Special properties of

The string form

defined by Eq. (3) is very special. Besides

it satisfies, for any

, the following special properties:

(4)

exists :

(5)

where

corresponds to the BV Laplacian w.r.to the half-density

. Eqs (4) and (5) are very special. They imply:

Therefore, the equivariant analogue of

in the Cartan model can be constructed as follows:

where:

The construction of the Cartan equivariant form is recast as a construction of a representation of

in functions on BV phase space. In other words, we need a morphism

(where

is the Lie superalgebra of infinitesimal BV-canonical transformations) such that

agrees with

(remember

is the same as

). We denote it like this: