  We consider the Compton amplitude for the scattering of a photon and a
(massless) ``electron/positron'' at one loop (i.e. genus one) in a
four-dimensional fermionic heterotic string model. Starting from the
bosonization of the world-sheet fermions needed to explicitly construct the
spin-fields representing the space-time fermions, we present all the steps of
the computation which leads to the explicit form of the amplitude as an
integral of modular forms over the moduli space.
