On complexes related with calculus of variations

We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a particular new class of forms. We establish relation of the complex of variational derivatives and the variational complex. Certain calculus of Lagrangians of multidimensional paths is developed. It is shown how covariant Lagrangians of higher order can be used to represent characteristic classes.Comment: LaTeX2e, 36 page

Geometry of differential operators, and odd Laplace operators

We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary weights appear naturally in the course of study. We show that for the algebra of densities it is possible to establish a one-to-one correspondence between operators and brackets generated by them. Everything is applicable to supermanifolds as well as to usual manifolds. In the super case the problem is closely connected with the geometry of the Batalin--Vilkovisky formalism in quantum field theory, namely the description of the generating operators for an odd bracket. We give a complete answer. This text is a concise outline of the main results. A detailed exposition is in \texttt{arXiv:math.DG/0212311}.Comment: LaTeX2e; 3 pages; Russian and English versions availabl

On odd Laplace operators

We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This includes earlier results for odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on $M$ is partitioned into orbits by a natural groupoid whose arrows correspond to the solutions of the quantum Batalin--Vilkovisky equations. We give a comparison with the situation for Riemannian and even Poisson manifolds. In particular, the square of odd Laplace operator happens to be a Poisson vector field defining an analog of Weinstein's ``modular class''.Comment: LaTeX2e, 18p. Exposition reworked and slightly compressed; we added a table with a comparison of odd Poisson geometry with Riemannian and even Poisson cases. Latest update: minor editin

On generalized symmetric powers and a generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory

The classical Kolmogorov-Gelfand theorem gives an embedding of a (compact Hausdorff) topological space X into the linear space of all linear functionals C(X)^* on the algebra of continuous functions C(X). The image is specified by algebraic equations: f(ab)=f(a)f(b) for all functions a, b on X; that is, the image consists of all algebra homomorphisms of C(X) to numbers. Buchstaber and Rees have found that not only X, but all symmetric powers of X can be embedded into the space C(X)^*. The embedding is again given by algebraic equations, but more complicated. Algebra homomorphisms are replaced by the so-called "n-homomorphisms", the notion that can be traced back to Frobenius, but which explicitly appeared in Buchstaber and Rees's works on multivalued groups. We give a further natural generalization of Kolmogorov-Gelfand-Buchstaber-Rees theory. Symmetric powers of a space X or of an algebra A are replaced by certain "generalized symmetric powers" Sym^{p|q}(X) and S^{p|q}A, which we introduce, and n-homomorphisms, by the new notion of "p|q-homomorphisms". Important tool of our study is a certain "characteristic function" R(f,a,z), which we introduce for an arbitrary linear map of algebras f, and whose functional properties with respect to the variable z reflect algebraic properties of the map f.Comment: LaTeX, 7 pages (3+4). In this new version we slightly edited the main text, and added to it an Appendix giving details of some constructions and a short direct proof of Buchstaber--Rees's main theore

Geometric constructions on the algebra of densities

The algebra of densities \Den(M) is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra \Den(M) is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra \Den(M). It allows a natural definition of bracket operations on vector densities of various weights on a (super)manifold $M$, similar to how the classical Fr\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from "vector fields" (derivations) on \Den(M) to "multivector fields". This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.Comment: LaTeX, 23 p

A short proof of the Buchstaber-Rees theorem

We give a short proof of the Buchstaber-Rees theorem concerning symmetric powers. The proof is based on the notion of a formal characteristic function of a linear map of algebras.Comment: 11 pages. LaTeX2