Direct derivation of the comma 3-Vertex in the full string basis

TL;DR

This paper derives the interacting comma 3-vertex for bosonic open strings directly in the full string basis, showing its equivalence to Witten's 3-vertex without using coherent states.

Contribution

It provides a direct derivation of the comma 3-vertex in the full string basis, confirming its equivalence to Witten's 3-vertex.

Findings

Derived the comma 3-vertex using half string overlap relations

Established the equivalence of comma 3-vertex and Witten's 3-vertex

Avoided the use of coherent states in the derivation

Abstract

The interacting comma 3-vertex for the bosonic open string in the full string basis is derived using the half string overlap relations directly. Thus avoiding the coherent states technique employed in earlier derivations. The resulting form of the interacting 3-vertex turns out to be precisely the desired expression obtained in terms of the full string oscillator modes. This derivation establishes that the comma 3-vertex and Witten's 3-vertex are identical and therefore are interchangeable.