Contents

# Contents

## Idea

A super Lie 2-algebra extension of the super Poincare Lie algebra in $D = 10$ for $N=2$ supersymmetry (as in type II supergravity).

The Chevalley-Eilenberg algebra of the $\mathfrak{sugra}_{typeII}$ (defining it) is that of the $D = 10$ $N = 2$ super Poincaré Lie algebra generated from $\{E, \Psi\}$ with one further generator $B$ in degree 2 and differential given schematically by

$d B = \big( \overline{\Psi} \Gamma^a \Gamma_{11} \Psi\big) \wedge E_a \,.$

This is (CAIB 99, equation (6.3)) for type IIA with $N = (1,1)$ and in (Sakaguchi 99 (2.4) and (2.25)) for type IIB with $N = (2,0)$.

It also makes sense to write $\mathfrak{string}_{IIA}$ and $\mathfrak{string}_{IIB}$ for these. See also at string Lie 2-algebra.

## Properties

The cocycles of the exceptional ∞-Lie algebra cohomology of $\mathfrak{sugra}_{typeII}$ induce the Green-Schwarz action functional infinity-Wess-Zumino-Witten theory-terms for the D-branes of type II superstring theory (CAIB 99, section 6.1 for IIA and (Sakaguchi 99, section 2) for IIB):

The brane scan.

The Green-Schwarz type super $p$-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

$\stackrel{d}{=}$$p =$123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9*
8$\ast$
7M2${}_{top}$
6F1${}_{little}$, S1${}_{sd}$S3
5$\ast$
4**
3*

(The first columns follow the exceptional spinors table.)

The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):

$\stackrel{d}{=}$$p =$123456789
11$\Psi^2 E^2$ on sIso(10,1)$\Psi^2 E^5 + \Psi^2 E^2 C_3$ on m2brane
10$\Psi^2 E^1$ on sIso(9,1)$B_2^2 + B_2 \Psi^2 + \Psi^2 E^2$ on StringIIA$\cdots$ on StringIIB$B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4$ on StringIIA$\Psi^2 E^5$ on sIso(9,1)$B_2^4 + \cdots + \Psi^2 E^6$ on StringIIA$\cdots$ on StringIIB$B_2^5 + \cdots + \Psi^2 E^8$ in StringIIA$\cdots$ on StringIIB
9$\Psi^2 E^4$ on sIso(8,1)
8$\Psi^2 E^3$ on sIso(7,1)
7$\Psi^2 E^2$ on sIso(6,1)
6$\Psi^2 E^1$ on sIso(5,1)$\Psi^2 E^3$ on sIso(5,1)
5$\Psi^2 E^2$ on sIso(4,1)
4$\Psi^2 E^1$ on sIso(3,1)$\Psi^2 E^2$ on sIso(3,1)
3$\Psi^2 E^1$ on sIso(2,1)

supergravity Lie 6-algebra$\to$ supergravity Lie 3-algebra $\to$ super Poincaré Lie algebra

## References

The type IIA supergravity Lie 2-algebra and its D-brane-Green-Schwarz action functional-type cocycles are discussed in section 6 of

The type IIB supergravity Lie 2-algebra and its D-brane-Green-Schwarz action functional-type cocycles are discussed in section 2 of

• Makoto Sakaguchi, IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)

The formulation in super L-infinity algebra theory is in

Last revised on November 8, 2019 at 07:10:42. See the history of this page for a list of all contributions to it.