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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

Definition 1: Symmetries of are called canonical transformations.
the group of canonical transformations will be denoted , and its Lie superalgebra
As in Classical Mechanics, is generated by Hamiltonians. Since is odd:

    ←    

each function on defines

an infinitesimal canonical transformation

(2)

Definition 2: A submanifold is Lagrangian if it is isotropic (i.e. ) and cannot be extended keeping this property ( in finite-dimensional case).

Important role is played by half-densities on .

Definition 3: For every Lagrangian submanifold , there is a natural restriction:

half-densities on   

volume elements 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:

(3)

Definition 5: The Quantum Master Equation is the following equation on a half-density :

(4)

It follows from the defining Eq. (3) that when satisfies the Master Equation, does not change under a small deformation of .

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