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.
In M-theory compactified compact 8-manifold fibers, tadpole cancellation for the supergravity C-field (see also at C-field tadpole cancellation) is equivalently the condition
where
$N_{M2}$ is the net number of M2-branes in the spacetime (whose worldvolume appears as points in $X^{(8)}$);
$G_4$ is the field strength/flux of the supergravity C-field
$p_1$ 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
is the Euler class (see this Prop. and this Prop., respectively), hence in these cases the condition is equivalently
where $\chi[X]$ is the Euler characteristic of $X$.
For references see there.
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)
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 $\mathbb{R}^{2,1} \times X$.
See also Morrison-Plesser 99, section 3.2.
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
the standard coset space-structures on the 7-sphere (see here)
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)
Edward Witten, Strong Coupling and the Cosmological Constant, Mod. Phys. Lett.A10:2153-2156, 1995 (arXiv:hep-th/9506101)
(on possible relation to the cosmological constant: Witten's Dark Fantasy)
Katrin Becker, Melanie Becker, M-Theory on Eight-Manifolds, Nucl. Phys. B477 (1996) 155-167 (arXiv:hep-th/9605053)
Savdeep Sethi, Cumrun Vafa, Edward Witten, Constraints on Low-Dimensional String Compactifications, Nucl. Phys. B480: 213-224, 1996 (arXiv:hep-th/9606122)
Discussion in terms of G-structures:
Chris Isham, Christopher Pope, Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings, Class. Quant. Grav. 5 (1988) 257 (spire:251240, doi:10.1088/0264-9381/5/2/006)
(focus on Spin(7)-structure)
Chris Isham, Christopher Pope, Nicholas Warner, Nowhere-vanishing spinors and triality rotations in 8-manifolds, Classical and Quantum Gravity, Volume 5, Number 10, 1988 (cds:185144, doi:10.1088/0264-9381/5/10/009)
(focus on Spin(7)-structure)
Cezar Condeescu, Andrei Micu, Eran Palti, M-theory Compactifications to Three Dimensions with M2-brane Potentials, JHEP 04 (2014) 026 (arXiv:1311.5901)
Daniël Prins, Dimitrios Tsimpis, IIA supergravity and M-theory on manifolds with $SU(4)$ structure, Phys. Rev. D 89.064030 (arXiv:1312.1692)
Elena Babalic, Calin Lazaroiu, Foliated eight-manifolds for M-theory compactification, JHEP 01 (2015) 140 (arXiv:1411.3148)
C. S. Shahbazi, M-theory on non-Kähler manifolds, JHEP 09 (2015) 178 (arXiv:1503.00733)
Elena Babalic, Calin Lazaroiu, Singular foliations for M-theory compactification, JHEP 03 (2015) 116 (arXiv:1411.3497)
Elena Babalic, Calin Lazaroiu, The landscape of $G$-structures in eight-manifold compactifications of M-theory, JHEP 11 (2015) 007 (arXiv:1505.02270)
Elena Babalic, Calin Lazaroiu, Internal circle uplifts, transversality and stratified $G$-structures, JHEP 11 (2015) 174 (arXiv:1505.05238)
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):
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, M-Theory on $Spin(7)$ Manifolds, Nucl. Phys. B625 (2002) 3-69 (arXiv:hep-th/0109025)
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, $\mathcal{N}=1\,$ 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 $G_2$ holonomy reductions and M-theory/IIA dualities, Phys. Rev. D79: 086009, 2009 (arXiv:0811.4422)
Adil Belhaj, Luis J. Boya, Antonio Segui, Holonomy Groups Coming From F-Theory Compactification, Int J Theor Phys (2010) 49: 681. (arXiv:0911.2125)
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 Pugh, Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds, JHEP 01 (2014) 112 (arXiv:1307.5858)
Federico Bonetti, Thomas Grimm, Eran Palti, Tom Pugh, F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit, JHEP02(2014)076 (arXiv:1309.2287)
Tom Pugh, M-theory on $Spin(7)$-manifolds and their F-theory duals (pdf)
Andreas Braun, Sakura Schaefer-Nameki, $Spin(7)$-Manifolds as Generalized Connected Sums and 3d $N=1$ Theories, JHEP 06 (2018) 103 (arXiv:1803.10755)
(generalization of compact twisted connected sum G2-manifolds)
See also
Relating M-theory on Spin(7)-manifolds with F-theory on Spin(7)-manifolds via Higgs bundles:
M-theory on HP^2, hence on a quaternion-Kähler manifold of dimension 8 with holonomy Sp(2).Sp(1), is considered in
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).
See also
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):
Edward Witten, p. 7 of: The Cosmological Constant From The Viewpoint Of String Theory, lecture at DM2000, in: David Kline (ed.) Sources and detection of dark matter and dark energy in the universe 2000, Springer 2001. 27-36. (arXiv:hep-ph/0002297, doi:10.1007/978-3-662-04587-9)
(see p. 7)
Edward Witten, Strong coupling and the cosmological constant, Mod. Phys. Lett. A 10:2153-2156, 1995 (arXiv:hep-th/9506101)
Edward Witten, Section 3 of Some Comments On String Dynamics, talk at Strings95 (arXiv:hep-th/9507121)
The realization of this scenario in F-theory on Spin(7)-manifolds:
Cumrun Vafa, Section 4.3 of: Evidence for F-Theory, Nucl. Phys. B469:403-418, 1996 (arxiv:hep-th/9602022)
Federico Bonetti, Thomas Grimm, Tom Pugh, Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds, JHEP 01 (2014) 112 (arXiv:1307.5858)
Federico Bonetti, Thomas Grimm, Eran Palti, Tom Pugh, F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit, JHEP 02 (2014) 076 (arXiv:1309.2287)
Jonathan Heckman, Craig Lawrie, Ling Lin, Gianluca Zoccarato, F-theory and Dark Energy, Fortschritte der Physik (arXiv:1811.01959, doi:10.1002/prop.201900057)
Jonathan Heckman, Craig Lawrie, Ling Lin, Jeremy Sakstein, Gianluca Zoccarato, Pixelated Dark Energy (arXiv:1901.10489)
Discussion of M-theory KK-compactified on the product manifold of two K3s:
Paul Aspinwall, An Analysis of Fluxes by Duality,(arXiv:hep-th/0504036, spire:679724)
Paul Aspinwall, Renata Kallosh, Fixing All Moduli for M-Theory on $K3 \times K3$, JHEP 0510:001, 2005 (arXiv:hep-th/0506014)
Andreas Braun, Arthur Hebecker, Christoph Ludeling, Roberto Valandro, Fixing D7 Brane Positions by F-Theory Fluxes, Nucl. Phys. B815:256-287, 2009 (arXiv:0811.2416)
Under duality between M-theory and type IIA string theory this translates to D6-branes wrapped on K3 (enhancon mechanism):
Review:
Satoshi Yamaguchi, Enhancon and Resolution of Singularity (arXiv:gr-qc/0108084)
Laur Järv, The enhancon mechanism in string theory, Durham 2002 (spire:899784, etheses:3981, pdf)
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