Superstring Scattering Amplitudes in Higher Genus

Abstract

Abstract. In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.

The Vanishing of Two-Point Functions for Three-Loop Superstring Scattering Amplitudes

“…All theta constants with characteristics are modular forms of weight one half with respect to a certain normal subgroup of finite index Γ(4, 8) ⊂ Sp(2g, Z), while all theta constants of the second order are modular forms of weight one half with respect to a bigger normal subgroup Γ(2, 4) ⊃ Γ (4,8).…”

“…The reformulation of this ansatz in terms of products of theta constants with characteristics in a syzygetic subspace given in [8] is as follows.…”

“…The genus g ≤ 3 ansatze were reformulated in terms of syzygetic subspaces by the first author in [8], where an ansatz for general genus was proposed, under the assumption on holomorphicity of certain 2 r -roots. Cacciatori, Dalla Piazza, and van Geemen in [3] give the genus 4 ansatz in terms of quadrics in the theta constants.…”

“…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 ansatz given in [8], theta functions, and specifically the theory of the Γ 00 linear system on Jacobians introduced by van Geemen and van der Geer [6]. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong [11,12,13,14,17,18], we give a new proof of the vanishing of the two-point function (which was proven by them).…”

“…At the two-loop level, where the amplitudes were computed by D'Hoker and Phong [11,12,13,14,17,18], 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 [8,24,3]. …”