Multiloop string amplitudes with B -field and noncommutative QFT

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.