nLab
type II supergravity Lie 2-algebra

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

String theory

Contents

Idea

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

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

dB=(Ψ¯Γ aΓ 11Ψ)E a. 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)N = (1,1) and in (Sakaguchi 99 (2.4) and (2.25)) for type IIB with N=(2,0)N = (2,0).

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

Properties

The cocycles of the exceptional ∞-Lie algebra cohomology of 𝔰𝔲𝔤𝔯𝔞 typeII\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 pp-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

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

(The first columns follow the exceptional spinors table.)

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

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