Oganes Khudaverdian (Joint Institute for Nuclear Research and UMIST)
khudian@umist.ac.uk The most powerful procedure when dealing with degenerate Lagrangians in Quantum Field Theory is the so-called BRST method (Becchi-Rouet-Stora-Tyutin). In the beginning of 1980s this procedure received a covariant Lagrangian formulation in terms of an odd Poisson bracket on the extended superspace of fields and antifields, the Batalin-Vilkovisky formalism. The generalization of the main ingredients of this formalism leads to a non-trivial "odd" analogue of the usual symplectic geometry. In odd symplectic superspace the canonically conjugate variables are of opposite parity.
We shall consider the basic invariant differential objects that arise in odd symplectic geometry. In particular, there is an odd analogue of the Laplace operator, which appears in the Batalin-Vilkovisky `Master Equation' DeS=0. We are going to describe a canonical odd invariant semidensity which can be considered as an exotic analogue of the classical Poincaré-Cartan integral invariant. It has also some amazing resemblance with the mean curvature of hypersurfaces in a Riemannian space.
In the talk we shall reveal the crucial role of semidensities in odd symplectic geometry. Developing differential calculus for semidensities we arrive at the general outlook of the previous results and obtain new invariants in higher order derivatives.
(The listeners are not expected to be experts in quantum physics or supermathematics.)