nLab moduli space of super Riemann surfaces

Contents

Context

Supergeometry

Higher geometry

Manifolds and cobordisms

Contents

Idea

The super-moduli space of super Riemann surfaces.

Just like the moduli space of Riemann surfaces is an orbifold, so the moduli space super-Riemann surfaces is a super-orbifold (e.g. Rabin 87, LeBrun-Rothstein 88, Witten 12, Codogni-Viviani 17).

Properties

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 𝔐 g,n S,n R\mathfrak{M}_{g, n_S, n_R} is generically not projected beyond low genus gg (the string’s loop order), depending on

  • the number n Sn_S of Neveu-punctures

  • the number n Rn_R of Ramond-punctures.

Specifically:

  • 𝔐 g,0,0\mathfrak{M}_{g, 0, 0} is not projected for g5g \geq 5,

    𝔐 g,n S1,0\mathfrak{M}_{g, n_S \geq 1, 0} is not projected for gn S+1g \geq n_S + 1

    [Donagi & Witten 2015]

  • 𝔐 g,0,2r2\mathfrak{M}_{g, 0, 2r \geq 2} is not projected for g5r+1g \geq 5r + 1

    [Donagi & Ott 2023].

On the other hand, g=2g = 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.

References

Super-Moduli space of super Riemann surfaces

On the moduli space of super Riemann surfaces (the supergeometric analog of the moduli space of Riemann surfaces):

Further discussion of supergeometric Teichmüller space:

In relation to fat graphs:

Last revised on August 17, 2023 at 09:09:54. See the history of this page for a list of all contributions to it.