Baranov-Schwarz

← prev up next →

Partial Baranov-Schwarz transform of

As usual, let

be the group of canonical transformations. Consider a submanifold

satisfying the following condition:

(1)

Here

means the derivative of the right shift.

For example, this is satisfied when is one dimensional, or when is a subgroup of .

Pick a Lagrangian submanifold

and consider the family of Lagrangian submanifolds:

For every

consider a submanifold

which is the common zero set of all Hamiltonians of elements of

restricted on

. It turns out that the union:

is a Lagrangian submanifold in

. Moreover:

Notice that any family of Lagrangian submanifolds can locally be considered as an orbit of some Lagrangian submanifold

by an abelian subgroup of

(the subgroup generated by gauge fermions). Therefore we can always assume that the condition (1) is satisfied. We conclude that:

Integration over a family of Lagrangian submanifolds is equivalent to the integration over a single "rotated" Lagrangian submanifold

Let us consider the -form component of . We have to integrate it over some -dimensional family of Lagrangian submanifolds. We parametrize the family by . Let us perform the Baranov-Schwarz transform by integrating over . This turns into an integral form. We get:
(2)
(Remember that we identify with the actual Hamiltonian, i.e. for us is a function on , and in particular a function on .) This means that actually we are integrating not over the whole Lagrangian submanifold , but over a submanifold of the codimension , which is defined by the system of equations:
But on the other hand, remember that we have to integrate over the family of the dimension . In other words, we lost integrations by inserting the -functions (2), but then regained them as integration over the family. This is equivalent to the “90 degree rotation” :