Ghost story. II. The midpoint ghost vertex

TL;DR

This paper constructs and analyzes midpoint ghost vertices in open string field theory, revealing spectral properties and confirming their role in defining the star product with ghost insertions at the midpoint.

Contribution

It introduces two explicit midpoint ghost vertices, analyzes their spectral properties, and confirms their consistency with the star product in open string field theory.

Findings

Neumann matrices commute among themselves and with matrix G

Spectral formulas for Neumann matrices are derived

Vertices correctly reproduce the expected wedge states with midpoint ghost insertions

Abstract

We construct the ghost number 9 three strings vertex for OSFT in the natural normal ordering. We find two versions, one with a ghost insertion at z=i and a twist-conjugate one with insertion at z=-i. For this reason we call them midpoint vertices. We show that the relevant Neumann matrices commute among themselves and with the matrix $G$ representing the operator K1. We analyze the spectrum of the latter and find that beside a continuous spectrum there is a (so far ignored) discrete one. We are able to write spectral formulas for all the Neumann matrices involved and clarify the important role of the integration contour over the continuous spectrum. We then pass to examine the (ghost) wedge states. We compute the discrete and continuous eigenvalues of the corresponding Neumann matrices and show that they satisfy the appropriate recursion relations. Using these results we show that the formulas for our vertices correctly define the star product in that, starting from the data of two ghost number 0 wedge states, they allow us to reconstruct a ghost number 3 state which is the expected wedge state with the ghost insertion at the midpoint, according to the star recursion relation.