Lightning review of BV formalism
The basic object is:
supermanifold with an odd nondegenerate closed 2-form (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:
Important role is played by half-densities on .
so we can integrate a half-density on over a Lagrangian submanifold; we just write:
Definition 4: there is a “canonical operator” acting on half-densities 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 .