Witten's Ghost Vertex Made Simple (bc and bosonized ghosts)

TL;DR

This paper simplifies the diagonalization of ghost vertices in string field theory, confirming eigenvalues and verifying key algebraic relations for combined matter and ghost systems, emphasizing the importance of proper normalization.

Contribution

It provides a simplified method for diagonalizing ghost vertices and verifies fundamental relations, introducing the normalization factor Z_N related to the partition function.

Findings

Eigenvalues agree with previous results

Verified descent and associativity relations

Identified the normalization factor Z_N

Abstract

First, we diagonalize the bc-ghost 3-string Neumann matrices using the technique described in hep-th/0304158. Their eigenvalues are in complete agreement with the previous authors. Second, we diagonalize the N-string gluing vertices for the bosonized ghost system. And third, we verify the descent and associativity relations for the combined bosonic matter+ghost gluing vertices. We find that in order for these relations to be true, the vertices must be normalized by the factor Z_N. Here Z_N is the partition function of the bosonic matter+ghost CFT on the gluing surface, which is the unit disc with the Neumann boundary conditions and the midpoint cone like singularity specifying by the angle excess \pi(N-2).