Family of Lagrangian submanifolds
Here we will describe a family of Lagrangian submanifolds which is a generalization of the construction used in bosonic string theory
1 Conormal bundle
Let be some family of submanifolds closed under the action of the gauge symmetry
“closed under the action” means that if then for any gauge transformation ,
For each , the odd conormal bundle of (denoted ) is a subbundle
of the odd cotangent bundle which consists of those covectors which evaluate to zero
on vectors tangent to . For each , the corresponding odd conormal bundle is a
Lagrangian submanifold. Given such a family , let us define a family of Lagrangian
submanifolds in the BV phase space in the following way: for every , the corresponding
Lagrangian submanifold is the odd conormal bundle of , times the space of ghosts:
2 Nondegeneracy
Let us ask the following question: under what conditions
the restriction of to each is nondegenerate?
Or, in case if it is degenerate, how can we characterize the degeneracy? We have:
The second term is the evaluation of the covector on the tangent vector .
Let us assume that the restriction of to any has a critical
point, and study the quadratic terms in the expansion of around a critical point.
To define the perturbation theory, we need already the quadratic terms to be nondegenerate.
Assuming that the critical point is at :
Suppose that all degeneracies of and of are due to symmetries. In other words:
Let be the quadratic part of :
The degeneracy is characterized by the isotropic subspace of which we denote :

Let us make the following assumptions:
The space is zero, in other words is transverse to the orbits of . This is a constraint on the choice of
The next term, , is also zero. This kernel being nonzero corresponds to “reducible” gauge symmetries.
But the last term is essentially nonzero.
It can be identified with the cotangent space to our family:
where is the moduli space of submanifolds . Therefore the quadratic part of
is degenerate. However this degeneration is removed by
the factor
Indeed, in this case:
When we integrate over , the differentials span the complement of in
. Since we require that the family be closed, defines a map
which we denote . With these notations:
In
the case of bosonic string
and contributes