nLab super multiplet

Idea

A supermultiplet is an irreducible representation of a supersymmetry supergroup or super Lie algebra, decomposed as a direct sum (branching) of irreps of the underlying ordinary group.

For the case of unitary representations of the super Poincaré group, then under the canonical inclusion of the ordinary Poincaré group into the super Poincaré group, the supermultiplets for super-isometries of super Minkowski spacetime decompose into a direct sum of irreducible unitary representations of the Poincaré group. Via the Wigner classification of fundamental particles as irreps of the Poincaré group, this gives a decomposition of supermultiplets into particle species. The particles in this direct sum related by odd supersymmetry transformations are called superpartners.

Similar statements hold for the super anti de Sitter group or superconformal group etc.

By Deligne's theorem on tensor categories the most general regular tensor categories are those spanned by supermultiplets for some supergroup.

Related concepts

• superconformal multiplet

• adinkra

References

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, chapter II.4 in Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Veeravalli Varadarajan, section 5.3 of Unitary representations of super Lie groups, 2006 (pdf)

See also:

• Wikipedia, Supermultiplet

Relation to $L_\infty$-algebras via pure spinor-formalism:

• Martin Cederwall, Simon Jonsson, Jakob Palmkvist, Ingmar Saberi, Canonical supermultiplets and their Koszul duals [arXiv:2304.01258]

See also:

• Ruben Minasian, Charles Strickland-Constable, Yi Zhang: On symmetries and dynamics of exotic supermultiplets, J. High Energ. Phys. 2021 174 (2021) [arXiv:2007.08888, doi:10.1007/JHEP01(2021)174]

Discussion via derived geometry on the pure spinor-variety:

• Chris Elliott, Fabian Hahner, Ingmar Saberi: The Derived Pure Spinor Formalism as an Equivalence of Categories, SIGMA 19 (2023) 022 [arXiv:2205.14133, doi:10.3842/SIGMA.2023.022]

• Richard Eager, Simone Noja, Raphael Senghaas, Johannes Walcher: Adinkras and Pure Spinors [arXiv:2404.07167]

• Simone Noja: The Geometry of Pure Spinor Superfield Formalism, talk at CQTS (Dec 2024) [pdf]