Microformal geometry and homotopy algebras

Microformal geometry is a new term. It refers to the new notion of microformal (or thick) morphisms of (super)manifolds, which generalize ordinary smooth maps. Thick morphisms act on smooth functions by pullbacks that unlike the familiar case are nonlinear transformations. Because of nonlinearity, they cannot be algebra homomorphisms as ordinary pullbacks. However, they are "nonlinear homomorphisms"; by that we mean a map of algebras such that its derivative at each point is an algebra homomorphism in the usual sense. This notion has originated from applications to homotopy Poisson structures, but should undoubtedly have more general significance. There are two parallel versions, "bosonic" and "fermionic". Bosonic thick morphisms are the classical limit of "quantum thick morphisms", which are special kind Fourier integral operators.

(Remark: there are multi-fold relations of all this activity with physics. Also, it is curious to note that an earlier version of the future Fourier integral operators was introduced by V. Fock -- the Fock of Fock space -- in 1959 who was trying to make precise Dirac's remark about a connection between canonical transformations in classical mechanics and unitary transformation in quantum mechanics, see Viestnik of Leningrad University №16, p. 67, 1959. Dirac discovered many things! There is one algebraic identity that I found, between commutator and Poisson bracket, $[a,b]f\{c,d\}=\{a,b\}f[c,d]$, see my paper Poisson envelope of a Lie algebra. "Noncommutative moment space" Funct. Anal. Appl. 29:3 (1995), 196-199, which -- as I have learned later, -- was previously found by Dirac without f in the middle, see either his book on Principles of quantum mechanics, or Fock's Foundations of quantum mechanics, who used it, with the physicist's type of logic, to claim the proportionality between Poisson bracket and commutator of operators in quantum mechanics! Which is of course wrong mathematically. So this identity remains as an important identity for non-commutative Poisson algebras. I recently learned that such identities are interesting for people doing operads and general theory of algebraic operations.)

This is a completely new area of research that excites me enormously!

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Last modified: 19 September (2 October) 2019 Ted Voronov