nLab non-projected super Riemann surface moduli -- section

Non-projected super-moduli of super Riemann surfaces

A supermanifold is called projected if it retracts onto its bosonic body. (That’s not the wording used in the literature, though.)

Since the computation of superstring scattering amplitudes involves a Berezin integral over the super moduli space of the given type of super Riemann surfaces, it is of interested to know when this moduli space of super Riemann surfaces is projected, as that allows to separate the bosonic from the fermionic sectors of this “path integral”.

However, it turns out that the super-moduli space of super Riemann surfaces $\mathfrak{M}_{g, n_S, n_R}$ is generically not projected beyond low genus $g$ (the string’s loop order), depending on

• the number $n_S$ of Neveu-punctures

• the number $n_R$ of Ramond-punctures.

Specifically:

• $\mathfrak{M}_{g, 0, 0}$ is not projected for $g \geq 5$,

  $\mathfrak{M}_{g, n_S \geq 1, 0}$ is not projected for $g \geq n_S + 1$

  [Donagi & Witten 2015]

• $\mathfrak{M}_{g, 0, 2r \geq 2}$ is not projected for $g \geq 5r + 1$

  [Donagi & Ott 2023].

On the other hand, $g = 2$ (stringy 2-loop) remains the highest order for which integration over the moduli space has actually been considered/performed, see D’Hoker & Phong 2002.