nLab extended supersymmetry



\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

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Lie theory

∞-Lie groupoids

∞-Lie algebroids

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\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras




The super-Poincaré Lie algebra has certain exceptional super Lie algebra extensions by “central charges” which commute with the generators of the super translation Lie algebra and form a tensor representation of the remaining Lorentz group generators.

Here “central charges” is in quotation marks (and typically is so in the literature) since these charges will be in the center of the super translation Lie algebra but not in general in the center of the full super Poincare Lie algebra.


The “M-theory Lie algebra” is a polyvector extension of the super Poincaré Lie algebra 𝔰𝔦𝔰𝔬 N=1(10,1)\mathfrak{siso}_{N=1}(10,1) by polyvectors of rank p=2p = 2 and p=5p=5 (the M2-brane and the M5-brane in the brane scan), see below Polyvector extensions as automorphism Lie algebras.

Similarly the type II supersymmetry algebra in type II supergravity.


Short multiplets

Extended supersymmetry algebras in general have “short” supermultiplets. These are called BPS states.

As automorphism Lie algebras of Lie nn-superalgebras

At least some of the polyvector extensions of the super Poincaré Lie algebra arise as the automorphism super Lie algebras of the Lie n-algebra extensions classified by the cocycles discussed above.

For instance the automorphisms of the supergravity Lie 3-algebra gives the “M-theory Lie algebra”-extension of super-Poincaré in 11-dimensions (FSS 13). This is also discussed at supergravity Lie 3-algebra – Polyvector extensions.



Introductions and lecture notes include

  • José D. Edelstein, Lecture 6: Extended supersymmetry, talk at Supersymmetry, Santiago de Compostela, November 20, 2012 (pdf)

  • Sven Krippendorf, Fernando Quevedo, Oliver Schlotterer, section 2.3 of Cambridge Lectures on Supersymmetry and Extra Dimensions (arXiv:1011.1491)

  • Theories with extended supersymmetry (pdf)

A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in

From super Lie nn-algebras

The derivation of the extended supersymmetry M-theory Lie algebra as the derivations of the supergravity Lie 3-algebra is, somewhat implicitly, in

A more explicit discussion of this in terms of super L-infinity algebra homotopy theory is in

Last revised on April 2, 2019 at 14:48:33. See the history of this page for a list of all contributions to it.