Dual Forms on Supermanifolds and Cartan Calculus

The complex of "stable forms" on supermanifolds is studied. Stable forms on $M$ are represented by certain Lagrangians of "copaths" (formal systems of equations, which may or may not specify actual surfaces) on $M\times\mathbb R^D$. Changes of $D$ give rise to stability isomorphisms. The Cartan--de Rham complex made of stable forms extends both in positive and negative degree and its positive half is isomorphic to the complex of forms defined as Lagrangians of paths. Considering the negative half is necessary, in particular, for homotopy invariance. We introduce analogs of exterior multiplication by covectors and of contraction with vectors. We find (anti)commutation relations for them. An analog of Cartan's homotopy identity is proved. Before stabilization it contains some "stability operator" $\sigma$.Comment: 21 pages, LaTeX, uses package diagrams.sty (= diagrams.tex) by Paul Taylor, available at ftp://ctan.tug.org/tex-archive/macros/generic/diagrams/taylor/ (or at any_mirror_of_CTAN/tex-archive/macros/generic/diagrams/taylor/

Graded manifolds and Drinfeld doubles for Lie bialgebroids

We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, etc. I give a construction of \textit{doubles} for \textit{graded} $QS$- and \textit{graded $QP$-manifolds} (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded $QS$-manifolds can be considered, roughly, as ``generalized Lie bialgebroids''. The double for them is closely related with the analog of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg. Lie bialgebroids as a generalization of Lie bialgebras, over some base manifold, were defined by Mackenzie and P. Xu. Graded $QP$-manifolds give an {odd version} for all this, in particular, they contain ``odd analogs'' for Lie bialgebras, Manin triples, and Drinfeld's double.Comment: LaTeX2e, 38 pages. Latest update (November 2002): text reworked, certain things added; this is the final version as publishe

Microformal geometry and homotopy algebras

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.Comment: LaTeX 2e. 47 p. Some editing of the expositio

Thick morphisms of supermanifolds and oscillatory integral operators

We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions