Batalin–Vilkovisky formalism and integration theory on manifolds

Abstract

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.

Semidensities on Odd Symplectic Supermanifolds

“…In terms of semidensities BV master equation (1.3b) gets an invariant formulation and the difference between conditions (1.3a, b, c) can be formulated exactly. (We note that in papers [17] and [24] was stated that conditions (1.3a), (1.3b) and (1.3c)) are equivalent, in spite of the fact that difference between these conditions was formulated in non-explicitly way in Theorem 5 of the paper [24]. )…”

“…Let us shortly sketch the results of [13,16,17,24]. If an odd symplectic supermanifold is provided with a volume form dv, then one can consider operator ∆ dv such that its action on a function on this supermanifold is equal (up to a coefficient) to the divergence of the Hamiltonian vector field corresponding to this function w.r.t.…”

“…The geometrical meaning of these objects and interpretation of BV master equation in its terms were studied in [13,16,17] and most notably by A.S.Schwarz in [24].…”

“…(see [16,24,17] supermanifold ΠT * M provided with a volume form dv = dv 2 satisfies conditions (1.1) and (1.2). In this case the action of operator ∆ dv on function on ΠT * M corresponds to the divergence operator on multivector fields on M .…”

“…In general, for the new volume form dv ′ neither condition (1.1) in some Darboux coordinates, nor condition (1.2) are true. The main essence of the geometrical formulation of BV formalism can be shortly expressed in the following two statements [16,24,17]:…”