Supermoduli Space Is Not Projected

TL;DR

This paper proves that for genus ≥ 5, the moduli space of super Riemann surfaces is not projected or split, indicating fundamental differences from classical moduli spaces and impacting superstring theory approaches.

Contribution

It establishes that the supermoduli space for higher genus is inherently non-projected, revealing its complex structure beyond elementary constructions.

Findings

Supermoduli space is not projected for genus ≥ 5

Implications for superstring perturbation theory at higher orders

Mathematical structure of supermoduli space is fundamentally complex

Abstract

We prove that for genus greater than or equal to 5, the moduli space of super Riemann surfaces is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.