superalgebra and (synthetic ) supergeometry
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
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 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.
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 is generically not projected beyond low genus (the string’s loop order), depending on
the number of Neveu-punctures
the number of Ramond-punctures.
Specifically:
is not projected for ,
is not projected for
[Donagi & Witten 2015]
is not projected for
[Donagi & Ott 2023].
On the other hand, (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,
1987
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 -extended supersymmetry – via adinkra symbols:
Charles Doran, Kevin Iga, Greg Landweber, Stefan Méndez-Diez, Geometrization of -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 -Extended 1-Dimensional Supersymmetry Algebras II, Adv. Theor. Math> Physics 22 3 (2018) [arXiv:1610.09983]
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]
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, 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:
Last revised on October 18, 2024 at 20:39:04. See the history of this page for a list of all contributions to it.