String Theory and Noncommutative Field Theories at One Loop

By exploiting the boundary state formalism we obtain the string correlator between two internal points on the one loop open string world-sheet in the presence of a constant background $B$-field. From this derivation it is clear that there is an ambiguity when one tries to restrict the Green function to the boundary of the surface. We fix this ambiguity by showing that there is a {\em unique} form for the correlator between two points on the boundary which reproduces the one loop field theory results of different noncommutative field theories. In particular, we present the derivation of one loop diagrams for $\phi^3_6$ and $\phi^4_4$ scalar interactions and for Yang--Mills theory. From the 2-point function we are able to derive the one loop $\beta$-function for noncommutative gauge theory.