On the Generalized Gluing and Resmoothing Theorem

TL;DR

This paper simplifies the proof of the generalized gluing and resmoothing theorem in string field theory, extending it from tree level to one-loop level and highlighting the importance of a sign factor for conformal field theory correlations.

Contribution

It provides a simplified proof for the tree level theorem and explicitly proves the extended version at one-loop level, including the crucial sign factor.

Findings

Simplified proof for the tree level theorem.

Explicit proof of the one-loop level extension.

Identification of a sign factor essential for conformal field theory.

Abstract

The generalized gluing and resmoothing theorem originally proved by LeClair, Peskin and Preitschopf, gives a powerful formula for the fused vertex obtained by contracting any two vertices in string field theories. Although the theorem is naturally expected to hold for the vertices at any loop level, the original proof was restricted to the vertices at tree level. Here we present a simplified proof for the tree level theorem and then prove explicitly the extended version at one-loop level. We also find that a non-trivial sign factor, which depends on the string states to be contracted, appears in the theorem. This sign factor turns out to be essential for reproducing correctly the conformal field theory correlation function on the torus.