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 $\mathbb{R}^{3,1\vert \mathbf{4}+ \mathbf{4}}$)

Textbook accounts:

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

specifically for D=4 supergravity:

• 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]

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

Discussion of superspace within philosophy of physics:

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