superalgebra and (synthetic ) supergeometry
Formalism
Definition
Spacetime configurations
Properties
Spacetimes
black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
Quantum theory
A supergeometric analog of anti-de Sitter spacetime (times an internal space). By the discussion at supersymmetry – Classification – Superconformal and super anti de Sitter symmetry this includes the following coset superspacetimes (super Klein geometries):
super anti de Sitter spacetime | |
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4 | |
5 | |
7 |
(Here denotes orthosymplectic super Lie groups.)
E.g. for this identification [D’Auria & Fré 1983] relies on the fact that and then on the exceptional isomorphism [e.g. Garret 2013, §2.8].
On the other hand, de Sitter spacetimes (instead of anti-de Sitter) do not have a standard extensions to supergeometry; but see arXiv:1610.01566.
Explicit formulas for super Cartan connections for and are given in dWPPS 98, p. 156, following Kallosh, Rahmfeld & Rajaraman 1998 and Claus & Kallosh 1999, see also Claus 1998 (reviewed in Wang 2023, §4.4).
See also the references at Green-Schwarz sigma-model – References – AdS backgrounds.
General discussion:
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, sections II.2.6, II.3.2-3, II.5, and V.4.4 in: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [ch II.2: pdf, ch II.3: pdf, ch II.5: pdf, ch V.4: pdf]
Renata Kallosh, J. Rahmfeld, Arvind Rajaraman, Near Horizon Superspace, JHEP 9809:002 (1998) [arXiv:hep-th/9805217, doi:10.1088/1126-6708/1998/09/002]
Piet Claus, Renata Kallosh, Superisometries of the Superspace, JHEP 9903:014 (1999) [arXiv:hep-th/9812087, doi:10.1088/1126-6708/1999/03/014]
Piet Claus, Super M-brane actions in and , Phys. Rev. D 59 (1999) 066003 [arXiv:hep-th/9809045, doi:10.1103/PhysRevD.59.066003]
Antoine Van Proeyen, sections 4.5, 4.7 of: Tools for supersymmetry [arXiv:hep-th/9910030]
Sergei Kuzenko, Gabriele Tartaglino-Mazzucchelli, Supertwistor realisations of AdS superspaces, The European Physical Journal C 82 2 (2022) 146 [doi:10.1140/epjc/s10052-022-10072-y, arXiv:2108.03907]
Specifically concerning super- (super-near horizon geometry of black M2-branes as in AdS4/CFT3-duality):
Riccardo D'Auria, Pietro Fré: Spontaneous generation of symmetry in the spontaneous compactification of supergravity, Physics Letters B 121 2–3 (1983) 141-146 [doi:10.1016/0370-2693(83)90903-6]
Mike Duff, Bengt Nilsson, Christopher Pope, pp. 33 in: Kaluza-Klein supergravity, Physics Reports 130 1–2 (1986) 1-142 [spire:229417, doi:10.1016/0370-1573(86)90163-8]
(no discussion of superspace, but of the -supersymmetry)
Bernard de Wit, Kasper Peeters, Jan Plefka, Alexander Sevrin, The M-theory two-brane in and , Physics Letters B 443 1-4 (1998) 153-158 [doi:10.1016/S0370-2693(98)01340-9, inspire:474621, arXiv:hep-th/9808052]
Gianguido Dall'Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante, §4.1 in: The singleton action from the supermembrane, Nucl. Phys. B 542 (1999) 157-194 [arXiv:hep-th/9807115, doi:10.1016/S0550-3213(98)00765-2]
Jaume Gomis, Dmitri Sorokin, Linus Wulff, The complete superspace for the type IIA superstring and D-branes, JHEP 0903:015 (2009) [arXiv:0811.1566]
Review:
Specifically for the superstring on super-:
The super 3-cocycle for the Green-Schwarz superstring on is originally due to
However, a supersymmetric trivialization of this cocycle seems to have been obtained (according to arxiv:1808.04470, p. 5 and equation (5.5), but check) in:
Last revised on July 29, 2024 at 19:52:27. See the history of this page for a list of all contributions to it.