nLab superspace

This entry is about the concept in supergeometry. For the concept in gravity/cosmology see at Wheeler superspace.

Idea

Physicists often refer to spaces in supergeometry, such as supermanifolds or super schemes, as superspaces. Hence a superspace can be an affine superspace (the affine counterpart of the super vector space over real or complex numbers), superscheme, supermanifold, etc.

Mostly however “superspace” is used for superspacetimes, and here mostly for super Minkowski spacetimes.

Related concepts

• super-manifold, super-orbifold

• super Lie group, super Lie algebra, super L-infinity algebra

• supersymmetry

• adinkra

References

The concept of superspace in physics (together with that of superfields) is due to

• Abdus Salam, John Strathdee, Supergauge Transformations, Nucl.Phys. B76 (1974) 477-482 (spire:89208)

• Abdus Salam, John Strathdee, Superfields and Fermi-Bose symmetry, Physical Review D11, 1521-1535 (1975) (doi:10.1142/9789812795915_0051)

(which considered superspace of dimension $d = 4$ with number of supersymmetries $N = 2$, hence the supermanifold $\mathbb{R}^{4\vert \mathbf{4}+ \mathbf{4}}$, or rather the super Minkowski spacetime $\mathbb{R}^{3,1\vert \mathbf{4}+ \mathbf{4}}$)

Monographs:

• Jim Gates Jr, Marcus Grisaru, Martin Roček, Warren Siegel, Superspace, or One thousand and one lessons in supersymmetry, Front. Phys. 58 (1983) 1-548 [arXiv:hep-th/0108200, inspire:195126]

• Ioseph L. Buchbinder, Sergei M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity – or: A Walk Through Superspace, IoP (1995) [doi:10.1201/9780367802530, oapen:20.500.12657/50874, inspire:1946734]

The formulation of supergravity on super spacetime supermanifolds:

• Julius Wess, Bruno Zumino, Superspace formulation of supergravity, Phys. Lett. B 66 (1977) 361-364 [doi:10.1016/0370-2693(77)90015-6]

• Richard Grimm, Julius Wess, Bruno Zumino, A complete solution of the Bianchi identities in superspace with supergravity constraints, Nuclear Phys. B 152 (1979) 255-265 [doi:10.1016/0550-3213(79)90102-0]

• Warren Siegel, S. James Gates Jr., Superfield supergravity, Nuclear Physics B 147 1–2 (1979) 77-104 [doi:10.1016/0550-3213(79)90416-4]

• S. James Gates Jr., Kellogg S. Stelle, Peter C. West Algebraic origins of superspace constraints in supergravity, Nuclear Physics B 169 3–4 (1980) 347-364 [doi:10.1016/0550-3213(80)90037-1]

• Paul Howe, Supergravity in superspace, Nuclear Physics B 199 2 (1982) 309-364 [doi:10.1016/0550-3213(82)90349-2]

• S. James Gates Jr., On-shell and conformal $N=4$ supergravity in superspace, Nuclear Physics B 213 3 (1983) 409-444 [doi:10.1016/0550-3213(83)90229-8]

• James L. Carr, S. James Gates Jr., Robert N. Oerter: $D = 10$, $N = 2a$ Supergravity in Superspace, Physics Letters B 189 1–2 (1987) 68-74 [doi:10.1016/0370-2693(87)91271-8, spire:235791]

• S. James Gates Jr., Liang Lu, Robert N. Oerter: Simplified $SU(2)$ spinning string superspace supergravity, Physics Letters B 218 1 (1989) 33-38 [doi:10.1016/0370-2693(89)90470-X, spire:263912]

  (for the superstring worldvolume)

See also:

• S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, Superspace approaches to $\mathcal{N} = 1$ supergravity, Handbook of Quantum Gravity, Springer (2023) [arXiv:2210.17088]

• S. M. Kuzenko, E. S. N. Raptakis, G. Tartaglino-Mazzucchelli, Covariant superspace approaches to $\mathcal{N}=2$ supergravity, in: Handbook of Quantum Gravity, Springer (2023) [arXiv:2211.11162]

The claim of the derivation of $D=11$ supergravity in supergeometry, by solving the torsion constraint and Bianchi identities on super spacetime supermanifolds is due to

• Eugène Cremmer, Sergio Ferrara, Formulation of Eleven-Dimensional Supergravity in Superspace, Phys. Lett. B 91 (1980) 61 [doi:10.1016/0370-2693(80)90662-0]

• Lars Brink, Paul Howe, Eleven-Dimensional Supergravity on the Mass-Shell in Superspace, Phys. Lett. B 91 (1980) 384 [doi:10.1016/0370-2693(80)91002-3]

and in the mild variation (using a manifestly duality-symmetric super-C-field flux density) due to

• Castellani, D’Auria & Fré 1991, §III.8.5

A proof of this claim is laid out in

• Grigorios Giotopoulos, Hisham Sati, Urs Schreiber, §3 of: Flux Quantization on 11d Superspace, Journal of High Energy Physics 2024 82 (2024) [arXiv:2403.16456, doi:10.1007/JHEP07(2024)082]

using further lemmas and then heavy computer algebra checks (here).

Further review:

• I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace

• Albert Schwarz, On the definition of superspace Teoret. Mat. Fiz., 1984, Volume 60, Number 1, Pages 37–42 (Mi tmf5111), (russian original)

• Urs Schreiber, geometry of physics – supergeometry

On T-duality formulated on superspace:

• M. Abou-Zeid, Chris Hull, Ulf Lindström, Martin Roček: T-Duality in $(2,1)$ Superspace, J. High Energ. Phys. 2019 138 (2019) [arXiv:1901.00662, doi:10.1007/JHEP06(2019)138]

• Willie Carl Merrell: Application of superspace techniques to effective actions, complex geometry and T-duality in String theory, PhD thesis, University of Maryland (2007) [hdl:1903/6865]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber: T-Duality from super Lie n-algebra cocycles for super p-branes, Adv. Theor. Math. Phys 22 5 (2018) [arXiv:1611.06536, doi:10.4310/ATMP.2018.v22.n5.a3]

• Daniel Butter, Falk Hassler, Christopher N. Pope, Haoyu Zhang: Generalized Dualities and Supergroups, J. High Energ. Phys. 2023 52 (2023) [arXiv:2307.05665, doi:10.1007/JHEP12(2023)052]

More mathematical discussion via supermanifolds:

• Johannes Moerland: Spinorial Superspaces and Super Yang-Mills Theories [arXiv:2411.06165]

Discussion of superspace within philosophy of physics:

• Tushar Menon, Taking up superspace – the spacetime structure of supersymmetric field theory (philsci:14682, pdf)