Ghost story. III. Back to ghost number zero

TL;DR

This paper explores the relationship between ghost number zero states and midpoint duals in string field theory, providing explicit spectral decompositions and revealing how the midpoint vertex simplifies the structure by avoiding noncommutativity.

Contribution

It introduces explicit relations and reconstruction formulas connecting gh=0 states and gh=3 midpoint duals, clarifying the structure of wedge states and the role of the midpoint vertex.

Findings

Derived explicit spectral decomposition for gh=0 Neumann matrices.

Identified the origin of noncommutativity due to imaginary poles in eigenvalues.

Showed how the midpoint vertex simplifies the structure by avoiding noncommutativity.

Abstract

After having defined a 3-strings midpoint-inserted vertex for the bc system, we analyze the relation between gh=0 states (wedge states) and gh=3 midpoint duals. We find explicit and regular relations connecting the two objects. In the case of wedge states this allows us to write down a spectral decomposition for the gh=0 Neumann matrices, despite the fact that they are not commuting with the matrix representation of K1. We thus trace back the origin of this noncommutativity to be a consequence of the imaginary poles of the wedge eigenvalues in the complex k-plane. With explicit reconstruction formulas at hand for both gh=0 and gh=3, we can finally show how the midpoint vertex avoids this intrinsic noncommutativity at gh=0, making everything as simple as the zero momentum matter sector.