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 Trudy Mat. Inst. Steklova, 2018, Volume 302, Pages 98–142 (Mi tm3934)

Microformal geometry and homotopy algebras

Th. Th. Voronovab

a School of Mathematics, University of Manchester, Manchester M13 9PL, UK
b Faculty of Physics, Tomsk State University, Novosobornaya pl. 1, Tomsk, 634050 Russia

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 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.

DOI: https://doi.org/10.1134/S0371968518030056

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 88–129

Bibliographic databases:

UDC: 515.16

Citation: Th. Th. Voronov, “Microformal geometry and homotopy algebras”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 98–142; Proc. Steklov Inst. Math., 302 (2018), 88–129

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3934
• https://doi.org/10.1134/S0371968518030056
• http://mi.mathnet.ru/eng/tm/v302/p98

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This publication is cited in the following articles:
1. A. Karabegov, “Formal oscillatory integrals and deformation quantization”, Lett. Math. Phys., 109:8 (2019), 1907–1937
2. T. T. Voronov, “Graded geometry, q-manifolds, and microformal geometry lms/epsrc durham symposium on higher structures in m-theory”, Fortschritte Phys.-Prog. Phys., 67:8-9, SI (2019), 1910023
3. H. Khudaverdian, T. Voronov, “Thick morphisms of supermanifolds, quantum mechanics, and spinor representation”, J. Geom. Phys., 148 (2020), 103540
4. R. Bandiera, Zh. Chen, M. Stienon, P. Xu, “Shifted derived Poisson manifolds associated with lie pairs”, Commun. Math. Phys., 375:3 (2020), 1717–1760
5. Shemyakova E., Voronov T., “On Differential Operators Over a Map, Thick Morphisms of Supermanifolds, and Symplectic Micromorphisms”, Differ. Geom. Appl., 74 (2021), 101704
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