RNS superstring measure for genus 3

Perturbative computation of scattering amplitudes in superstring theory involves a certain volume form, called the super Mumford form, on the moduli space of super Riemann surfaces of given genus. It is analogous to the Mumford form, which appears when scattering amplitudes in bosonic string theory are computed perturbatively (the Polyakov measure is the "modulus squared" of the Mumford form).
Explicit formulas for the super Mumford form are known for genus 1 (since 1980's) and for genus 2 (derived by D'Hoker and Phong in the beginning of 2000's). Cacciatori, Dalla Piazza and van Geemen proposed an ansatz for the "top component" of the super Mumford form for genus 3 in 2008, but Witten in 2015 presented arguments against their ansatz. Namely, this ansatz has no poles, and Witten argued that the "top component" should have a pole at the hyperelliptic locus. In the talk I will present a new formula for the "top component" of the super Mumford form for genus 3. This formula agrees with Witten's result: it has a pole at the hyperelliptic locus, and the order of this pole coincides with the order computed by Witten. Moreover, this formula is not an ansatz, it follows from first principles, except for the values of three coefficients (complex numbers) appearing in it, which are only conjectured so far. The talk will be based on our recent paper https://arxiv.org/abs/2505.02950.