# nLab M-theory on G2-manifolds

## Surveys, textbooks and lecture notes

#### Gravity

gravity, supergravity

# Contents

## Idea

The Kaluza-Klein reduction of 11-dimensional supergravity on G2 manifolds yields an effective $N=1$ 4-dimensional supergravity. This construction is the lift to M-theory of the KK-compactification of string theory on Calabi-Yau manifolds.

Specifically for discussion of obtaining or approximating the standard model of particle physics by this procedure see at G2-MSSM.

## Details

### The C-field

In compactifications with weak G2 holonomy it is the defining 4-form $\phi_4$ (the one which for strict G2 manifolds is the Hodge dual of the associative 3-form) which is the flux/field strength of the supergravity C-field. See for instance towards the end of (Bilal-Serendinger-Sfetos) for a derivation.

### Singularities

For realistic field content after Kaluza-Klein compactification one needs to consider not smooth (weak) G2-manifolds but orbifolds of these. see the first page of (Acharya-Denef-Hofman-Lambert) for discussion of phenomenology for such orbifold $G_2$ models and further pointers and see (Achary 98) for general discussion of orbifolds with $G_2$-structure.

## References

• Michael Atiyah, Edward Witten $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (pdf)

• Mirjam Cvetic, Gary Gibbons, H. Lü and C.N. Pope, Supersymmetric M3-branes and G2 Manifolds (pdf)

Discussion of Freund-Rubin compactification on $\mathbb{R}^4 \times X_7$ “with flux”, hence non-vanishing supergravity C-field and how they preserve one supersymmetry if $X_7$ is of weak G2 holonomy with $\lambda$ = cosmological constant = C-field strength on $\mathbb{R}^4$ is in

• Adel Bilal, J.-P. Derendinger, K. Sfetsos, (Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry, Nucl.Phys. B628 (2002) 112-132 (arXiv:hep-th/0111274)

Surveys include

• Mike Duff, M-theory on manifolds of G2 holonomy: the first twenty years (arXiv:hep-th/0201062)

• Sergei Gukov, M-theory on manifolds with exceptional holonomy, Fortschr. Phys. 51 (2003), 719–731 (pdf)

• Bobby Acharya, M Theory, $G_2$-manifolds and Four Dimensional Physics, Classical and Quantum Gravity Volume 19 Number 22 (pdf)

• Thomas House, Andrei Micu, M-theory Compactifications on Manifolds with $G_2$ Structure (arXiv:hep-th/0412006)

• Adil Belhaj, M-theory on G2 manifolds and the method of (p, q) brane webs (2004) (web)

Compactificaton on orbifolds of $G_2$-manifolds, introducing (orbifold-) singularities necessary for realistic effective QFTs is discussed in

The corresponding membrane instanton corrections to the superpotential? are discussed in

The hierarchy problem in the context of $G_2$-compactifications is discussed in

A survey of the corresponding string phenomenology is in

• Bobby Acharya, $G_2$-manifolds at the CERN Large Hadron collider and in the Galaxy, talk at $G_2$-days (2012) (pdf)

Revised on January 11, 2013 20:26:15 by Urs Schreiber (89.204.138.251)