Multiloop noncommutative open string theory and their QFT limit

The multiloop amplitudes for open bosonic string in presence of a constant B-field are derived from first principles. The basic ingredients of the con struction are the commutation relations for the string modes and the Reggeo n vertex describing the interaction among three generic string states. The modifications due to the presence of the B-field affect non-trivially only the zero modes. This makes it possible to write in a simple and elegant way the general expression for multiloop string amplitudes in presence of a co nstant B-field. The field theory limit of these string amplitudes is also c onsidered, and we show that it reproduces exactly the Feynman diagrams of n oncommutative field theories.