nLab super Riemann surface

Idea

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

Examples

A Riemann surface with spin structure that is also a branched cover of $\mathbb{C}P^{(1 \vert 1)}$ 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.

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 $\mathfrak{M}_{g, n_S, n_R}$ 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.

Related concepts

• superstring

• string scattering amplitude

• picture changing operator

• adinkra

References

General

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,

  1987

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

See also

• Eric D'Hoker, Duong Phong, Complex geometry and supergeometry, Current Developments in Mathematics 2005 (2007): 1-40 (euclid:1223654523)

Discussion of super Riemann surfaces induced by supermultiplets for $N$-extended $d = 1$ supersymmetry – via adinkra symbols:

• Charles Doran, Kevin Iga, Greg Landweber, Stefan Méndez-Diez, Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras, Adv. Theor. Math. Phys. 19 5 (2015) 1043-1113 [arXiv:1311.3736, doi:10.4310/ATMP.2015.v19.n5.a4, pdf]

• Charles Doran, Kevin Iga, Jordan Kostiuk, Stefan Méndez-Diez, Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II, Adv. Theor. Math> Physics 22 3 (2018) [arXiv:1610.09983]

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):

• Jeffrey Rabin, Supermanifolds and Super Riemann Surfaces, In: H.C. Lee et. al (eds.) Super Field Theories, NATO Science Series (Series B: Physics), vol 160. Springer (1987) (doi:10.1007/978-1-4613-0913-0_34, pdf)

• Claude LeBrun, Mitchell Rothstein, Moduli of super Riemann surfaces, Comm. Math. Phys. Volume 117, Number 1 (1988), 159-176 (euclid:cmp/1104161598)

• Edward Witten, Notes On Super Riemann Surfaces And Their Moduli, Pure and Applied Mathematics Quarterly Volume 15 (2019) Number 1 Special Issue on Super Riemann Surfaces and String Theory [doi:10.4310/PAMQ.2019.v15.n1.a2arXiv:1209.2459]

• Ron Donagi, Edward Witten: Supermoduli Space Is Not Projected, Proc. Symp. Pure Math. 90 (2015) 19-72 [spire:1231519, arXiv:1304.7798]

• Supermoduli Workshop: May 18 – 22, 2015, videos of lecture courses by Pierre Deligne, Eric D'Hoker, Ron Donagi and Edward Witten

• Giulio Codogni, Filippo Viviani, Moduli and Periods of Supersymmetric Curves, Adv. Theor. Math. Phys. 23 (2019) 2, 345-402 (arXiv:1706.04910, doi:10.4310/ATMP.2019.v23.n2.a2)

• Ugo Bruzzo, Daniel Hernández Ruipérez, The supermoduli of SUSY curves with Ramond punctures, RACSAM 115 144 (2021) [doi:10.1007/s13398-021-01078-4, arXiv:1910.12236]

• Nadia OttThe Supermoduli Space of Genus Zero Susy Curves with Ramond Punctures, PhD thesis, University of Minnesota (2020) [proquest:28094035]

• Nadia Ott, Alexander A. Voronov: The supermoduli space of genus zero SUSY curves with Ramond punctures, Journal of Geometry and Physics 185 (2023) 104726 [arXiv:1910.05655, doi:10.1016/j.geomphys.2022.104726]

• Dimitri Skliros: Moving NS Punctures on Super Spheres, SIGMA 20 (2024) 090 [doi:10.3842/SIGMA.2024.090, arXiv:2307.06355]

• Ron Donagi, Nadia Ott, Supermoduli Space with Ramond punctures is not projected [spire:2688635, arXiv:2308.07957]

• Dimitry Leites, Alexander S. Tikhomirov: Non-split superstrings of dimension $(1|2)$ [arXiv:2503.02416]

Further discussion of supergeometric Teichmüller space:

• Robert Penner, Anton Zeitlin, Decorated Super-Teichmüller Space (arXiv:1509.06302)

• Ivan C.H. Ip, Robert Penner, Anton Zeitlin, $\mathcal{N}=2$ Super-Teichmüller Theory, Advances in Mathematics 336 (2018) 409-454 (arXiv:1605.08094)

• Anton Zeitlin, Super-Teichmüller spaces and related structures (arXiv:1811.09939)

In relation to fat graphs:

• Albert S. Schwarz, Anton M. Zeitlin, Super Riemann surfaces and fatgraphs, Universe 9 9 (2023) 384 [doi:10.3390/universe9090384, arXiv:2307.02706]