RNS superstring measure for genus 3

TL;DR

This paper introduces a new, first-principles formula for the RNS superstring measure at genus 3, using invariant theory and a novel algebraic parametrization of the moduli space, with conjectured coefficients supported by evidence.

Contribution

It presents a new formula for the genus 3 superstring measure based on invariant theory and a unique algebraic parametrization, differing from previous approaches.

Findings

The measure is a linear combination of three known functions.

The formula has a polar singularity along the hyperelliptic locus.

Conjectured coefficients are supported by evidence.

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.