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.

Related concepts

• superstring

• string scattering amplitude

• picture changing operator

• adinkra

References

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)

Further discussion of the moduli space of super Riemann surfaces includes the following:

• 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 (arXiv:1209.2459, doi:10.4310/PAMQ.2019.v15.n1.a2)

• Ron Donagi, Edward Witten, Supermoduli Space Is Not Projected, Proc. Symp. Pure Math. 90 (2015) 19-72 (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)

See also:

• Ugo Bruzzo, Daniel Hernández Ruipérez, The supermoduli of SUSY curves with Ramond punctures (arXiv:1910.12236)

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

• Charles Doran, Kevin Iga, Greg Landweber, Stefan Méndez-Diez, Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras (arXiv:1311.3736)

• Charles Doran, Kevin Iga, Jordan Kostiuk, Stefan Méndez-Diez, Geometrization of $\mathcal{N}$-Extended 1-Dimensional Supersymmetry Algebras II (arXiv:1610.09983)

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)