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

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

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/

Lattice Study of the Extent of the Conformal Window in Two-Color Yang-Mills Theory

We perform a lattice calculation of the Schr\"odinger functional running coupling in SU(2) Yang-Mills theory with six massless Wilson fermions in the fundamental representation. The aim of this work is to determine whether the above theory has an infrared fixed point. Due to sensitivity of the $SF$ renormalized coupling to the tuning of the fermion bare mass we were unable to reliably extract the running coupling for stronger bare couplings