LaTeX 2e. 47 p. Some editing of the exposition

References in corpus (8)

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• $Q$-Manifolds and Mackenzie Theory
• On volumes of classical supermanifolds
• "Nonlinear pullbacks" of functions and $L_{\infty }$-morphisms for homotopy Poisson structures
• Symplectic Microgeometry III: Monoids
• Quantum microformal morphisms of supermanifolds: an explicit formula and further properties
• Integration of Lie Algebroid Comorphisms
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Cited by in corpus (7)

• Graded Geometry, $Q$-Manifolds, and Microformal Geometry
• Formal oscillatory integrals and deformation quantization
• On a Batalin--Vilkovisky operator generating higher Koszul brackets on differential forms
• Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology
• Thick morphisms of supermanifolds, quantum mechanics, and spinor representation
• Quantization of (-1)-Shifted Derived Poisson Manifolds
• A Graphical Calculus for Classical and Quantum Microformal Morphisms