On this page:
1 PDF on the group of canonical transformations
2 PDF on the space of Lagrangian submanifolds
3 PDF on an equivalence class of actions

Summary

We want to define some pseudo-differential form which can serve as string measure.

There are several closely related definitions:

    1 PDF on the group of canonical transformations

    2 PDF on the space of Lagrangian submanifolds

    3 PDF on an equivalence class of actions

1 PDF on the group of canonical transformations

Let denote the central extension of the algebra of Hamiltonian vector fields by constant Hamiltonians, and the corresponding central extension of the group of canonical transformations.

Possible subtlety: it is not clear to us if this central extension exists globally; for now let us assume that we are working in the vicinity of the unit element of

We will start by constructing this version of . The definition actually depends on the choice of a fixed Lagrangian submanifold . Strictly speaking, we can characterize as a map of the following type:

(1)

(which associates a PDF on to every Lagrangian submanifold).

2 PDF on the space of Lagrangian submanifolds

There is a natural map:

coming from the action of on . A natural question is, does descend to a PDF on ? The answer is essentially “yes”, although some minor modifications are needed. Essentially:

is a PDF on the space of Lagrangian submanifolds

However, does not have interesting integration cycles. In order to get interesting integration cycles, we have to also descend to the factorspace of over some symmetries.

3 PDF on an equivalence class of actions

In BV formalism the choice of a Lagrangian submanifold is closely related to the choice of a quantization scheme. In other words, it is essentially the choice of a representative in a class of physically equivalent theories. Given and , the restriction gives a physical action functional which we use in the path integral. A different choice of gives a BRST equivalent action functional.

Therefore it would be natural to try to interpret as a PDF on such an equivalence class. This, however, is not straightforward. The space of Lagrangian submanifolds is actually larger than the space of action functionals; to descend to the space of action functionals one has to take the factorspace over the symmetries of . Generally speaking, does not descend to this factorspace.