Summary of BV

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