# Microformal geometry and homotopy algebrasCitation formats

## Standard

In: Proceedings of the Steklov Institute of Mathematics, Vol. 302, No. 1, 2019, p. 88-129.

Research output: Contribution to journalArticlepeer-review

## Harvard

Voronov, T 2019, 'Microformal geometry and homotopy algebras', Proceedings of the Steklov Institute of Mathematics, vol. 302, no. 1, pp. 88-129. https://doi.org/10.1134/S0081543818060056

## APA

Voronov, T. (2019). Microformal geometry and homotopy algebras. Proceedings of the Steklov Institute of Mathematics, 302(1), 88-129. https://doi.org/10.1134/S0081543818060056

## Vancouver

Voronov T. Microformal geometry and homotopy algebras. Proceedings of the Steklov Institute of Mathematics. 2019;302(1):88-129. https://doi.org/10.1134/S0081543818060056

## Author

Voronov, Theodore. / Microformal geometry and homotopy algebras. In: Proceedings of the Steklov Institute of Mathematics. 2019 ; Vol. 302, No. 1. pp. 88-129.

## Bibtex

@article{3c4c69c05b204503a9f22f3c6b7fe1a4,
title = "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 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.",
author = "Theodore Voronov",
year = "2019",
doi = "10.1134/S0081543818060056",
language = "English",
volume = "302",
pages = "88--129",
journal = "Steklov Institute of Mathematics. Proceedings ",
issn = "0081-5438",
publisher = "MAIK Nauka - Interperiodica",
number = "1",

}

## RIS

TY - JOUR

T1 - Microformal geometry and homotopy algebras

AU - Voronov, Theodore

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

UR - https://rdcu.be/bflnC

U2 - 10.1134/S0081543818060056

DO - 10.1134/S0081543818060056

M3 - Article

VL - 302

SP - 88

EP - 129

JO - Steklov Institute of Mathematics. Proceedings

JF - Steklov Institute of Mathematics. Proceedings

SN - 0081-5438

IS - 1

ER -