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 and at F-theory on Spin(7)-manifolds and at Witten's Dark Fantasy.

(Bonetti-Grimm-Pugh 13a, Bonetti-Grimm-Pugh 13b, reviewed in Pugh)

### 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 $\mathbb{R}^{2,1} \times X$.

### Relation to J-twisted Cohomotopy

On a spin-manifold of dimension 8 a choice of topological Spin(7)-structure is equivalently a choice of cocycle in J-twisted Cohomotopy cohomology theory. This follows (FSS 19, 3.4) from

1. the standard coset space-structures on the 7-sphere (see here)

2. the fact that coset spaces $G/H$ are the homotopy fibers of the maps $B H \to B G$ of the corresponding classifying spaces (see here)

## References

### In terms of $G$-structure

Discussion in terms of G-structures:

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

Discussion of KK-compactification on 8-dimensional Spin(7)-manifolds (see also at M-theory on G2-manifolds and at F-theory on Spin(7)-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 possible proof of confinement in the resulting 4d effective field theory (see there for more).

### Witten’s Dark Fantasy

For more on the following see at Witten's Dark Fantasy:

An argument for non-perturbative non-supersymmetric 4d string phenomenology with fundamentally vanishing cosmological constant, based on 3d M-theory on 8-manifolds decompactified at strong coupling to 4d via duality between M-theory and type IIA string theory (recall the super 2-brane in 4d):

The realization of this scenario in F-theory on Spin(7)-manifolds:

### M-theory on $K3 \times K3$

Discussion of M-theory KK-compactified on the product manifold of two K3s:

Under duality between M-theory and type IIA string theory this translates to D6-branes wrapped on K3 (enhancon mechanism):

Review:

Last revised on January 24, 2023 at 16:41:55. See the history of this page for a list of all contributions to it.