BV from BRST

#### 1` `Definition of the BV phase space

The BV phase space is:

The zero section is a Lagrangian submanifold. It comes with the integration measure,
which lifts to a half-density on of the form , where:

where is as defined in Eq. (1).

We can imagine a more general situation when we have a functional with an odd symmetry nilpotent off-shell. But, just to describe the “standard BRST formalism”, we explicitly break the fields into and .

#### 2` `Lifting the symmetries to BV phase space

We have realized the gauge algebra as symmetries of the BRST configuration space
as left shifts on .
Since the BV phase space is the odd cotangent bundle, we can further lift them to the BV phase space.
The symmetry corresponding to the infinitesimal left shift (2) is generated by the BV Hamiltonian:

Notice that in the BRST case coincides with .