Special properties of

The string form defined by Eq. (3) is very special. Besides it satisfies, for any , the following special properties:


exists :


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:


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: