RNS superstring measure for genus 3

TL;DR

The paper derives a principled formula for the genus-3 RNS superstring measure by combining invariant theory with an algebraic parametrization of the moduli of even spin genus-3 curves. It shows that the ratio $\pi_*\psi_3/\varphi_3$ is a linear combination of three degree-18 invariants, which can be rewritten as a genus-3 Siegel modular form $\Xi^{(3)}$ involving a determinant-based parametrization $A(\tau)$ and theta constants; three coefficients are fixed by a conjectured genus-1/2 factorization condition and supported by computer verifications. The approach also clarifies pole structures along the hyperelliptic locus, recovers the known pole behavior of $\pi_*\psi_3$ (as predicted by Witten), and demonstrates the vanishing of the sum over even spin structures for genus-3, which aligns with expectations for the vacuum amplitude in type II theory. By connecting invariant theory with modular forms and exploiting an algebraic parametrization, the work provides a principled alternative to previous Ansätze for genus-3, with potential implications for higher-genus amplitudes and broader understanding of the superstring measure. The paper also outlines paths to prove the modular-analytic conjectures and to analyze boundary contributions to the 0-point function beyond interior vanishing.

Abstract

We propose a new formula for the RNS supersting measure for genus 3. Our derivation is based on invariant theory. We follow Witten's idea of using an algebraic parametrization of the moduli space (which he applied to re-derive D'Hoker and Phong's formula for the RNS superstring measure for genus 2); but the particular parametrization that we use has not been applied to superstring theory before. We prove that the superstring measure is a linear combinaition (with complex coefficients) of three known functions. Furthermore, we conjecture the values of the coefficients of this linear combination and provide evidence for this conjecture. Unlike the Ansatz of Cacciatori, Dalla Piazza and van Geemen from 2008, our formula has a polar singularity along the hyperelliptic locus; the existence of this singularity was established by Witten in 2015. Moreover, our formula is not an Ansatz but follows from first principles, except for the values of the three coefficients.