The closed-string 3-loop amplitude and S-duality

Abstract

The low energy limit of the closed-string 3-loop amplitude (including its overall coefficient) is computed for four external massless states using the pure spinor formalism and the result is compared with a prediction of Green and Vanhove based on SL(2, Z) duality. Agreement is found provided the three-loop amplitude prescription includes a symmetry factor 1/3. We argue for its inclusion in order to compensate a Z(3) symmetry present in genus-three Riemann surfaces.