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Microformal geometry and homotopy algebras
Differential Geometry · Submitted 25 Nov 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. | contribution 1.00 | 0.95 | 0.09 | asserted 1.00 |
| s1 | They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. | method 0.66 | 0.72 | 0.04 | asserted 1.00 |
| s2 | The result is a formal category so that its composition law is also specified by a formal power series. | contribution 0.45 | 0.80 | 0.06 | asserted 1.00 |
| s3 | A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. | contribution 0.55 | 0.65 | 0.07 | asserted 0.99 |
| s4 | 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. | contribution 0.44 | 0.65 | 0.06 | asserted 1.00 |
| s5 | This suggests an abstract notion of a "nonlinear algebra homomorphism" and the corresponding extension of the classical "algebraic-functional" duality. | interpretation 1.00 | 0.52 | 0.11 | hedged possibility 0.99 |
| s6 | There is a parallel fermionic version. | contribution 0.68 | 0.36 | 0.02 | asserted 0.95 |
| s7 | 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. | result 0.56 | 0.86 | 0.38 | asserted 1.00 |
| s8 | We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. | contribution 0.66 | 0.93 | 0.09 | asserted 0.98 |
| s9 | 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. | result 0.91 | 0.94 | 0.17 | asserted 1.00 |
| s10 | We apply this construction to higher Koszul brackets on differential forms and to triangular $L_{\infty}$-bialgebroids. | contribution 0.92 | 0.94 | 0.07 | asserted 0.99 |
| s11 | 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. | contribution 0.95 | 0.90 | 0.07 | asserted 0.98 |
| s12 | 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. | result 0.96 | 0.94 | 0.14 | asserted 1.00 |
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Original 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.
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4 arXiv papers cite this in 14 sentences
Papers citing it · 10 sentences without a confident label
Uses its method or tool 1 paper
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This is achieved by a formal version of the stationary phase formula [ref] .
Graded Geometry, $Q$-Manifolds, and Microformal Geometry · Quantum pullbacks and quantum thick morphisms · 0.94
Cites it as background 2 papers
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The key references are: [ref] , [ref] , [ref] for main ideas and constructions, also [ref] ; and [ref] and [ref] for further development and applications.
Graded Geometry, $Q$-Manifolds, and Microformal Geometry · Microformal geometry. Classical thick morphisms · 1.00
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In 2014 Ted Voronov introduced the notion of a thick morphism (see [ref] , [ref] ) of manifolds, which generalises ordinary maps.
Non-linear homomorphisms of algebras of functions are induced by thick morphisms · Introduction · 0.99
What it cites · 1 sentence without a confident label
Cites as background 1 paper
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In a remarkable series of papers [ref] , see also [ref] and [ref] , Cattaneo, Dherin and Weinstein developed a functorial framework for the deformation quantization of Poisson manifolds, based on the consideration of germs of symplectic manifolds near Lagrangian submanifolds.
Integration of Lie Algebroid Comorphisms · Introduction · 0.98
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