Multiloop String Amplitudes with B-Field and Noncommutative QFT

TL;DR

This paper develops a method to compute multiloop string amplitudes with a constant B-field, revealing how noncommutative field theories emerge from string theory in the field theory limit.

Contribution

It introduces a simplified way to incorporate B-field effects into multiloop string amplitudes using modified Reggeon vertices, connecting string theory with noncommutative QFT.

Findings

Derived the open string Green function on the annulus with B-field.

Modified the three Reggeon vertex to include B-field effects.

Reproduced noncommutative scalar field theory results in the field theory limit.

Abstract

The multiloop amplitudes for the bosonic string in presence of a constant B-field are built by using the basic commutation relations for the open string zero modes and oscillators. The open string Green function on the annulus is obtained from the one loop scattering amplitude among N tachyons. For higher loops, it is necessary to use the so called three Reggeon vertex, which describes the emission from the open string of another string and not simply of a tachyon. We find that the modifications to the three (and multi) Reggeon vertex due to the B-field only affect the zero modes and can be written in a simple and elegant way. Therefore we can easily sew these vertices together and write the general expression for the multiloop N-Reggeon vertex, which contains any loop string amplitude, in presence of the B-field. The field theory limit is also considered in some examples at two loops and reproduces exactly the results of a noncommutative scalar field theory.