Contents

# Contents

## Idea

The KK-compactification of M-theory on fibers which are 8-manifolds. In the low-energy limiting 11-dimensional supergravity this is KK-compactification to 3d supergravity.

Typically this is considered with a reduction of the structure group on the compactification fiber from Spin(8) to Spin(7), in which case one speaks of M-theory on Spin(7)-manifolds (see the references below). Further reduction to G2-structure yields M-theory on G2-manifolds.

## Properties

$N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \underset{ I_8(X^8) }{ \underbrace{ \tfrac{1}{48}\big( p_2 - \tfrac{1}{2}p_1^2 \big)[X^{8}] } } \;\;\;\; \in \mathbb{Z} \,,$

where

1. $N_{M2}$ is the net number of M2-branes in the spacetime (whose worldvolume appears as points in $X^{(8)}$);

2. $G_4$ is the field strength/flux of the supergravity C-field

3. $p_1$ is the first Pontryagin class and $p_2$ the second Pontryagin class combining to I8, all regarded here in rational homotopy theory.

If $X^{8}$ has

or

then

$\tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi$

is the Euler class (see this Prop. and this Prop., respectively), hence in these cases the condition is equivalently

$N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,,$

where $\chi[X]$ is the Euler characteristic of $X$.

For references see there.

### Relation to F-theory

If the 8-dimensional fibers themselves are elliptic fibrations, then M-theory on these 8-manifolds is supposedly T-dual to F-theory KK-compactified to $3+1$ spacetime-dimensions.

In particular, if there is an M2-brane filling the base 2+1-dimensional spacetime, this is supposedly T-dual to a 3+1-dimensional spacetime filling D3-brane in F-theory (e.g. Condeescu-Micu-Palti 14, p. 2)

For more on this see at F/M-theory on elliptically fibered Calabi-Yau 4-folds.

### Black M2-branes and Exotic 7-spheres

The discovery of exotic 7-spheres proceeded via 8-manifolds $X$ with boundary homeomorphic to the 7-sphere $\partial X \simeq_{homeo} S^7$, but not necessarily diffeomorphic to $S^7$ with its canonical smooth structure (for more see there).

Hence when regarded from the point of view of M-theory on 8-manifolds, exotic 7-spheres arise as near horizon limits of peculiar black M2-brane spacetimes $\mathbb{R}^{2,1} \times X$.

## References

### With $Spin(7)$-structure

Discussion of KK-compactification on 8-dimensional Spin(7)-manifolds (see also at M-theory on G2-manifolds):

• Mirjam Cvetic, Gary Gibbons, H. Lu, Christopher Pope, New Complete Non-compact Spin(7) Manifolds, Nucl.Phys.B620:29-54, 2002 (arXiv:hep-th/0103155)

• Jaydeep Majumder, Type IIA Orientifold Limit of M-Theory on Compact Joyce 8-Manifold of Spin(7)-Holonomy, JHEP 0201 (2002) 048 (arXiv:hep-th/0109076)

• Ralph Blumenhagen, Volker Braun, Superconformal Field Theories for Compact Manifolds with Spin(7) Holonomy, JHEP 0112:013, 2001 (arXiv:hep-th/0111048)

• Sergei Gukov, James Sparks, David Tong, Conifold Transitions and Five-Brane Condensation in M-Theory on $Spin(7)$ Manifolds, Class.Quant.Grav.20:665-706, 2003 (arXiv:hep-th/0207244)

• Melanie Becker, Dragos Constantin, Sylvester James Gates Jr., William D. Linch III, Willie Merrell, J. Phillips, M-theory on Spin(7) Manifolds, Fluxes and 3D, N=1 Supergravity, Nucl.Phys. B683 (2004) 67-104 (arXiv:hep-th/0312040)

• Dragos Constantin, M-Theory Vacua from Warped Compactifications on Spin(7) Manifolds, Nucl.Phys.B706:221-244, 2005 (arXiv:hep-th/0410157)

• Dragos Constantin, Flux Compactification of M-theory on Compact Manifolds with Spin(7) Holonomy, Fortsch.Phys. 53 (2005) 1272-1329 (arXiv:hep-th/0507104)

• Dimitrios Tsimpis, M-theory on eight-manifolds revisited: N=1 supersymmetry and generalized $Spin(7)$ structures, JHEP 0604 (2006) 027 (arXiv:hep-th/0511047)

• S. Salur, O. Santillan, New Spin(7) holonomy metrics admiting G2 holonomy reductions and M-theory/IIA dualities, Phys.Rev.D79:086009, 2009 (arXiv:0811.4422)

• Thomas Bruun Madsen, Spin(7)-manifolds with three-torus symmetry, J.Geom.Phys.61:2285-2292, 2011 (arXiv:1104.3089)

• Federico Bonetti, Thomas Grimm, Tom G. Pugh, Non-Supersymmetric F-Theory Compactifications on Spin(7) Manifolds, JHEP01(2014)112 (arXiv:1307.5858)

• Federico Bonetti, Thomas Grimm, Eran Palti, Tom G. Pugh, F-Theory on Spin(7) Manifolds: Weak-Coupling Limit, JHEP02(2014)076 (arXiv:1309.2287)

• Tom Pugh, M-theory on $Spin(7)$-manifold duals and their F-theory duals (pdf)

• Andreas Braun, Sakura Schaefer-Nameki, Spin(7)-Manifolds as Generalized Connected Sums and 3d $N=1$ Theories, JHEP06(2018)103 (arXiv:1803.10755)

(generalization of compact twisted connected sum G2-manifolds)

### With $Sp(2)\cdot Sp(1)$-structure

M-theory on HP^2, hence on a quaternion-Kähler manifold of dimension 8 with holonomy Sp(2).Sp(1), is considered in

• Michael Atiyah, Edward Witten, p. 75 onwards in $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

and argued to be dual to M-theory on G2-manifolds in three different ways, which in turn is argued to lead to a a possible proof of confinement in the resulting 4d effective field theory (see there for more).

### M2-brane spacetimes

Last revised on April 24, 2019 at 04:22:16. See the history of this page for a list of all contributions to it.