nLab extended supersymmetry

Idea

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.

Examples

The “M-theory Lie algebra” is a polyvector extension of the super Poincaré Lie algebra $\mathfrak{siso}_{N=1}(10,1)$ by polyvectors of rank $p = 2$ and $p=5$ (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.

Properties

Short multiplets

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

As automorphism Lie algebras of Lie $n$-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.

Related concepts

• extended super Minkowski spacetime

References

General

Introductions and lecture notes:

• 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 (2010) [arXiv:1011.1491, spire:875723]

• Theories with extended supersymmetry (pdf)

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

• Dmitry Vladimirovich Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Antoine Van Proeyen, Polyvector Super-Poincaré Algebras Commun.Math.Phys. 253 (2004) 385-422 (arXiv:hep-th/0311107)

Understanding as extension of Noether charges of probe brane sigma-models:

• José de Azcárraga, Jerome Gauntlett, José M. Izquierdo, Paul Townsend: Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys. Rev. Lett. 63 (1989) 2443 [doi:10.1103/PhysRevLett.63.2443, spire:26393]

From super Lie $n$-algebras

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

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields