arXiv subjects
Publications and source records attributed to O. M. Khudaverdian.
The action of Batalin-Vilkovisky Delta-operator on semidensities in an odd symplectic superspace is defined. This is used for the construction of integral invariants on surfaces embedded in an odd symplectic superspace and for more clear interpretation of the Batalin-Vilkovisky formalism geometry.
For a given configuration space $M$ and Lie algebra $g$ whose action is defined on $M$, the space $V_{0.0}$ of weakly $g$-invariant Lagrangians (i.e. Lagrangians whose motion equations left hand sides are $g$-invariant) is studied. The problem is reformulated in the terms of the double complex of Lie algebra cochains with values in the complex of Lagrangians. Calculating the cohomology of this complex using the method of spectral sequences, we come to the hierarchy in the space $V_{0.0}$: The double filtration $V_{s.r}$ ($s=0,1,2,3,4;r=0,1$) and the homomorphisms on every space $V_{s.r}$ are constructed. These homomorphisms take values in cohomologies of the Lie algebra $g$ and configuration space $M$. On one hand every space $V_{s.r}$ is the kernel of the corresponding homomorphism, on the other hand this space is defined by its physical properties.
The divergence-like operator on an odd symplectic superspace which acts invariantly on a specially chosen odd vector field is considered. This operator is used to construct an odd invariant semidensity in a geometrically clear way. The formula for this semidensity is similar to the formula of the mean curvature of hypersurfaces in Euclidean space.
It is a review of some results in Odd symplectic geometry related to the Batalin-Vilkovisky Formalism
The correspondence between the BV-formalism and integration theory on supermanifolds is established. An explicit formula for the density on a Lagrangian surface in a superspace provided with an odd symplectic structure and a volume form is proposed.
We introduce some new classes of algebras and estabilish in these algebras Campbell--Hausdorff like formula. We describe the application of these constructions to the problem of the connectivity of the Feynman graphs corresponding to the Green functions in Quantum Fields Theory.
An invariant definition of the operator $Δ$ of the Batalin-Vilkovisky formalism is proposed. It is defined as the divergence of a Hamiltonian vector field with an odd Poisson bracket (antibracket). Its main properties, which follow from this definition, as well as an example of realization on Kählerian supermanifolds, are considered. The geometrical meaning of the Batalin-Vilkovisky formalism is discussed.