nLab super Riemann surface




Complex geometry

Manifolds and cobordisms



A super Riemann surface is the analog of a Riemann surface in supergeometry.


A Riemann surface with spin structure that is also a branched cover of P (1|1)\mathbb{C}P^{(1 \vert 1)} is canonically endowed with the structure of a super Riemann surface with Ramond punctures (Donagi-Witten 15). This implies that adinkras encode certain very special super Riemann surfaces Doran & Iga & Landweber & Mendez-Diez 13.


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.


  • 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.



The concept of super Riemann surfaces originates with the following articles:

  • Daniel Friedan, Notes On String Theory and 2-Dimensional Conformal Field Theory. 1986

  • M.A. Baranov, Albert Schwarz, On the Multiloop Contribution to the String Theory Int.J.Mod.Phys., A2:1773, 1987

  • Yuri Manin, Critical Dimensions of String Theories and the Dualizing Sheaf on the Moduli Space of (Super) Curves, Funct.Anal.Appl., 20:244-245,


  • Steve Giddings, P. Nelson, The Geometry of super Riemann surfaces, Commun. Math. Phys., 116, (1988), 607

See also

Discussion of super Riemann surfaces induced by supermultiplets for NN-extended d=1d = 1 supersymmetry – via adinkra symbols, is due to

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 09:10:44. See the history of this page for a list of all contributions to it.