On this page:
1 Definitions
2 Expansion of in powers of
    ← prev  up  next → 

Equivariant half-densities

    1 Definitions

    2 Expansion of in powers of

1 Definitions

Suppose that we have the following data:

  • A subgroup of the group of canonical transformations of an odd symplectic manifold ; the Lie algebra of consists of vector fields of the form where runs over Hamiltonians of elements of .

  • A map

    (index C is for “Cartan model”)

    such that:

    (20)

    (21)

Then the following equation is a cocycle of the -equivariant Cartan complex on :

This is proven using the interpretation of as an intertwining operator.

Notice the compatibility of Eqs. (20) and (21):

Here we used the relation between and Lie derivative. also, since is even, is odd and therefore .

It seems that the main ingredient in this approach is the choice of a group . (In the case of bosonic string this is the group of diffeomorphisms.) The rest of the formalism is built around . We suspect that is more or less unambiguously determined by the choice of ; even is already unambigously determined. Indeed, we will now see that the constraints arizing from Eqs. (20) and (21) are very tight.

2 Expansion of in powers of

Let us expand in powers of :

we use angular brackets to emphasize linear depnendence: is some linear function of

Eq. (21) implies that satisfies the Master Equation. In this Section we will use to define the odd Laplace operator on functions:

Eq. (20) implies that is -invariant; using the relation between and Lie derivative we derive:

Therefore the Hamiltonians generating should be all -closed. Moreover, Eq. (21) implies:

so they are actually all -exact. Eq. (20) implies:

(22)

The image of , as a linear statistics-reversing map , will be called . Notice that the inverse map to is . To summarize:

    ← prev  up  next →