Supermoduli Space Is Not Projected
Ron Donagi, Edward Witten
TL;DR
The authors prove that the moduli space of genus-$g$ super Riemann surfaces ${\mathfrak{M}}_g$ is not projected (and in particular not split) for $g\ge 5$, via the nonvanishing of the leading obstruction class $\omega_2$ in $H^1({\mathcal{SM}}_g,\mathrm{Hom}(\wedge^2 T_{-}, T_{+}))$. They further show ${\mathfrak{M}}_{g,1}$ is non-projected for even spin structure when $g\ge 2$, and deduce non-projectedness of ${\mathfrak{M}}_{g,n}$ for certain $n$ by embedding compact spin families and tracing obstructions through compatible restriction maps. The core method combines Green’s non-abelian obstruction theory for splittings, compatibility lemmas for submanifolds, and explicit branched-cover and blowup operations that generate non-split, non-projected deformations; these constructions reveal a rich stacky structure essential to supermoduli and have implications for higher-genus superstring perturbation theory where projection-based simplifications fail. Collectively, these results demonstrate that the supermoduli space has a life of its own, not reconstructible from the underlying bosonic moduli, and that projecting perturbative approaches used at low genus do not generalize to higher orders. The work leverages branched covers, ramification patterns, and relative spin structures to produce compact families that expose the nontrivial obstruction classes, with broader relevance to the geometry of superstacks and spin geometry in string theory.
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.