The vanishing of two-point functions for three-loop superstring scattering amplitudes

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen [2] vanishes. Our proof uses the reformulation of the ansatz given in [8], theta functions, and specifically the theory of the Γ₀₀ linear system, introduced by van Geemen and van der Geer [6], on Jacobians. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong [11-14, 17, 18], we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera [3, 8, 25].