# Geometry of extended null supersymmetry in M theory

## Abstract

For supersymmetric spacetimes in 11 dimensions admitting a null Killing spinor, a set of explicit necessary and sufficient conditions for the existence of any number of arbitrary additional Killing spinors is derived. The necessary and sufficient conditions are comprised of algebraic relationships, linear in the spinorial components, between the spinorial components and their first derivatives, and the components of the spin connection and four-form. The integrability conditions for the Killing spinor equation are also analyzed in detail, to determine which components of the field equations are implied by arbitrary additional supersymmetries and the four-form Bianchi identity. This provides a complete formalism for the systematic and exhaustive investigation of all spacetimes with extended null supersymmetry in 11 dimensions. The formalism is employed to show that the general bosonic solution of 11 dimensional supergravity admitting a G{sub 2} structure defined by four Killing spinors is either locally the direct product of R{sup 1,3} with a seven-manifold of G{sub 2} holonomy, or locally the Freund-Rubin direct product of AdS{sub 4} with a seven-manifold of weak G{sub 2} holonomy. In addition, all supersymmetric spacetimes admitting a (G{sub 2}xR{sup 7})xR{sup 2} structure are classified.

- Authors:

- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 20776784

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 73; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.73.045012; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; FIELD EQUATIONS; GEOMETRY; MATHEMATICAL SOLUTIONS; MATRICES; QUANTUM FIELD THEORY; SMOOTH MANIFOLDS; SPACE-TIME; SPIN; SPINORS; SUPERGRAVITY; SUPERSYMMETRY

### Citation Formats

```
Conamhna, Oisin A.P. Mac.
```*Geometry of extended null supersymmetry in M theory*. United States: N. p., 2006.
Web. doi:10.1103/PhysRevD.73.045012.

```
Conamhna, Oisin A.P. Mac.
```*Geometry of extended null supersymmetry in M theory*. United States. doi:10.1103/PhysRevD.73.045012.

```
Conamhna, Oisin A.P. Mac. Wed .
"Geometry of extended null supersymmetry in M theory". United States.
doi:10.1103/PhysRevD.73.045012.
```

```
@article{osti_20776784,
```

title = {Geometry of extended null supersymmetry in M theory},

author = {Conamhna, Oisin A.P. Mac},

abstractNote = {For supersymmetric spacetimes in 11 dimensions admitting a null Killing spinor, a set of explicit necessary and sufficient conditions for the existence of any number of arbitrary additional Killing spinors is derived. The necessary and sufficient conditions are comprised of algebraic relationships, linear in the spinorial components, between the spinorial components and their first derivatives, and the components of the spin connection and four-form. The integrability conditions for the Killing spinor equation are also analyzed in detail, to determine which components of the field equations are implied by arbitrary additional supersymmetries and the four-form Bianchi identity. This provides a complete formalism for the systematic and exhaustive investigation of all spacetimes with extended null supersymmetry in 11 dimensions. The formalism is employed to show that the general bosonic solution of 11 dimensional supergravity admitting a G{sub 2} structure defined by four Killing spinors is either locally the direct product of R{sup 1,3} with a seven-manifold of G{sub 2} holonomy, or locally the Freund-Rubin direct product of AdS{sub 4} with a seven-manifold of weak G{sub 2} holonomy. In addition, all supersymmetric spacetimes admitting a (G{sub 2}xR{sup 7})xR{sup 2} structure are classified.},

doi = {10.1103/PhysRevD.73.045012},

journal = {Physical Review. D, Particles Fields},

number = 4,

volume = 73,

place = {United States},

year = {Wed Feb 15 00:00:00 EST 2006},

month = {Wed Feb 15 00:00:00 EST 2006}

}