Odd symplectic geometry in the BV-formalism

Abstract: Odd symplectic geometry was considered by physicists as an exotic counterpart of even symplectic geometry. Batalin and Vilkovisky changed this point of view by the seminal work considering the quantisation of general theory in Lagrangian framework, where they considered odd symplectic superspace of fields and antifields. [In the case of Lie group of symmetries BV receipt is reduced to the standard Faddeev-Popov method.]

The main ingredient of the theory, the exponent of the master action, is defined by the function $f$ such that $\Delta f=0$, where $\Delta$ is second order differential operator of the second order: $\Delta=\frac{\partial^2}{\partial x^i \partial\theta_i}$, ($x^i,\theta_j$ are the Darboux coordinates of an odd symplectic superspace.) This operator has no analogy in the standard symplectic geometry.

I consider in this talk the main properties of the BV-formalism geometry.

The $\Delta$-operator is defined in geometrical clear way, and this operator depends on the volume form.

It is suggested the canonical operator $\Delta$ on half-densities. This operator is the proper framework for BV geometry. We also study the groupoid property of BV master-equation; this leads us to the notion of BV groupoid. We also discuss some constructions of invariants for odd symplectic structure.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

( video )

Geometry of differential equations seminar