← prev  up  next → 

General properties of string worldsheet theory

When does a sigma-model qualify as a string worldsheet theory? The usual answer is that the action should be invariant under a nilpotent symmetry , and the energy-momentum tensor should be exact: .

But when we consider the pure spinor superstring, introducing a metric in the worldsheet theory appears somewhat unnatural. (Given that we can consider an arbitrary BRST-exact deformation of the worldsheet action, why should we specifically restrict ourselves to considering the variation of the metric only?) This would suggest, that in the pure spinor formalism the energy-momentum tensor and the -ghost should not be considered fundamental concepts. But which objects should we consider instead?

Let us introduce the concept of pseudosymmetry. In some field theory, suppose that the action depends on some coupling constants. Suppose that we have a field transformation which is not strictly speaking a symmetry of the action, but becomes a symmetry if we allow it to also act on those coupling constants

Such “pseudosymmetries” do not lead to conserved charges. However, they often lead to interesting Ward identities.

Let us cross-breed this concept with the BRST structure in the following way: consider only those pseudosymmetries whose corresponding variations of the coupling constants amount to adding BRST-exact terms. If is only nilpotent on-shell, then such a definition of the pseudosymmetries will strictly speaking fail, as they would not form a group. However, the BV formalism comes to resque and provides the correct definition as symmetries of the BV action not necessarily preserving the chosen Lagrangian submanifold.

We propose to define the pure-spinor-like string worldsheets as sigma-models equipped with the following additional data:

  1. nilpotent symmetry (this is a strict symmetry, not pseudo!) and

  2. action of the group of diffeomorphisms of the worldsheet as pseudosymmetries

  3. (requires BV) the BV Hamiltonian of an infinitesimal diffeomorphism (= vector field) is of the form

  4. (requires BV) for two vector fields and :

Notice that the -ghost is not on the list. (However, the last axiom resembles the nilpotency of the -ghost.)

Our set of properties is motivated by the necessity to have closed cycles in the moduli space of Lagrangian submanifold. It seems probable that closed cycles only exist if we can consider the factorspace by diffeomorphisms. Therefore, we should have diffeomorphisms. Moreover, the construction of the integration measure on the moduli space of Lagrangian submanifold does not requires specifically the -ghost. However, we do want it to be equivariant with respect to diffeomorphisms; this requires our th axiom.

Another reason to want diffeomorphisms is that they justify the concept of unintegrated vertex operators.

Suppose that we are given the “usual” data: . Can we construct our data from this standard data? We do not have a complete answer to this question. In the special case when off-shell, it turns out that the -ghost provides a clue on how to construct the action of diffeomorphisms. We must stress, however, that in the pure spinor formalism only on-shell.

    ← prev  up  next →