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Partial integration of half-density

When some fields are integrated out, we get an “effective action” for the remaining fields.

Let be an odd symplectic supermanifold, and be a half-density.

Suppose that we are given a submanifold which happens to be a fiber bundle with fibers isotropic submanifolds over some base . This can be thought of as a family of isotropic submanifolds. For every we have the corresponding fiber, an isotropic submanifold which we denote . Moreover, we require:

(i.e. the degenerate subspace of the restriction of to is the tangent space of the fiber). Since , this condition implies that the Lie derivative of along the fiber vanishes, and therefore defines an odd symplectic form on the base which we will denote .

Finally, we require certain maximality property:

cannot be embedded in any larger submanifold of where would still be isotropic

(1)

Let be a Lagrangian submanifold of . Under the above conditions, it can be lifted to a Lagrangian submanifold in as , where is a natural projection:

We then define a half-density on so that for every Lagrangian submanifold and every function :

Suppose that we are given a function on whose restriction on is constant along . Then it defines a function on which we denote :

Notice that in this case the flux is tangent to .

It is enough to prove that for any constant on : . From the maximality of , as defined in Eq. (1), follows that is tangent to . Since is constant along , it follows that indeed .

We have:

(2)

Equivalently, for two functions and on both constant along :

In order to prove Eq. (2), notice that any vector field tangent to can be written as where are some functions on and are some functions on constant on . The commutator with of such a vector field is — again tangent to . This means that the flow of brings fibers to fibers, which is equivalent to Eq. (2).

We will now prove that for any whose restriction on is constant along :

(3)

Indeed, for any “test function” :

(4)

(5)

Equality of Lines (4) and (5) implies Eq. (3). In particular:
if satisfies the Quantum Master Equation on , then satisfies the Quantum Master Equation on
Indeed, satisfying the Quantum Master Equation is equivalent to the statement that for any functions and :

(6)

When satisfies the QME on , considering Eq. (6) with both and constant along the fiber of and using Eq. (3) proves that also satisfies the QME.

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