# Microformal geometry and homotopy algebras

Research output: Contribution to journalArticlepeer-review

## 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 Schr\"odinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of certain quantum pullbacks'', which are defined as special form Fourier integral operators.
Original language English 88-129 42 Proceedings of the Steklov Institute of Mathematics 302 1 3 Jan 2019 https://doi.org/10.1134/S0081543818060056 Published - 2019

## Fingerprint

Dive into the research topics of 'Microformal geometry and homotopy algebras'. Together they form a unique fingerprint.
• ### Thick morphisms and homotopy bracket structures

Theodore Voronov (Invited speaker)

13 Aug 2018

Activity: Talk or presentationInvited talk

• ### Microformal geometry and homotopy algebras

Theodore Voronov (Invited speaker)

25 May 2018

Activity: Talk or presentationInvited talk

• ### Thick morphisms of supermanifolds and homotopy algebras

Theodore Voronov (Invited speaker)

7 Mar 2018

Activity: Talk or presentationInvited talk