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1 BRST analysis
2 BV analysis
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BV Hamiltonian Of Diffeomorphisms

Here we will discuss the BV Hamiltonian corresponding to diffeomorphisms.

    1 BRST analysis

    2 BV analysis

1 BRST analysis

It was shown in BerkovitsMazzucato that this composite -ghost is holomorphic on-shell modulo BRST exact terms; in other words:

(7)

where is some functional and is some field transformation. Eq. (7) defines up to:

(8)

with some antisymmetric .

We are tempted to say that is well-defined on-shell. However, in our situation does not preserve the equations of motion. Therefore, we would not be able to type it as a vector field on the classical phase space. The best thing we can say about is that it is a vector field on the space of fields defined up to an ambiguity of the form (8).

In some sense, is the analogue of in bosonic string.

Tentative Definition 1: we will tentatively define the action of diffeomorphisms on the worldsheet sigma-model as follows:

(9)

— the anticommutators of two field transformations. This definition automatically implies:

(10)

There are two problems here:

  • in the pure spinor formalism, only on-shell; therefore the validity of (10) as a motivation for (9) is dubious

  • the ambiguity (8) translates into some ambiguity in and in particular precludes us from even asking “”

A possible naive argument: “the energy-momentum generates diffeomorphisms; but therefore generates satisfying (9)”. However it is not clear to us in which sense the energy-momentum tensor generates diffeomorphisms.

2 BV analysis

We are tempted to define the BV Hamiltonian of a diffeomorphism as follows:

(11)

Here stands for the terms quadratic and higher order in the antifields. We can change:

(If starts with the terms quadratic in the antifields, this corresponds to Eq. (8)). We should be able to fix this ambiguity so that the resulting satisfy the algebra of vector fields on the worldsheet:

Open problem: how do we see that such a choice of exists?

Eqs. (11) and (9) suggest that the first two terms in the antifield expansion of are:

(12)

(we did not explicitly compute ). This would be analogous to the generator of diffeomorphisms for topologically twisted theories.

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