Lightning review of BV formalism
The basic object is:
supermanifold with an odd nondegenerate closed 2form (like )
is called BV phase space
the only difference with Classical Mechanics is that is odd
the group of canonical transformations will be denoted , and its Lie superalgebra
As in Classical Mechanics, is generated by Hamiltonians. Since is odd:
Definition 2: A submanifold is Lagrangian if it is isotropic (i.e. ) and cannot be extended keeping this property ( in finitedimensional case).
Important role is played by halfdensities on .


so we can integrate a halfdensity on over a Lagrangian submanifold; we just write:
Definition 4: there is a “canonical operator”
acting on halfdensities on which has the following characteristic
property. For any function , when a Lagrangian submanifold is carried by the Hamiltonian flux of
, the velocity of change of is:
It follows from the defining Eq. (3) that when satisfies the Master Equation,
does not change under a small deformation of .