BV from BRST
1 Definition of the BV phase space
The BV phase space is:
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The zero section 
is a Lagrangian submanifold. It comes with the integration measure,
which lifts to a half-density on 
of the form 
, where:
is a Lagrangian submanifold. It comes with the integration measure,
which lifts to a half-density on of the form , where:
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where 
is as defined in Eq. (1).
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:
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:
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Notice that in the BRST case 
coincides with 
.
coincides with .