One-loop diagrams in superstring theory

In this chapter we discuss one-loop amplitudes in superstring theory. The operator methods used for the calculations of one-loop amplitudes with on-shell external states in chapter 8 will be applied to the calculation of the corresponding diagrams in superstring theories. The resulting amplitudes are expressed as integrals over world sheets that are closed and orientable in the case of the type II or heterotic theories. Type I theories are based on unoriented open and closed strings, and as a result their world sheets need not be orientable and can have boundaries. As in the case of the bosonic theory, the operator approach automatically gives the correct measure factors by elementary algebraic manipulations. In approaches that explicitly compute the sum over geometries, considerable care is required to correctly define and evaluate infinite determinants that give the measure. Nonetheless, such approaches have a number of advantages and certainly appear preferable for the study of multiloop amplitudes.