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1 Conormal bundle
2 Non-degeneracy

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

    2 Non-degeneracy

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 Non-degeneracy

Let us ask the following question: under what conditions the restriction of to each is non-degenerate? 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 non-degenerate.

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:
  1. The space is zero, in other words is transverse to the orbits of . This is a constraint on the choice of

  2. 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