String Field Theory Projectors for Fermions of Integral Weight

TL;DR

This paper develops a framework for fermionic string field theory projectors using the Moyal basis, classifies all such projectors, and explores their properties, including their relation to surface states and the structure of the star product.

Contribution

It introduces a classification of fermionic projectors in string field theory using the Moyal basis and generalizes the Moyal-Weyl map to the fermionic case, revealing new insights into surface states and star products.

Findings

Classified all fermionic projectors factorizing into ppa-subspaces.

Recovered the full set of bosonic projectors for squeezed states.

Identified a class of ghost number-homogeneous projectors including generalized butterfly states.

Abstract

The interaction vertex for a fermionic first order system of weights (1,0) such as the twisted bc-system, the fermionic part of N=2 string field theory and the auxiliary \eta\xi system of N=1 strings is formulated in the Moyal basis. In this basis, the Neumann matrices are diagonal; as usual, the eigenvectors are labeled by \kappa\in\R. Oscillators constructed from these eigenvectors make up two Clifford algebras for each nonzero value of \kappa. Using a generalization of the Moyal-Weyl map to the fermionic case, we classify all projectors of the star-algebra which factorize into projectors for each \kappa-subspace. At least for the case of squeezed states we recover the full set of bosonic projectors with this property. Among the subclass of ghost number-homogeneous squeezed state projectors, we find a single class of BPZ-real states parametrized by one (nearly) arbitrary function of \kappa. This class is shown to contain the generalized butterfly states. Furthermore, we elaborate on sufficient and necessary conditions which have to be fulfilled by our projectors in order to constitute surface states. As a byproduct we find that the full star product of N=2 string field theory translates into a canonically normalized continuous tensor product of Moyal-Weyl products up to an overall normalization. The divergent factors arising from the translation to the continuous basis cancel between bosons and fermions in any even dimension.