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
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:
, 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:
to each is non-degenerate?
Or, in case if it is degenerate, how can we characterize the degeneracy? We have:
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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 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 
:
:
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Suppose that all degeneracies of 
and of 
are due to symmetries. In other words:
and of are due to symmetries. In other words:
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Let 
be the quadratic part of :
be the quadratic part of :
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The degeneracy is characterized by the isotropic subspace of which we denote 
:
which we denote :
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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:
is essentially nonzero.
It can be identified with the cotangent space to our family:
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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:
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:
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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:
, the differentials span the complement of in
. Since we require that the family be -closed, defines a map
which we denote . With these notations:
