Let denote the BV phase space. In “physics notations”:

Given an embedding , each gives an “exact symmetry” . Let be the corresponding group, . Let us promote to the “extended” BV phase space, in the following way:

The space can be understood as the space of maps . Let and be the coordinates of . We define the cohomological vector field . The “denominator” in Eq. (39) consists of maps constant in (i.e. function of only).

Consider the coset space where the “denominator” in Eq. (39) consists of maps constant in (i.e. function of only). The algebra of functions on this coset space, with the differential is the Weil algebra of :

We define the “extended Master Action” as follows:

Here is the BV Hamiltonian of , in other words:

It satisfies the Master Equation . We assume that there is no anomaly, and satisfies the Quantum Master Equation. Essentially,

where is the differential of the Weil complex . Notice that:

— the -invariant subspace. Indeed the invariance under , , follows from the fact that the dependence on is through the exponential factor . Conceptually, the construction of equivariant half-density is very straightforward. Given a representation , we consider the BV Hamiltonian of the Weil differential in , and take its exponential. Notice that the differential is in the Cartan model, while the equivariant half-density of Eq. (41) is in the Weil model.

We observe:

In other words, this canonical transformation brings the action to the “direct sum” of (a function on ) and (a function on ).

We can now derive the integration prescription of 〚Averaging procedure using 〛. Let us put . This means that we consider a slightly smaller BV phase space, namely the second factor is just , and :

For any function on , such that (“unintegrated vertex operator”), the half-density satisfies the Quantum Master Equation:

where is the action of on . We will now explain that the half-density defines a closed PDF on . The construction goes as follows. It is enough to define, for any surface , the integral . We define it in the following way. We define the Lagrangian submanifold in , where and the corresponding antifield run over the odd conormal bundle of , and let run freely. We define as the integral of the half-density over this Lagrangian submanifold times a Lagrangian submanifold in . Eq (43) implies that is a closed form.

On the other hand, the canonical transformation of Eq. (42) implies that this is equivalent to:

This is Eq. (38).