M-theory on G2-manifolds

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#Contents#
* table of contents
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## Idea

The [[Kaluza-Klein reduction]] of [[11-dimensional supergravity]] on [[G2 manifolds]] (notably [[Freund-Rubin compactifications]] and variants) yields an [[effective field theory|effective]] $N=1$ [[4-dimensional supergravity]] with [[gauge fields]] (arising from the KK-modes of the [[graviton]]) and charged [[fermions]] (arising from the KK-models of the [[gravitino]]). This construction is thought to lift to [[M-theory]] as the analog of the KK-compactification of [[heterotic string theory]] on [[Calabi-Yau manifolds]] (see at _[[string phenomenology]]_), and of [[F-theory on CY4-manifolds]].

[[!include N=1 susy compactifications -- table]]

In order for this to yield [[phenomenology|phenomenologically]] interesting effective physics the compactification space must be a [[G2-orbifold]] (hence a [[Riemannian orbifold]] of [[special holonomy]]), its [[stabilizer groups]] will encode the [[nonabelian group|nonabelian]] [[gauge group]] of the effective theory by "[[geometric engineering of quantum field theory]]" ([Acharya 98](#Acharya98), [Atiyah-Witten 01, section 6](#AtiyahWitten01)), see [below](#EnhancedGaugeGroups). Specifically for discussion of [[string phenomenology]] obtaining or approximating the [[standard model of particle physics]] by this procedure see at _[[G2-MSSM]]_.

## Details
 {#G2Manifolds}

### Vacuum solutions
 {#VacuumSolutionsAndTorsion}

Genuine [[G2-manifold]]/[[orbifold]] fibers, these having vanishing [[Ricci curvature]], correspond to [[vacuum]] solutions of the [[Einstein equations]] of [[11d supergravity]], i.e. with vanishing [[field strength]] of the [[gravitino]] and the [[supergravity C-field]] (see e.g. [Acharya 02, p. 9](#Acharya02)).
(If one includes non-vanishing $C$-field strength one finds "weak $G_2$-holonomy" instead, see [below](#TheCField)).

Notice that vanishing [[gravitino]] [[field strength]] (i.e. [[covariant derivative]]) means that the [[torsion of a Cartan connection|torsion]] of the super-[[vielbein]] is in each [[tangent space]] the canonical torsion of the [[super Minkowski spacetime]]. This [[supergravity torsion constraint|torsion constraint]] already just for the bosonic part $(E^a)$ of the super-vielbein $(E^a, E^\alpha)$ implies (together with the [[Bianchi identities]]) the [[equations of motion]] of supergravity, hence here the vacuum [[Einstein equations]] in the 11d [[spacetime]].

### Complexified moduli space
{#ComplexifiedModuli}

For vanishing [[field strength]] of the [[supergravity C-field]], the formal linear combination

$$
  \tau \coloneqq C_3 + i \phi_3
$$

of the (flat) supergravity $C$-field $C_3$ and the the 3-form $\phi_3$ of the $G_2$-structure is the natural [[holomorphic function|holomorphic coordinate]] on the [[moduli space]] of the [[KK-compactification]] of a $G_2$-manifold, in M-theoretic higher analogy of the complexified Kähler classes of CY compactifications of 10d string theory ([Harvey-Moore 99, (2.7)](#HarveyMoore99), [Acharya 02, (32) (59) (74)](#Acharya02), [Grigorian-Yau 08, (4.57)](#GrigorianYau08), [Acharya-Bobkov 08, (4)](#AcharyaBobkov08)).

Notice that restricted to [[associative submanifolds]] this combination becomes $C_3 + i vol$, which also governs the [[membrane instanton]]-contributions ("[[complex volume]]").

### Nonabelian gauge groups and chiral fermions at orbifold singularities
 {#EnhancedGaugeGroups}

The [[KK-compactification]] of vacuum [[11-dimensional supergravity]] on a _[[smooth manifold|smooth]]_ [[G2-manifold]] $Y$ results in a [[effective field theory|effective]] [[N=1 D=4 super Yang-Mills theory]] with [[abelian group|abelian]] [[gauge group]] $U(1)^{b_2(Y)}$ and with $b_3(Y)$ [[complex scalar fields]] which are neutral (not charged) under this gauge group (with $b_\bullet(Y)$ the [[Betti numbers]] of $Y$) (e.g. [Acharya 02, section 2.3](#Acharya02)). This is of course unsuitable for [[phenomenology]].

But when $Y$ is a $G_2$-[[orbifold]] then:

1. at an [[ADE singularity]] there is _[[enhanced gauge symmetry]] in that the [[gauge group]] (which a priori is copies of the [[abelian group]] $U(1)$ of the [[supergravity C-field]]) becomes [[nonabelian group|nonabelian]] ([Acharya 98](#Acharya98), [Acharya 00](#Acharya00), review includes [Acharya 02, section 3](#Acharya02), [BBS 07, p. 422, 436](#BBS07), [Ibáñez-Uranga 12, section 6.3.3](string+phenomenology#IbanezUranga12), [Wijnholt 14, part III](string+phenomenology#Wijnholt14) (from which the graphics below is grabbed));

1. at a (non-orbifold) [[conical singularity]] [[chiral fermions]] appear ([Witten 01, p. 3](#Witten01), [Atiyah-Witten 01](#AtiyahWitten01), [Acharya-Witten 01](#AcharyaWitten01), [Berglund-Brandhuber 02](#BerglundBrandhuber02),  [Bourjaily-Espahbodi 08](#BourjailyEspahbodi08)).

| [[ADE-singularity]] |  [[conical singularity]] |
|---------------------|--------------------------|
|  [[gauge enhancement]] | [[chiral fermions]]   |

The [[conical singularities]] are supposed/assumed to be isolated ([Witten 01, section 2](#Witten01)), while the [[ADE singularities]] are supposed/assumed to be of [[codimension]]-4 in the 7-dimensional fibers ([Witten 01, section 3](#Witten01), [Barrett 06](#Barrett06)).

In the absence of a proper microscopic definition of M-theory, the first point is argued for indirectly in at least these ways:

1. The fact that under [[KK-compactification]] to [[type IIA string theory]] the singularity becomes special points of intersecting [[D6-branes]] for which the gauge enhancement is known ([Sen 97](#Sen97), [Witten 01, p. 1](#Witten01), based on [Cvetic-Shiu-Uranga 01](#CveticShiuUranga01)).

1. The [[duality in string theory|duality]] between M-theory compactified on [[K3]] and [[heterotic string theory]] on a 3-torus ([Acharya-Witten 01](#AcharyaWitten01)). Here it is fairly well understood that at the degeneration points of the K3-[[moduli space]] enhanced nonabelian gauge symmetry appears (e.g. [Acharya-Gukov 04, section 5.1](#AcharyaGukov04)). This comes down ([Intriligator-Seiberg 96](ALE+space#IntriligatorSeiberg96)) to the fact that an [[ADE singularity]] $\mathbb{C}^2/\Gamma$ generically constitutes a point in the [[moduli space]] of [[vacua]] in the [Higgs branch](N%3D2+D%3D4+super+Yang-Mills+theory#ModuliSpacesOfVacua) of a super Yang-Mills theory.

1. {#Blowup}

 The [[blow-up]] of an [[ADE-singularity]] happens to be a union of [[2-spheres]] touching pairwise in one point, such as to form the [[Dynkin diagram]] of the [[simple Lie group]] which under the [[ADE classification]] corresponds to the given [[orbifold]] isotropy group.

   (graphics grabbed from [HSS18](http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes))

   [[M2-branes]] may [[wrapped brane|wrap]] these 2-cycles and since before blow-up they are of vanishing size, this corresponds to [[double dimensional reduction]] under which the M2-branes become [[strings]] stretching between coincident [[D-branes]]. These are well-understood to be the quanta of nonabelian gauge [[Chan-Paton gauge fields]] located on the D-branes, and hence these same nonabelian degrees of freedom have had to be present already at the level of the M2-branes. This is due to ([Sen 97](#Sen97)), for more see at _[[M-theory lift of gauge enhancement on D6-branes]]_.

1. In the [[F-theory]] description the ADE singularity maps to the locus where the F-theory [[elliptic fibration]] degenerates with 2-cycles in the elliptic fibers shrinking to 0. Via [[double dimensional reduction]] this manifestly takes the [[M2-brane]] [[wrapped brane|wrapping]] these elliptic fibers to an [[open string]] stretching between [[D7-branes]]. This yields at least $SU(N)$ gauge symmetry by the usual string theory argument about [[Chan-Paton gauge fields]].

Also notice that at least the $SU(N)$-enhancement of the [[effective field theory]] at $\mathbb{Z}_k$-singularities matches the $SU(N)$-enhancement of the [[worldvolume]] theory of $N$-coincident [[M2-branes]] sitting at the orbifold singularity: this is the statement of the [[ABJM model]].

### Solutions with non-vanishing $C$-field strength
 {#TheCField}

{#FluxBreaksSuSy} **Claim:** _There is no warped [[KK-compactification]] of M-theory on $X_4 \times F_7$ which retains at least $N = 1$ [[supersymmetry]] in 4d while at the same time having non-vanishing $G_4$-[[flux]] ([[field strength]] of the [[supergravity C-field]]). In other words, non-vanishing flux always breaks the supersymmetry._

e.g. ([Acharya-Spence 00](#AcharyaSpence00)) see the Introduction of ([Beasley-Witten 02](#BeasleyWitten02))

$\,$

In compactifications with [[weak G2 holonomy]] it is the defining 4-form $\phi_4$ (the one which for strict [[G2 manifolds]] is the [[Hodge star operator|Hodge dual]] of the [[associative 3-form]]) which is the [[flux]]/[[field strength]] of the [[supergravity C-field]]. See for instance ([Bilal-Serendinger-Sfetos 02, section 6](#BilalDerendingerSfetos)):

Consider a [[KK-compactification]]-Ansatz $X_{11} = (X_4,g_4) \times (X_7,g_7)$ and

* $F_4 = f vol_{X_4}$;

* $F_7 = \tilde g e_7^\ast \phi_4$

where $e_4$, $e_7$ are given [[vielbein]] fields on $X_4$ and $X_7$ and $\phi_4$ is the [[Hodge star operator|Hodge dual]] of the [[associative 3-form]]. Then the [[Einstein equations]] of [[11-dimensional supergravity]] give

$$
  R_4 = - \frac{1}{3}\left(f^2 + \frac{7}{2} \tilde g^2\right) g_4
$$

$$
  R_7 = \frac{1}{6}\left(f^2 + 5 \tilde g^2\right) g_7
$$

(where $g_4$, $g_7$ is the [[pseudo-Riemannian metric|metric tensor]]) saying that both spaces are [[Einstein manifolds]] ([BSS 02, (5.4)](#BilalDerendingerSfetos)). The [[equations of motion]] for the [[supergravity C-field]] is

$$
  \tilde g\left(
     d \phi - f \star\phi
  \right)
  = 0
$$

for $\phi = e_7^\ast \phi_3$ the pullback of the [[associative 3-form]] ([BSS 02, (5.5)](#BilalDerendingerSfetos)), saying that $\phi \propto \star F_7$ exhibits [weak G2-holonomy](G2+manifold#WeakG2Holonomy) with weakness parameter given by the component of the  [[C-field]] on $X_4$.

### Confinement?
 {#Confinement}

An idea for a strategy towards a proof of [[confinement]] in [[N=1 D=4 super Yang-Mills theory]] via two different but conjecturally equivalent realizations as [[M-theory on G2-manifolds]] has been given in [Atiyah-Witten 01, section 6](#AtiyahWitten01), review is in [Acharya-Gukov 04, section 5.3](#AcharyaGukov04).

The idea here is to consider a [[KK-compactification]] of [[M-theory]] on [[fibers]] which are [[G2-manifolds]] that locally around a special point
are of the form

$$
  X_{1,\Gamma}
  \;\coloneqq\;
  \big( S^3 / \Gamma \big) \times Cone\big(S^3\big)
  \phantom{AA}
  \text{or}
  \phantom{AA}
  X_{2,\Gamma}
  \;\coloneqq\;
  S^3 \times Cone\big(S^3/\Gamma\big)
$$

where

* $\Gamma$ is a [[finite subgroup of SU(2)]] that [[action|acts]] canonically by left-multiplication on $S^3 \simeq $ [[SU(2)]];

* $Cone(\cdots)$ denotes the [[metric cone]] construction.

This means that $X_{1,\Gamma}$ is a [[smooth manifold]], but $X_{2,\Gamma}$, as soon as $\Gamma$ is not the [[trivial group]], $\Gamma \neq 1$, is an [[orbifold]] with an [[ADE singularity]].

Now the lore of [[M-theory on G2-manifolds]] predicts that [[KK-compactification]]

1.  on $X_{1,\Gamma}$ yields a 4d theory without massless fields (since there are no massless modes on the [[covering space]] $S^3$ of $X_{1,\Gamma}$)

1. on the [[ADE-singularity]] $X_{2,\Gamma}$ yields [[non-abelian group|non-abelian]] [[Yang-Mills theory]] in 4d coupled to [[chiral fermions]].

both of these [[duality in string theory|dual]] by thinking of them in two different ways as [[M-theory on 8-manifolds|M-theory on the 8-manifold]] [[HP^2]] ([Atiyah-Witten 01, p. 75 onwards](#AtiyahWitten01)).

So in the first case a [[mass gap]] is manifest, while non-abelian gauge theory is not visible, while in the second case it is the other way around.

But if there were an argument that [[M-theory on G2-manifolds]] is in fact equivalent for compactification both on $X_{1,\Gamma}$ and on $X_{2,\Gamma}$. To the extent that this is true, it looks like an argument that could demonstrate confinement in non-abelian 4d gauge theory.

This approach is suggested in [Atiyah-Witten 01, pages 84-85](#AtiyahWitten01). An argument that this equivalence is indeed the case is then provided in sections 6.1-6.4, based on an argument in [Atiyah-Maldacena-Vafa 00](#AtiyahMaldacenaVafa00).

### Relation to intersecting D-brane models

relation to [[intersecting D-brane models]]: see [there](intersecting+D-brane+model#ReferencesLiftToMTheory)

## Related concepts

* [[G2]]

* [[G2 manifold]], [[generalized G2-manifold]]

* [[G2-MSSM]]

* [[Freund-Rubin compactification]]

* [[exceptional generalized geometry]]

* [[topological M-theory]], [[Hitchin functional]]

* [[7d Chern-Simons theory]]

[[!include KK-compactifications of M-theory -- table]]

## References
 {#References}

### General

The first discussion of [[KK-compactification]] of [[11d supergravity]] on  [[fibers]] with [[G2-holonomy]] is due to:

* {#AwadaDuffPope83} M. A. Awada, [[Mike Duff]], [[Christopher Pope]], _$N=8$ Supergravity Breaks Down to $N=1$_, Phys. Rev. Lett. 50, 294 (1983) ([doi:10.1103/PhysRevLett.50.294](https://doi.org/10.1103/PhysRevLett.50.294))

* [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere_, Phys. Rev. Lett. 50, 2043 (1983) ([doi:10.1103/PhysRevLett.50.2043](https://doi.org/10.1103/PhysRevLett.50.2043), [erratum](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.846))

  > (compactification on a [[squashed sphere|squashed]] [[7-sphere]] with $G_2$-holonomy)

More generally, the [[KK-compactification]] of [[11d supergravity]] of [[fibers]] of [[special holonomy]] was originally considered in

* [[Edward Witten]], _Search for a realistic Kaluza-Klein theory_, Nuclear Physics B Volume 186, Issue 3, 10 August 1981, Pages 412-428 (doi:10.1016/0550-3213(81)90021-3)

* {#DauriaFreCastellani91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], chapter V.6 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[George Papadopoulos]], [[Paul Townsend]], _Compactification of D=11 supergravity on spaces of exceptional holonomy_, Phys. Lett. B357:300-306,1995 ([arXiv:hep-th/9506150](http://arxiv.org/abs/hep-th/9506150))

Dedicated [[string phenomenology]] for the case of compactification on [[G2-manifolds]] (or rather [[orbifolds]]) goes back to:

* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv. Theor. Math. Phys. **3** (1999) 227-248 [[arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205)]

* {#Acharya00} [[Bobby Acharya]], _On Realising $N=1$ Super Yang-Mills in M theory_ ([arXiv:hep-th/0011089](http://arxiv.org/abs/hep-th/0011089))

* {#AcharyaSpence00} [[Bobby Acharya]], B. Spence, _Flux, Supersymmetry and M theory on 7-manifolds_ ([arXiv:hep-th/0007213](http://arxiv.org/abs/hep-th/0007213))

* {#Acharya02} [[Bobby Acharya]], _M Theory, $G_2$-manifolds and four-dimensional physics_,  Classical and Quantum Gravity Volume 19 Number 22, 2002  ([doi:10.1088/0264-9381/19/22/301](https://iopscience.iop.org/article/10.1088/0264-9381/19/22/301/meta), [pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))

* {#AtijayMaldacenaVafa00} [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], _An M-theory Flop as a Large N Duality_, J. Math. Phys.42:3209-3220, 2001 ([arXiv:hep-th/0011256](https://arxiv.org/abs/hep-th/0011256))

* [[Sergei Gukov]]. *Solitons, Superpotentials and Calibrations*. Nuclear Physics B Volume 574, Issues 1–2, 15 May 2000, Pages 169-188. ([doi](https://doi.org/10.1016/S0550-3213(00)00053-5))

* {#BeasleyWitten02} [[Chris Beasley]], [[Edward Witten]], _A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of $G_2$ Holonomy_, JHEP 0207:046,2002 ([arXiv:hep-th/0203061](http://arxiv.org/abs/hep-th/0203061))

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177), [doi:10.4310/ATMP.2002.v6.n1.a1]( https://dx.doi.org/10.4310/ATMP.2002.v6.n1.a1))

* {#Witten01} [[Edward Witten]], _Anomaly Cancellation On Manifolds Of $G_2$ Holonomy_ ([arXiv:hep-th/0108165](http://arxiv.org/abs/hep-th/0108165))

* {#AcharyaWitten01} [[Bobby Acharya]], [[Edward Witten]], _Chiral Fermions from Manifolds of $G_2$ Holonomy_ ([arXiv:hep-th/0109152](http://arxiv.org/abs/hep-th/0109152), [spire:563029](https://inspirehep.net/literature/563029))

* {#BerglundBrandhuber02} Per Berglund, Andreas Brandhuber, _Matter from $G_2$-manifolds_, Nucl. Phys. B641 (2002) 351-375 ([arXiv:hep-th/0205184](https://arxiv.org/abs/hep-th/0205184))

* {#Barrett06} Adam B. Barrett, _M-Theory on Manifolds with $G_2$ Holonomy_, 2006 ([arXiv:hep-th/0612096](http://arxiv.org/abs/hep-th/0612096))

* {#BourjailyEspahbodi08} Jacob L. Bourjaily, Sam Espahbodi, _Geometrically Engineerable Chiral Matter in M-Theory_ ([arXiv:0804.1132](https://arxiv.org/abs/0804.1132))

Expanding on [Atiyah-Witten 01](#AtiyahWitten01):

* [[Bobby Acharya]], Lorenzo Foscolo, Marwan Najjar, Eirik Eik Svanes, _New $G_2$-conifolds in M-theory and their Field Theory Interpretation_ ([arXiv:2011.06998](https://arxiv.org/abs/2011.06998))

See also:

* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

* [[Mirjam Cvetic]], [[Gary Gibbons]], H. Lü  and [[Christopher Pope]], _Supersymmetric M3-branes and G2 Manifolds_ ([pdf](http://cdsweb.cern.ch/record/503160/files/0106026.pdf))

* {#AcharyaDenefHofmanLambert03} [[Bobby Acharya]], F. Denef, C. Hofman, [[Neil Lambert]], _Freund-Rubin Revisited_ ([arXiv:hep-th/0308046](http://arxiv.org/abs/hep-th/0308046))

More discussion of the non-abelian gauge group enhancement at [[orbifold]] singularities includes

* {#CveticShiuUranga01} [[Mirjam Cvetic]], [[Gary Shiu]], [[Angel Uranga]], _Chiral Four-Dimensional $N=1$ Supersymmetric Type IIA Orientifolds from Intersecting D6-Branes_, Nucl. Phys. B615:3-32,2001 ([arXiv:hep-th/0107166](http://arxiv.org/abs/hep-th/0107166))

* {#HalversonMorrison15} [[James Halverson]], [[David Morrison]], _On Gauge Enhancement and Singular Limits in $G_2$ Compactifications of M-theory_ ([arXiv:1507.05965](http://arxiv.org/abs/1507.05965))

* [[Antonella Grassi]], [[James Halverson]], Julius L. Shaneson, _Matter From Geometry Without Resolution_, Journal of High Energy Physics October 2013, 2013:205 ([arXiv:1306.1832](http://arxiv.org/abs/1306.1832))

* [[Neil Lambert]], Miles Owen, _Charged Chiral Fermions from M5-Branes_ ([arXiv:1802.07766](https://arxiv.org/abs/1802.07766))

* {#BCHS19} [[Andreas Braun]], Sebastjan Cizel, Max Hubner, [[Sakura Schafer-Nameki]], _Higgs Bundles for M-theory on G2-Manifolds_ ([arXiv:1812.06072](https://arxiv.org/abs/1812.06072))

* Max Hubner, _Local $G_2$-Manifolds, Higgs Bundles and a Colored Quantum Mechanics_ ([arXiv:2009.07136](https://arxiv.org/abs/2009.07136))

Discussion in relation to the [[duality between M-theory and heterotic string theory]]:

* [[Bobby Acharya]], Alex Kinsella, [[David Morrison]], *Non-Perturbative Heterotic Duals of M-Theory on $G_2$ Orbifolds* ([arXiv:2106.03886](https://arxiv.org/abs/2106.03886))

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:

* {#BilalDerendingerSfetos} [[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](http://arxiv.org/abs/hep-th/0111274))

* {#HouseMicu04} Thomas House, Andrei Micu, _M-theory Compactifications on Manifolds with $G_2$ Structure_ ([arXiv:hep-th/0412006](http://arxiv.org/abs/hep-th/0412006))

Further discussion of [[membrane instantons]] in this context includes

* [[Gottfried Curio]], _Superpotentials for M-theory on a G_2 holonomy manifold and Triality symmetry_, JHEP 0303:024,2003 ([arXiv:hep-th/0212211](http://arxiv.org/abs/hep-th/0212211))

Survey and further discussion includes

* [[Michael Duff]], _M-theory on manifolds of G2 holonomy: the first twenty years_ ([arXiv:hep-th/0201062](http://arxiv.org/abs/hep-th/0201062))

* [[Sergei Gukov]], _M-theory on manifolds with exceptional holonomy_, Fortschr. Phys. 51 (2003), 719–731 ([pdf](http://research.physics.unc.edu/string/gukov.pdf))

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

* Adil Belhaj, _M-theory on G2 manifolds and the method of (p, q) brane webs_ (2004) ([web](http://iopscience.iop.org/0305-4470/37/18/011))

* Adam B. Barrett, _M-Theory on Manifolds with $G_2$ Holonomy_ ([arXiv:hep-th/0612096](http://arxiv.org/abs/hep-th/0612096))

* [[James Halverson]], [[David Morrison]], _The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy_ ([arXiv:1412.4123](http://arxiv.org/abs/1412.4123))

* Aaron Kennon, _$G_2$-Manifolds and M-Theory Compactifications_ ([arXiv:1810.12659](https://arxiv.org/abs/1810.12659))

The corresponding [[membrane]] [[instanton]] corrections to the [[superpotential]] are discussed in

* {#HarveyMoore99} [[Jeffrey Harvey]], [[Greg Moore]], _Superpotentials and Membrane Instantons_ ([arXiv:hep-th/9907026](http://arxiv.org/abs/hep-th/9907026))

* {#BBS07} [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 333 of _String Theory and M-Theory: A Modern Introduction_, 2007

Discussion of [[duality in string theory|duality]] with [[F-theory on CY4-manifolds]] includes

* {#GukovYauZaslow02} [[Sergei Gukov]], [[Shing-Tung Yau]], [[Eric Zaslow]], _Duality and Fibrations on $G_2$ Manifolds_ ([arXiv:hep-th/0203217](http://arxiv.org/abs/hep-th/0203217))

* {#Belhaj02} Adil Belhaj, _F-theory Duals of M-theory on $G_2$ Manifolds from Mirror Symmetry_ ([arXiv:hep-th/0207208](http://arxiv.org/abs/hep-th/0207208))

* [[Mariana Graña]], [[Carlos S. Shahbazi]], [[Marco Zambon]], _$Spin(7)$-manifolds in compactifications to four dimensions_, JHEP11(2014)046 [[arXiv:1405.3698](http://arxiv.org/abs/1405.3698)]

Discussion of [[duality in string theory|duality]] with [[heterotic string theory on CY3-manifolds]]:

* {#BraunSchaeferNameki17} [[Andreas Braun]], Sakura Schaefer-Nameki, _Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality_, JHEP04(2018)126 ([arXiv:1708.07215](https://arxiv.org/abs/1708.07215))

The [[moduli space]] is discussed in

* {#GrigorianYau08} [[Sergey Grigorian]], [[Shing-Tung Yau]], _Local geometry of the $G_2$ moduli space_, Commun.Math.Phys.287:459-488,2009 ([arXiv:0802.0723](http://arxiv.org/abs/0802.0723))

* {#AcharyaBobkov08} [[Bobby Acharya]], Konstantin Bobkov, _Kähler Independence of the G2-MSSM_, HEP 1009:001,2010 ([arXiv:0810.3285](http://arxiv.org/abs/0810.3285))

* [[Spiro Karigiannis]], [[Naichung Conan Leung]]_, _Hodge Theory for $G_2$-manifolds: Intermediate Jacobians and Abel-Jacobi maps_, Proceedings of the London Mathematical Society (3) 99, 297-325 (2009) ([arXiv:0709.2987](http://arxiv.org/abs/0709.2987) [talk slides pdf](http://www.math.uwaterloo.ca/~karigian/talks/g2modulispace.pdf)

### Phenomenology
 {#ReferencesPhenomenology}

Popular exposition of the [[G2-MSSM]] [[phenomenology]] is in

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017 (doi:0.1088/978-1-6817-4489-6)

Further discussion of [[string phenomenology]] in terms of $M$-theory on $G_2$-manifolds, beyond the original ([Acharya 98](#Acharya98), [Atiyah-Witten 01](#AtiyahWitten01), [Acharya-Witten 01](#AcharyaWitten01)), includes

* {#AcharyaKaneKumar12} [[Bobby Acharya]], [[Gordon Kane]], [[Piyush Kumar]], _Compactified String Theories -- Generic Predictions for Particle Physics_, Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](http://arxiv.org/abs/1204.2795))

* {#Acharya12} [[Bobby Acharya]], _$G_2$-manifolds at the CERN Large Hadron collider and in the Galaxy_, talk at _$G_2$-days_ (2012) ([pdf](http://www.mth.kcl.ac.uk/~tbmadsen/acharya.pdf))

* {#Kane16} [[Gordon Kane]], _String/M-theories About Our World Are Testable in the traditional Physics Way_ ([arXiv:1601.07511](http://arxiv.org/abs/1601.07511), [video recording](https://videoonline.edu.lmu.de/en/node/7485))

Discussion of [[moduli stabilization]] for stabilization via "[[flux]]" (non-vanishing bosonic  [[field strength]] of the [[supergravity C-field]]) is in

* {#Acharya02} [[Bobby Acharya]], _A Moduli Fixing Mechanism in M theory_ ([arXiv:hep-th/0212294](http://arxiv.org/abs/hep-th/0212294))

and [[moduli stabilization]] for fluxless compactifications via [[nonperturbative effects]], claimed to be sufficient and necessary to solve the [[hierarchy problem]], is discussed in

* {#AcharyaBobkovKaneKumarVaman06} [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, _An M theory Solution to the Hierarchy Problem_, Phys.Rev.Lett.97:191601,2006 ([arXiv:hep-th/0606262](http://arxiv.org/abs/hep-th/0606262))

* {#AcharyaBobkovKaneKumarShao07} [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _Explaining the Electroweak Scale and Stabilizing Moduli in M Theory_, Phys.Rev.D76:126010,2007 ([arXiv:hep-th/0701034](http://arxiv.org/abs/hep-th/0701034))

* {#AcharyaKumarBobbkovKaneShaoWatson08} [[Bobby Acharya]], [[Piyush Kumar]], Konstantin Bobkov, [[Gordon Kane]], Jing Shao, Scott Watson, _Non-thermal Dark Matter and the Moduli Problem in String Frameworks_,JHEP 0806:064,2008 ([arXiv:0804.0863](http://arxiv.org/abs/0804.0863))

and specifically for the [[G2-MSSM]] in

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics_ ([arXiv:0801.0478](http://arxiv.org/abs/0801.0478))

the [[strong CP problem]] is discussed in

* {#SvrcekWitten06} Peter Svrcek, [[Edward Witten]], section 6 of _Axions In String Theory_, JHEP 0606:051,2006 ([arXiv:hep-th/0605206](http://arxiv.org/abs/hep-th/0605206))

* [[Bobby Acharya]], Konstantin Bobkov, [[Piyush Kumar]], _An M Theory Solution to the Strong CP Problem and Constraints on the Axiverse_, JHEP 1011:105,2010 ([arXiv:1004.5138](http://arxiv.org/abs/1004.5138))

and realization of [[GUTs]] in

* {#Witten02} [[Edward Witten]], _Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting_, ([arXiv:hep-ph/0201018](http://arxiv.org/abs/hep-ph/0201018))

* [[Bobby Acharya]], Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, _$SO(10)$ Grand Unification in M theory on a $G_2$ manifold_ ([arXiv:1502.01727](http://arxiv.org/abs/1502.01727))

The phenomenology of compactifications on [[compact twisted connected sum G2-manifolds]] ([Kovalev 00](G2+manifold#Kovalev00)) is in

* {#GHKY17} Thaisa C. da C. Guio, [[Hans Jockers]], [[Albrecht Klemm]], Hung-Yu Yeh, _Effective action from M-theory on twisted connected sum $G_2$-manifolds_ ([arXiv:1702.05435](https://arxiv.org/abs/1702.05435), [talk video](https://lecture2go.uni-hamburg.de/l2go/-/get/v/21906))

Discussion of the [[cosmological constant]] in these models includes

* Beatriz de Carlos, Andre Lukas, Stephen Morris, _Non-perturbative vacua for M-theory on G2 manifolds_, JHEP 0412:018, 2004 ([arxiv:hep-th/0409255](https://arxiv.org/abs/hep-th/0409255))

which concludes that with taking [[non-perturbative effects]] from [[membrane instantons]] into account one gets 4d vacua with vanishing and negative [[cosmological constant]] ([[Minkowski spacetime]] and [[anti-de Sitter spacetime]]) but not with positive [[cosmological constant]] ([[de Sitter spacetime]]). They close by speculating that [[M5-brane]]
instantons might yield [[de Sitter spacetime]].

Suggestion that [[higher curvature corrections]] allow [[de Sitter spacetime]] vacua:

* Johan Blåbäck, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, _Constructing stable de Sitter in M-theory from higher curvature corrections_ ([arXiv:1902.04053](https://arxiv.org/abs/1902.04053))

See also

* {#BCHS18} [[Andreas Braun]], Sebastjan Cizel, Max Hubner, Sakura Schafer-Nameki, _Higgs Bundles for M-theory on $G_2$-Manifolds_ ([arXiv:1812.06072](https://arxiv.org/abs/1812.06072))

Discussion of [[Yukawa couplings]] among 3 [[generations of fundamental fermions]]:

* Eric Gonzalez, [[Gordon Kane]], Khoa Dang Nguyen, [[Malcolm Perry]], _Quark and lepton mass matrices from localization in M-theory on $G_2$ orbifold_ ([arXiv:2002.11820](https://arxiv.org/abs/2002.11820))

On [[neutrino]] masses:

* Eric Gonzalez, [[Gordon Kane]], Khoa Nguyen, *Neutrino mass matrices from localization in M-theory on G2 orbifold* ([arXiv:2107.12893](https://arxiv.org/abs/2107.12893))

[[!redirects G2 compactifications of M-theory]]


M-theory on hyperbolic manifolds

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## Idea

The [[Kaluza-Klein compactification]] of [[11-dimensional supergravity]]/[[M-theory]] on [[fibers]] which are [[hyperbolic manifolds]].

This appears naturally as the [[near horizon geometry]] of [[S-branes]], see there. Then there are [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]] yielding the [[3d-3d correspondence]] ([Gang-Kim-Lee 15, (2.2)](#GangKimLee15)).

## Related concepts

* [[S-brane]]

[[!include KK-compactifications of M-theory -- table]]

## References

* A. Kehagias, J. G. Russo, _Hyperbolic Spaces in String and M-Theory_, JHEP 0007 (2000) 027 ([arXiv:hep-th/0003281](https://arxiv.org/abs/hep-th/0003281))

* Chiang-Mei Chen, Pei-Ming Ho, Ishwaree P. Neupane, Nobuyoshi Ohta, John E. Wang, _Hyperbolic Space Cosmologies_, JHEP 0310:058, 2003 ([arXiv:hep-th/0306291](https://arxiv.org/abs/hep-th/0306291))

* A. A. Bytsenko, M. E. X. Guimaraes, R. Kerner, _Orbifold Compactification and Solutions of M--Theory from Milne Spaces_, Eur. Phys. J. C39:519-524, 2005 ([arXiv:hep-th/0501008](https://arxiv.org/abs/hep-th/0501008))

* Andrey A. Bytsenko, Maria Emília X. Guimarães, José Abdalla Helayël-Neto, _Hyperbolic Space Forms and Orbifold Compactification in M-Theory_, PoS WC2004 (2005) pp.017 ([arXiv:hep-th/0502031](https://arxiv.org/abs/hep-th/0502031), [cds:819828](https://cds.cern.ch/record/819828/))

* Domenico Orlando, _M-theory compactifications on hyperbolic spaces_, Fortsch. Phys. 55:793-797, 2007 ([arXiv:hep-th/0702013](https://arxiv.org/abs/hep-th/0702013))

* Domenico Orlando, Seong Chan Park, _Compact hyperbolic extra dimensions: a M-theory solution and its implications for the LHC_, JHEP 1008:006, 2010 ([arXiv:1006.1901](https://arxiv.org/abs/1006.1901))

See also

* Alan S. Cornell, _Hyperbolic extra-dimensions in particle physics and beyond_ ([arXiv:1506.05598](https://arxiv.org/abs/1506.05598))

Discussion on [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-space]] in the context of [[3d-3d correspondence]]:

* {#GangKimLee15} Dongmin Gang, Nakwoo Kim, Sangmin Lee, _Holography of 3d-3d correspondence at Large N_, JHEP04 (2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))

Discussion in a context of [[cosmology]] with an eye towards [[de Sitter spacetimes]]:

* G. Bruno De Luca, [[Eva Silverstein]], Gonzalo Torroba, *Hyperbolic compactification of M-theory and de Sitter quantum gravity* ([arXiv:2104.13380](https://arxiv.org/abs/2104.13380))

[[!redirects M-theory an a hyperbolic manifold]]


M-theory on Sp(1)-manifolds

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## Idea

The [[KK-compactification]] of [[M-theory]]/[[string theory]] on [[spacetime]] [[manifolds]] with [[G-structure]] for $G = $ [[Sp(1)]], notably on the [[product space]] $K3 \times T2$ of a [[K3-surface]] with a [[2-torus]].

## Related concepts

[[!include KK-compactifications of M-theory -- table]]

## References

For [[type IIA string theory]]:

* Bastiaan Spanjaard, _Compactifications of IIA Supergravity on $SU(2)$-Structure Manifolds_, 2008 ([pdf](http://www.desy.de/uni-th/stringth/Works/PhDthesis_Bastiaan_Spanjaard.pdf))

* Thomas Danckaert, _Type IIA orientifolds on $SU(2)$-structure Manifolds_, 2010 ([pdf](http://www-library.desy.de/preparch/desy/thesis/desy-thesis-10-046.pdf))

* Thomas Danckaert, [[Jan Louis]], _Type IIA orientifold compactification on SU(2)-structure manifolds_, JHEP 1001:105, 2010 ([arXiv:0911.5697](https://arxiv.org/abs/0911.5697))

For [[heterotic string theory]]:

* [[Jan Louis]], Danny Martinez-Pedrera, [[Andrei Micu]], _Heterotic compactifications on SU(2)-structure backgrounds_, JHEP 0909:012, 2009 ([arXiv:0907.3799](https://arxiv.org/abs/0907.3799))

[[!redirects M-theory on SU(2)-manifold]]

[[!redirects string theory on Sp(1)-manifolds]]
[[!redirects string theory on SU(2)-manifolds]]

[[!redirects heterotic string theory on Sp(1)-manifolds]]
[[!redirects heterotic string theory on SU(2)-manifolds]]

[[!redirects type II string theory on Sp(1)-manifolds]]
[[!redirects type II string theory on SU(2)-manifolds]]

[[!redirects type IIA string theory on Sp(1)-manifolds]]
[[!redirects type IIA string theory on SU(2)-manifolds]]

[[!redirects type IIB string theory on Sp(1)-manifolds]]
[[!redirects type IIB string theory on SU(2)-manifolds]]


M-theory supersymmetry algebra

[[!redirects M-theory super Lie algebra]]

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## Idea

A [[super Lie algebra]] which is a [polyvector](super+Poincare+Lie+algebra#PolyvectorExtensions) [[Lie algebra extension|extension]] of the [[super Poincaré Lie algebra]] ([[supersymmetry]]) in $D = 10+1$ for  $N=1$ [[supersymmetry]] by [[charges]] corresponding to the [[M2-brane]] and the [[M5-brane]] ("[[extended supersymmetry]]").

## Properties

### As the algebra of conserved currents of the M-branes

In ([AGIT 89](#AGIT89)) it is shows that the M-algebra-like polyvector extensions arise as the algebras of [[conserved currents]] of the [[Green-Schwarz super p-brane sigma-models]].

By the discussion at [conserved current -- In higher prequantum geometry](conserved+current#InHigherPrequantumGeometry) this means that this is the degree-0 piece in the [[Heisenberg Lie n-algebra]] which is induced by regarding the WZW-curvature terms as super [[n-plectic forms]] on $\mathbb{R}^{10,1|32}$.

### As the Lie algebra of derivations of the SuGra Lie 3-algebra

In ([Castellani 05](#Castellani05)) it is
implicitly shown ([FSS 13](#FSS13)) that the M-extension arises  as the [[derivations]]/automorphisms of the [[supergravity Lie 3-algebra]]/[[supergravity Lie 6-algebra]] (see there for the details).

### As an 11-dimensional boundary condition for the M2-brane
 {#AsAn11DimensionalBoundaryCondition}

The original construction in ([D'Auria-Fré 82](#DAuriaFre82)) asks for a [[super Lie algebra]] [[Lie algebra extension|extension]] $\mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g}$ of [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert 32}$ such that the 4-[[cocycle]] $\mu_4 = \overline{\psi} \wedge \Gamma^{a b} \psi \wedge e_a \wedge e_b$ for the M2-brane trivializes when pulled back to this:

$$
  \array{
    & & \mathbb{R}^{10,1\vert 32} \rtimes \mathfrak{g}
    \\
    & \swarrow && \searrow
    \\
    \ast && \swArrow_{\simeq} && \mathbb{R}^{10,1\vert 32}
    \\
    & \searrow && \swarrow_{\mathrlap{\mu_4}}
    \\
    && B^{3}\mathbb{R}
  }
  \,.
$$

(In the language of [[local prequantum field theory]] this identifies a [[boundary condition]] for the WZW term of the M2-brane.)

They find, see also ([Bandos-Azcarraga-Izquierdo-PiconVarela 04](#BandosAzcarragaIzquierdoPiconVarela04)) that a solution for $\mathfrak{g}$ includes a fermionic extension of the M-theory super Lie algebra.

## Related concepts

* [[type II super Lie algebra]]

* [[BPS state]]

* [[supergravity Lie 3-algebra]]

* [[supergravity Lie 6-algebra]]

* [[M2-brane]], [[M5-brane]]

* [[M-theory]]

## References

### From the M2-Cocyle
 {#ReferencesFromTheM2Cocycle}

Two versions of a fermionic extension of the Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ on which the [[M2-brane]] [[Lie algebra cohomology|4-cocycle]] trivializes were first found in

* {#DAuriaFre82}  [[Riccardo D'Auria]], [[Pietro Fré]], last pages of _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140

and then a 1-parameter family of such was discovered in

* {#BandosAzcarragaIzquierdoPiconVarela04} [[Igor Bandos]], [[José de Azcárraga]], J.M. Izquierdo, M. Picon, O. Varela, _On the underlying gauge group structure of D=11 supergravity_, Phys.Lett.B596:145-155,2004 ([arXiv:hep-th/0406020](http://arxiv.org/abs/hep-th/0406020))

* [[Igor Bandos]], [[José de Azcárraga]], Moises Picon, Oscar Varela, _On the formulation of $D=11$ supergravity and the composite nature of its three-from field_, Annals Phys. 317 (2005) 238-279 ([arXiv:hep-th/0409100](https://arxiv.org/abs/hep-th/0409100))

That a limiting case of this is given by the [[orthosymplectic super Lie algebra]] $\mathfrak{osp}(1\vert 32)$:

* {#FernandezIzquierdoOlmo15} J.J. Fernandez, [[José M. Izquierdo]], M.A. del Olmo, _Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators_, Nuclear Physics B **897** (2015) 87-97 [[arXiv:1504.05946](http://arxiv.org/abs/1504.05946)]

Further discussion is in

* {#AndrianopoliDAuriaRavera16} [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _Hidden Gauge Structure of Supersymmetric Free Differential Algebras_, JHEP 1608 (2016) 095 ([arXiv:1606.07328](https://arxiv.org/abs/1606.07328))

* [[Laura Andrianopoli]], [[Riccardo D'Auria]], [[Lucrezia Ravera]], _More on the Hidden Symmetries of 11D Supergravity_ ([arXiv:1705.06251](https://arxiv.org/abs/1705.06251))

where the algebra is referred to as the _DF-algebra_, in honor of [D'Auria-Fré 82](#DAuriaFre82).

All this is reviewed in

* {#Azcarraga05} [[José de Azcárraga]], section 5 of _Superbranes, D=11 CJS supergravity and enlarged superspace coordinates/fields correspondence_, AIP Conf. Proc. 767:243-267, 2005 ([arXiv:hep-th/0501198](https://arxiv.org/abs/hep-th/0501198))

also

* [[Lucrezia Ravera]], *On the hidden symmetries of $D=11$ supergravity* ([arXiv:2112.00445](https://arxiv.org/abs/2112.00445))

Another, alternative "weak [[decomposable differential form|decomposition]]" of the M2-brane [[extended super-Minkowski spacetime]] was found in

* {#Ravera18a} [[Lucrezia Ravera]], section 3 of _Hidden Role of Maxwell Superalgebras in the Free Differential Algebras of D=4 and D=11 Supergravity_ ([arXiv:1801.08860](https://arxiv.org/abs/1801.08860))

with the interesting difference that for this splitting [[super Lie algebra]]  is non-abelian, in fact an extension of the Lie algebra of the [[Spin group]] $Spin(10,1)$ ([Ravera 18, (3.5)-(3-6)](#Ravera18)).

See also:

* {#Ravera18b} [[Lucrezia Ravera]], _Group Theoretical Hidden Structure of Supergravity Theories in Higher Dimensions_ ([arXiv:1802.06602](https://arxiv.org/abs/1802.06602))

That the underlying bosonic [[body]] of this super Lie algebra happens to be the [[typical fiber]] of what would be the 11-d [[exceptional generalized tangent bundle]], namely the [level-2 truncation of the l1-representation](E11#FundamentalRepresentationAndBraneCharges) of [[E11]] according to ([West 04](E11#West04)) was highlighted in the review

* [[Silvia Vaula]], _On the underlying $E_{11}$ symmetry of the $D= 11$ Free Differential Algebra_, JHEP 0703:010, 2007 ([arXiv:hep-th/0612130](http://arxiv.org/abs/hep-th/0612130))

For analogous discussion in [[7d supergravity]] and [[4d supergravity]], see the references there.

### Alternative

From a different perspective the M-theory algebra extensions were (apparently independently) introduced in

 * [[Jan-Willem van Holten]], [[Antoine Van Proeyen]], _$N=1$ supersymmetry algebras in $d=2,3,4 \,mod\, 8$_ J.Phys. A15, 3763 (1982).

 * [[Paul Townsend]], _p-Brane Democracy_ ([arXiv:hep-th/9507048](http://arxiv.org/abs/hep-th/9507048))

with further amplification including

* {#Townsend97a} [[Paul Townsend]], _M(embrane) theory on $T^0$_, Nucl.Phys.Proc.Suppl.68:11-16,1998 ([arXiv:hep-th/9708034](http://arxiv.org/abs/hep-th/9708034))

* {#Townsend97b} [[Paul Townsend]], _M-theory from its superalgebra_, Cargese lectures 1997 ([arXiv:hep-th/9712004](http://arxiv.org/abs/hep-th/9712004))

In their global form, where differential forms are replaced by their de Rham cohomology classes on curved superspacetimes, these algebras were identified (for the case including the 2-form piece but not the 5-form piece) as the algebras of [[conserved currents]] of the [[Green-Schwarz super p-brane sigma-models]] in

* {#AGIT89} [[José de Azcárraga]], [[Jerome Gauntlett]], J.M. Izquierdo, [[Paul Townsend]], _Topological Extensions of the Supersymmetry Algebra for Extended Objects_, Phys.Rev.Lett. 63 (1989) 2443 ([spire](https://inspirehep.net/record/26393?ln=en))

reviewed in

* [[José de Azcárraga]], José M. Izquierdo, section 8.8 of _[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]_, Cambridge monographs of mathematical physics, (1995)

The generalization of this including also the contribution of the [[M5-brane]] was considered in

* {#SorokinTownsend97} [[Dmitri Sorokin]], [[Paul Townsend]], _M-theory superalgebra from the M-5-brane_, Phys.Lett. B412 (1997) 265-273 ([arXiv:hep-th/9708003](http://arxiv.org/abs/hep-th/9708003))

Further detailed discussion along these lines producing also the [[type II supersymmetry algebras]] is in

* Hanno Hammer, _Topological Extensions of Noether Charge Algebras carried by D-p-branes_, Nucl.Phys. B521 (1998) 503-546 ([arXiv:hep-th/9711009](http://arxiv.org/abs/hep-th/9711009))

The full extension was named "M-algebra" in

* [[Ergin Sezgin]], _The M-Algebra_, Phys.Lett. B392 (1997) 323-331 ([arXiv:hep-th/9609086](http://arxiv.org/abs/hep-th/9609086))

In ([D'Auria-Fré 82](#DAuriaFre82)) the motivation is from the formulation of the fields of [[11-dimensional supergravity]] as connections with values in the [[supergravity Lie 3-algebra]], see at _[[D'Auria-Fré formulation of supergravity]]_. Realization of the M-theory super Lie algebra as the algebra of [[derivations]] of the [[supergravity Lie 3-algebra]] is in

* {#Castellani05} [[Leonardo Castellani]], _Lie derivatives along antisymmetric tensors, and the M-theory superalgebra_, J. Phys. Math. Volume 3 (2011), 1-7. ([arXiv:hep-th/0508213](http://arxiv.org/abs/hep-th/0508213), [Euclid](http://projecteuclid.org/euclid.jpm/1359468398))

with amplification in

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ (2013)

* [[nLab:Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lie n-algebras of BPS charges]]_

Discussion of a formulation in terms of [[octonions]] (see also at _[[division algebra and supersymmetry]]_) includes

* A. Anastasiou, [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, _An octonionic formulation of the M-theory algebra_ ([arXiv:1402.4649](http://arxiv.org/abs/1402.4649))

Arguments that the charges of the M-theory super Lie algebra may be identified inside [[E11]] are given in

* [[Peter West]], _$E_{11}$, $SL(32)$ and Central Charges_, Phys.Lett.B575:333-342,2003 ([arXiv:hep-th/0307098v2](http://arxiv.org/abs/hep-th/0307098v2))

* [[Paul Cook]], around p. 75 of _Connections between Kac-Moody algebras and M-theory_ ([arXiv:0711.3498](http://arxiv.org/abs/0711.3498))

[[!redirects M-theory Lie algebra]]

[[!redirects M-Lie algebra]]
[[!redirects M-theory algebra]]

[[!redirects M-theory supersymmetry Lie algebra]]
[[!redirects M-theory supersymmetry super Lie algebra]]

[[!redirects M-theory super Lie algebra extension]]
[[!redirects M-theory super Lie algebra extensions]]

[[!redirects DF-algebra]]
[[!redirects DF-algebras]]

[[!redirects D'Auiria-Fré algebra]]
[[!redirects D'Auiria-Fré algebras]]

[[!redirects D'Auiria-Fre algebra]]
[[!redirects D'Auiria-Fre algebras]]


M-type

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## Idea

The notion of _M-type_ is the [[duality|formal dual]] of that of [[W-type]]: a certain [[coinductive type]].

The [[categorical semantics]] of an $M$-type is a [[terminal coalgebra of an endofunctor]].

## See also

* [[M-cotopos]]

## References

* [[Benno van den Berg]], de Marchi _Non-well-founded trees in categories_, APAL 146 (2007) 40–59. [link](https://arxiv.org/abs/math/0409158)

* Benedikt Ahrens, Paolo Capriotti and Régis Spadotti, _Non-wellfounded trees in Homotopy Type Theory_, [link](https://hott.github.io/M-types/)

* Andrea Vezzosi, _On Induction, Coinduction and Equality in Martin-Löf and Homotopy Type Theory_ [thesis](https://research.chalmers.se/en/publication/?id=504152)

* R.E. Møgelberg, N. Veltri, _Bisimulation as path type for guarded recursive types_ POPL 2019. [arXiv](https://arxiv.org/abs/1810.13261)

[[!redirects M-types]]


M-wave

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## Idea

One of the types of [[BPS states]] of [[11-dimensional supergravity]] is a [[pp-wave spacetime|pp-]][[gravitational wave]]-solution called the _M-wave_ ([Hull 84](#Hull84)).
Survey includes ([Philip 05, p. 94](#Philip05), [Bandos 12](#Bandos12)).

Under the [[duality between M-theory and type IIA string theory]] the M-wave becomes the [[black brane|black]] [[D0-brane]] under [[double dimensional reduction]] ([Bergshoeff-Townsend 96](#BergshoeffTwonsend96)).

The [[bound states]] of M-waves and [[M5-branes]] are argued to correspond to [[self-dual string]] [[instanton]] solutions in the [[6d (2,0)-superconformal QFT]] [[worldsheet]] theory of the M5 in ([Chu-Isono 12](#ChuIsono13)).

## Properties

### Spin structure

That the spinor-to-vector pairing on the M-wave takes values in just one of the two light rays is made fully explicit in [Philip 05, p. 94](#Philip05)

This is the pairing of chiral spinors in $D = 1 + 1$

### Relation to M2-Branes at ADE-Singularities

The M-wave may [[brane intersection|intersect]] [[black brane|black]] [[M2-branes]] at [[ADE-singularities]] (show in [BPST 09, bottom of p. 13](#BPST09) via [[boundary conditions]] for the [[BLG model]]/[[ABJM model]]).

## Related concepts

[[!include table of branes]]

## References

* {#Hull84} [[Chris Hull]], _Exact pp wave solutions of eleven-dimensional supergravity_, Phys. Lett. B 139 (1984) 39 ([spire](http://inspirehep.net/record/14499?ln=en))

* {#BergshoeffTownsend96} E. Bergshoeff, [[Paul Townsend]], _Super D-branes_, Nucl.Phys. B490 (1997) 145-162 ([arXiv:hep-th/9611173](http://arxiv.org/abs/hep-th/9611173))

* {#Townsend97} [[Paul Townsend]], first pages of _M-Theory from its superalgebra_, Cargese Lectures 1997 ([arXiv:hep-th/9712004](https://arxiv.org/abs/hep-th/9712004))

* Takeshi Sato, Section 2.3 of: :_Superalgebras in Many Types of M-Brane Backgrounds and Various Supersymmetric Brane Configurations_,  	Nucl. Phys. B548 (1999) 231-257 ([arXiv:hep-th/9812014](https://arxiv.org/abs/hep-th/9812014))

* [[José Figueroa-O'Farrill]], [[Joan Simón]], section 2 of _Supersymmetric Kaluza-Klein reductions of M-waves and MKK-monopoles_, Class. Quant. Grav.19:6147-6174, 2002 ([arXiv:hep-th/0208108](https://arxiv.org/abs/hep-th/0208108))

* {#Philip05} Simon Philip, _Plane-wave limits and homogeneous M-theory backgrounds_, 2005 ([pdf](https://www.era.lib.ed.ac.uk/bitstream/handle/1842/15645/Philip2005.Pdf?sequence=1), [[PhilipPlaneWave05.pdf:file]])

* {#BPST09} [[David Berman]], Malcolm J. Perry, [[Ergin Sezgin]], Daniel C. Thompson, _Boundary Conditions for Interacting Membranes_, JHEP 1004:025, 2010 ([arXiv:0912.3504](https://arxiv.org/abs/0912.3504))

* {#Bandos12} [[Igor Bandos]], _Action for the eleven dimensional multiple M-wave system_, 2012 ([pdf](http://strings.na.infn.it/files/Igor.pdf))

* {#ChuIsono13} [[Chong-Sun Chu]], Hiroshi Isono, _Instanton String and M-Wave in Multiple M5-Branes System_ ([arXiv:1305.6808](http://arxiv.org/abs/1305.6808), slides: [[IsonoSlides.pdf:file]])

* Yoshifumi Hyakutake, _Quantum M-wave and Black 0-brane_, JHEP09(2014)075 ([arXiv:1407.6023](http://arxiv.org/abs/1407.6023))

As an M-theoretic [[orientifold]]:

* {#HananyKol00} [[Amihay Hanany]], [[Barak Kol]], section 3.3 of _On Orientifolds, Discrete Torsion, Branes and M Theory_, JHEP 0006 (2000) 013 ([arXiv:hep-th/0003025](https://arxiv.org/abs/hep-th/0003025))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], Prop. 4.7 of _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. 2019 ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

[[!redirects M-waves]]


M. F. Bourdeau

* [inspire page](https://inspirehep.net/authors/2356568)

## Selected writings

On the [[CP^N sigma-model|$\mathbb{C}P^N$ sigma-model]] and its [[quantum cohomology ring]]:

* {#BourdeauDouglas94} [[M. F. Bourdeau]], [[Michael R. Douglas]], *Topological-Antitopological Fusion and the Large $N$ $\mathbb{C}P^N$ Model*, Nucl. Phys. B **420** (1994) 243-267 [[doi:hep-th/9312095](https://arxiv.org/abs/hep-th/9312095), doi:10.1016/0550-3213(94)90380-8]

category: people


M. L. Laplaza

## Selected writings

Proof of [[coherence theorems]] (cf. *[[coherence theorem for monoidal categories]]*)...

for [[rig categories]]:

* {#Laplaza72} [[M. L. Laplaza]], _Coherence for distributivity_, Lecture Notes in Mathematics **281** Springer (1972) 29-72 [[pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/laplaza.pdf)]

for [[compact closed categories]]:

* {#KellyLaplaza80} [[Max Kelly]], [[M. L. Laplaza]], *Coherence for compact closed categories*, Journal of Pure and Applied Algebra, **19** 193-213 (1980) [doi:10.1016/0022-4049(80)90101-2, [pdf](https://core.ac.uk/download/pdf/82696829.pdf)]

category: people


M. Ronen Plesser

* [webpage](https://math.duke.edu/people/m-ronen-plesser)

## selected writings

* C.E. Beasley and [[M. Ronen Plesser]], _Toric duality is Seiberg duality_, J. High Energy Phys. 12 (2001) 001 [hep-th/0109053].
JHEP02(2004)070

  (on [[Seiberg duality]])

* {#MorrisonPlesser99} [[David Morrison]], [[M. Ronen Plesser]], section 3.2 of _Non-Spherical Horizons, I_, Adv.Theor.Math.Phys.3:1-81, 1999 ([arXiv:hep-th/9810201](https://arxiv.org/abs/hep-th/9810201))

  (on [[M-theory on 8-manifolds]] with [[exotic 7-sphere]] [[boundaries]])

## related $n$Lab entries

* [[Seiberg duality]]

* [[M-theory on 8-manifolds]]

  [[8-manifold]], [[exotic 7-sphere]]

category: people

[[!redirects Ronen Plesser]]
[[!redirects M. Plesser]]


M. Takeyama

## Selected writings

On [[bireflective subcategories]] with [[ambidextrous adjoints]]:

* {#FOPTST99} [[Peter Freyd]], [[Peter O’Hearn]],  [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], *Bireflectivity*, Theoretical Computer Science **228** 1–2 (1999) 49-76 [doi:10.1016/S0304-3975(98)00354-5]

category: people


M. Zahid Hasan

* [personal page](https://phy.princeton.edu/people/m-zahid-hasan)

* [research group page](https://physics.princeton.edu/zahidhasangroup/) (on [[topological phases of matter|topological phases]] of [[quantum materials]])

## Selected writings

On [[topological insulators]]:

* [[M. Zahid Hasan]], [[Charles Kane]], *Colloquium: Topological insulators*, Rev. Mod. Phys. **82** (2010) 3045 $[$[doi:10.1103/RevModPhys.82.3045](https://doi.org/10.1103/RevModPhys.82.3045)$]$

* [[M. Zahid Hasan]], [[Joel E. Moore]], *Three-Dimensional Topological Insulators*, Ann. Review.Condensed Matter Physics **2** (2011) 55-78 $[$[arXiv:1011.5462](https://arxiv.org/abs/1011.5462), [doi:10.1146/annurev-conmatphys-062910-140432](https://doi.org/10.1146/annurev-conmatphys-062910-140432)$]$

On [[experiment|experimental]] realization of [[topological crystalline insulators]]:

* [[M. Zahid Hasan]] et al., *Observation of a topological crystalline insulator phase and topological phase transition in Pb1−xSnxTe*, Nature Communications **3** 1192 (2012) $[$[doi:10.1038/ncomms2191](https://doi.org/10.1038/ncomms2191)$]$

* [[M. Zahid Hasan]] et al., *Observation of Dirac Node Formation and Mass Acquisition in a Topological Crystalline Insulator*, Science **341** 1496 (2013) $[$[doi:10.1126/science.1239451](https://www.science.org/doi/10.1126/science.1239451)$]$

* [[M. Zahid Hasan]] et al., *Mapping the unconventional orbital texture in topological crystalline insulators*, Nature Physics **10** (2014) 572–577 $[$[doi:10.1038/nphys3012](https://doi.org/10.1038/nphys3012)$]$

category: people

[[!redirects M. Z. Hazan]]
[[!redirects M. Hasan]]


M.V.N. Murthy

* [institute page](https://www.imsc.res.in/m_v_n_murthy)

* [institute page](https://www.imsc.res.in/~murthy/)

* [GoogleScholar page](https://scholar.google.co.in/citations?user=hgONYSgAAAAJ&hl=en)

## Selected writings

On [[anyon]]-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]]:

* {#MurthyShankar09} [[M.V.N. Murthy]], [[Ramamurti Shankar]], *Exclusion Statistics: From Pauli to Haldane* (1999, 2009) $[$[dspace:123456789/334](https://dspace.imsc.res.in/xmlui/handle/123456789/334) [pdf](https://www.imsc.res.in/xmlui/bitstream/handle/123456789/334/MR120.pdf?sequence=1)$]$

* {#DMV03} G. Date, [[M.V.N. Murthy]], [[Radhika Vathsan]], *Classical and Quantum Mechanics of Anyons* $[$[arXiv:cond-mat/0302019](https://arxiv.org/abs/cond-mat/0302019)$]$

category: people

[[!redirects M. V. N. Murthy]]


M2-brane

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The theory of [[11-dimensional supergravity]] contains a higher [[gauge field]] -- the [[supergravity C-field]] -- that naturally couples to higher electrically charged 2-[[branes]], [[membranes]] ([Bergshoeff-Sezgin-Townsend 87](#BergshoeffSezginTownsend87)). By _[[double dimensional reduction]]_, these turn into the [[superstrings]] of [[type IIA string theory]] ([Duff-Howe-Inami-Stelle 87](#DuffHoweInamiStelle87)). (See at _[[duality between M-theory and type IIA string theory]]_.)

When in ([Witten95](#Witten95)) it was argued that the 10-dimensional [[target space]] theories of the five types of [[string theory|superstring theories]] are all limiting cases of one single 11-dimensional target space theory that extends [[11-dimensional supergravity]] ([[M-theory]]), it was natural to guess that this supergravity membrane accordingly yields a 3-dimensional [[sigma-model]] that reduces in limiting cases to the [[string]] [[sigma-models]].

But there were two aspects that make this idea a little subtle, even at this vague level: first, there is no good theory of the [[quantization]] of the membrane [[sigma-model]], as opposed to the well understood quantum [[string]]. Secondly, that hypothetical "theory extending 11-dimensional supergravity" ("M-theory") has remained elusive enough that it is not clear in which sense the membrane would relate to it in a way analogous to how the string relates to its [[target space]] theories (which is fairly well understood).

Later, with the [[BFSS matrix model]] some people gained
more confidence in the idea, by identifying the corresponding degrees
of freedom in a special case ([Nicolai-Helling 98](#NicolaiHelling98), [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)). See also at _[[membrane matrix model]]_.

In a more modern perspective, the M2-brane [[worldvolume]] theory appears under _[AdS4-CFT3 duality](AdS-CFT#AdS4CFT3)_ as a [[holographic principle|holographic dual]] of a [[4-dimensional Chern-Simons theory]]. Indeed, its [[Green-Schwarz action functional]] is entirely controled by the [[super Lie algebra|super]]-[[Lie algebra cocycle|Lie algebra 4-cocycle]] of [[super Minkowski spacetime]] given by the [[brane scan]]. This exhibits the M2-brane worldvolume theory as a 3-dimensional [[higher dimensional WZW model]].

## Definition

There are two different incarnations of the M2-brane. On the one hand it is defined as a [[Green-Schwarz sigma model]] with [[target space]] a [[spacetime]] that is a solution to the [[equations of motion]] of [[11-dimensional supergravity]]. One would call this the "fundamental" M2 in analogy with the "fundamental string", if only there were an "M2-perturbation series" which however is essentially ruled out.

On the other hand the M2 also appears as a [[black brane]], hence as a solution to the [[equations of motion]] of [[11-dimensional supergravity]] with [[singularity]] that looks from outside like a charged 2 dimensional object.

### As a Green-Schwarz sigma model

See at _[[Green-Schwarz sigma model]]_ and _[[brane scan]]_.

### As a black brane
 {#AsABlackBrane}

#### In the absence of 4-flux

As a [[black brane]] solution to the [[equations of motion]] of [[11-dimensional supergravity]], with vanishing [[C-field]] [[flux]], the M2 is the [[spacetime]] $\mathbb{R}^{2,1} \times (\mathbb{R}^8-\{0\})$ with [[pseudo-Riemannian metric]] being

$$
  g = H^{-2/3} g_{\mathbb{R}^{2,1}} + H^{1/3}g_{\mathbb{R}^8-\{0\}}
$$

where

$$
  H = \alpha + \frac{\beta}{r^6}
$$

for $(\alpha,\beta) \in \mathbb{R}^2 \setminus \{(0,0)\}$;

and the [[field strength]] of the [[supergravity C-field]] is

$$
  F = d vol_{\mathbb{R}^{2,1}} \wedge \mathbf{d} H^{-1}
  \,.
$$

For $\alpha \beta \neq 0$ this is a 1/2 [[BPS state]] of 11d sugra.

In the above [[coordinates]] the metric is ill-defined at $r = - \beta^{1/6} \alpha$, but in fact it may be smoothly continued through this point ([Duff-Gibbons-Townsend 94, section 3](#DuffGibbonsTownsend94)), which is a [[event horizon]]. An actual singularity is at $r = 0$.

The [[near horizon geometry]] of this spacetime is the [[Freund-Rubin compactification]] [[anti de Sitter spacetime|AdS4]]$\times$[[7-sphere|S7]]. For more on this see at _[[AdS-CFT]]_.

[[!include black branes in supergravity -- table]]

{#BlackM2AtADESingularity} More generally, one may classify those solutions of [[11-dimensional supergravity]] of the form $AdS_4 \times X_7$ for some [[closed manifold]] $X_7$, that are at least 1/2 [[BPS states]]. One finds ([Medeiros-Figueroa 10](#MedeirosFigueroa10)) that all these are of the form $AdS_4 \times S^7/G_{ADE}$, where $S^7 / G_{ADE}$ is an [[orbifold]] of the [[7-sphere]] (a [[spherical space form]] in the smooth case, see [there](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure)) by a [[finite subgroup of SU(2)]] $G_{ADE} \hookrightarrow SU(2)$, i.e. a finite group in the [[ADE-classification]]

[[!include ADE -- table]]

Here for $ 5 \leq \mathcal{N} \leq 8$-supersymmetry then the action of $G_{ADE}$ on $S^7$ is via the canonical action of $SU(2)$ as in the [[quaternionic Hopf fibration]] ([Medeiros-Figueroa 10](#MedeirosFigueroa10)), while for $\mathcal{N} = 4$ then there is an extra twist to the action  ([MFFGME 09](#MFFGME09)). See the table [below](#WorldvolumeTheory).

#### In the presence of 4-flux
 {#BlackM2BranesInThePresenceOfG4Flux}

In the presence of non-vanishing [[C-field]] [[flux]] $G_4$, the electric [[flux density]] of [[M2-branes]] is not $G_7 \coloneqq \star G_4$ alone, but receives corrections, first due to the quadratic [[C-field]] self-interaction in [[D=11 supergravity]], but then also due to the [[shifted C-field flux quantization]] expected in [[M-theory]]:

\linebreak

**The [[11d supergravity]] literature** states the corrected 7-flux to be the following combination, also known as the _[[Page charge]]_ (due to [Page 83 (8)](#Page83), [Duff-Stelle 91 (43)](#DuffStelle91), reviewed e.g. in [BLMP 13, p. 21](#BaggerLambertMukhiPapageorgakis13)):

\[
  \label{FirstCorrectionToG7}
  \widetilde G'_7
  \;\coloneqq\;
  G_7 + \tfrac{1}{2} C_3 \wedge G_4
  \,,
\]

where the second term subtracts the electric flux induced by the self-intersection of the [[field]], and also ensures that the full expression is a [[closed differential form]] if the naive [[11d supergravity]] [[equations of motion]] hold:

$$
  d \widetilde G_7
  \;=\;
  0
  \,.
$$

But in fact (eq:FirstCorrectionToG7) does not quite make general sense, for two reasons:

1. In general $G_4 = 0$ is not an admissible condition and is not the actual vanshing of the [[C-field]], due to the [[shifted C-field flux quantization]].

1. Even if $G_4$ happens to be integrally quantizaed (if $\tfrac{1}{4}p_1$ is integral) the appearance of a globally defined [[C-field]] potential $C_3$ in (eq:FirstCorrectionToG7),means that the total [[flux]] actually does vanish after all.

\linebreak

**Charge-quantized $\widetilde G_7$-flux with [[shifted C-field flux quantization]]** ([FSS 19b, Prop. 4.3](#FSS19b), [FSS 19c, Section 4](#FSS19c))

Both of these issues are solved if the [[C-field]] is taken to be [[charge quantization|charge quantized]] in [[J-homomorphism|J-]][[twisted Cohomotopy]] ([[Hypothesis H]]). This gives the corrected formula

\[
  \label{FullCorrectionToG7}
  \widetilde G_7
  \;\coloneqq\;
  h^\ast G_7 + \tfrac{1}{2} H_3 \wedge h^\ast \widetilde G_4
\]

where

1. the expression lives on the [[homotopy pullback]] of the [[Sp(2)]]-[[parametrized homotopy theory|parametrized]] [[quaternionic Hopf fibration]]

   $$
     h \coloneqq h_{\mathbb{H}}\sslash Sp(2)
   $$

   to [[spacetime]], along the [[twisted Cohomotopy]]-[[cocycle]] that represents the [[C-field]] under [[Hypothesis H]];

1. $\widetilde G_4 \coloneqq h^\ast G_4 + \tfrac{1}{4}h^\ast p_1(\nabla)$ is the integral [[shifted C-field flux quantization|shifted C-field]] pulled back to that 3-[[spherical fibration]] over spacetime;

1. $d H_3 = h^\ast G_4 - \tfrac{1}{4}h^\ast p_1(\nabla)$ trivializes not the C-field itself, but its pullback, and not absolutely but relative to the background charge implied by [[shifted C-field flux quantization]].

With the corrected 7-flux in [[twisted Cohomotopy]] it becomes true that

1. the integral of $G_7$ around the [[7-sphere]] linking a [[black brane|black]] [[M2-brane]] is always [[integer]] ([FSS 19c, Theorem 4.6](#FSS19c));

1. this integer satisfies the [[C-field tadpole cancellation condition]] ([FSS 19b, Section 4.6](#FSS19b)).

## Properties

### Quantization via BFSS matrix model
 {#QuantizationViaBFSSMatrixModel}

A regularized [[quantization]] of the [[Green-Schwarz sigma-model]] for the [[M2-brane]] yields the [[BFSS matrix model]] ([Nicolai-Helling 98](#NicolaiHelling98), [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)).

In this correspondence, [[matrix]] blocks around the diagonal correspond to blobs of [[membrane]], while off-diagonal matrix elements correspond to thin tubes of membrane connecting these blobs.

> graphics grabbed from [Dasgupta-Nicolai-Plefka 02](#DasguptaNicolaiPlefka02)

### Worldvolume theory -- BLG and ABJM
 {#WorldvolumeTheory}

The [[worldvolume]] [[QFT]] of [[black brane|black]] M2-branes is a [[3d superconformal gauge field theory]]:

[[!include superconformal symmetry -- table]]

Specifically, [[worldvolume]] [[quantum field theory]] of M2-branes sitting at [[ADE singularities]] (as [above](#BlackM2AtADESingularity)) is supposed to be described by [[ABJM theory]] and, for the special case of $SU(2)$ gauge group, by the [[BLG model]]. See also at _[[gauge enhancement]]_.

[[!include 7d spherical space forms -- table]]

### AdS4-CFT3 duality

Under [[AdS-CFT duality]] the M2-brane is given by
_[AdS4-CFT3 duality](AdS-CFT#AdS4CFT3)_. ([Maldacena 97, section 3.2](#Maldacena97), [Klebanov-Torri 10](#KlebanovTorri10)).

\linebreak

### M2/M5 bound states
 {#M2M5BoundStates}

For _[[M2-M5 brane bound states]]_, i.e.
 [[bound states]] of [[M2-branes]] with [[M5-branes]] ([[dyon|dyonic]] M2-branes and [[giant gravitons]]), see the
references [below](#ReferencesDyonic).

For the [[type II string theory]]-version see at _[[NS5-brane]]_ the sectoin _[NS5/D4/D2 bound states](NS5-brane#NS5D4D2BoundStates)_.

\linebreak

[[!include M2-M5 brane bound states in the BMN matrix model -- subsection]]

\linebreak

## Related concepts

* [[membrane instanton]], [[SM2-brane]]

* [[fractional M2-brane]]

* [[triple membrane junction]]

* [[C-field tadpole cancellation]]

* [[BLG model]], [[ABJM model]], [[membrane matrix model]]

  [[M2-brane 3-algebra]]

* [[string theory]]

* [[11-dimensional supergravity]], [[M-theory]]

* [[string]], [[superstring]]

* [[D-brane]]

* [[M5-brane]], [[6d (2,0)-superconformal QFT]]

* [[M9-brane]]

* [[topological membrane]]

* [[supergravity Lie 3-algebra]], [[M-theory super Lie algebra]]

[[!include table of branes]]

## References
 {#References}

[[!include fundamental M2-brane sigma-model -- references]]

[[!include quantization of M2-brane on Minkowski spacetime to BFSS matrix model -- references]]

### As a black brane
 {#ReferencesAsABlackBrane}

The [[black brane|black]] membrane solution of [[11-dimensional supergravity]] was found in

* {#DuffStelle91} [[Mike Duff]], [[Kellogg Stelle]], *Multi-membrane solutions of $D = 11$ supergravity*, Phys. Lett. B **253** 1-2 (1991) 113-118 [[spire:299386](http://inspirehep.net/record/299386), doi:10.1016/0370-2693(91)91371-2]

Its regularity across the [[event horizon]] is due to

* {#DuffGibbonsTownsend94} [[Mike Duff]], [[Gary Gibbons]], [[Paul Townsend]], _Macroscopic superstrings as interpolating solitons_, Phys. Lett. B332:321-328, 1994 ([arXiv:hep-th/9405124](https://arxiv.org/abs/hep-th/9405124))

Further discussion of such _[[Freund-Rubin compactifications]]_:

* {#Page83} [[Don Page]], *Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity*, Phys. Rev. D **28** 12 (1983) 2976-2982 [[spire:14480](http://inspirehep.net/record/14480), [doi:10.1103/PhysRevD.28.2976](https://doi.org/10.1103/PhysRevD.28.2976)]

The [[Horava-Witten theory|Horava-Witten]]-[[orientifold]] of the black M2, supposedly yielding the [[black brane|black]] [[heterotic string]] is discussed in

* {#LalakLukasOvrut97} Zygmunt Lalak, André Lukas, [[Burt Ovrut]], _Soliton Solutions of M-theory on an Orbifold_, Phys. Lett. B425 (1998) 59-70 ([arXiv:hep-th/9709214](https://arxiv.org/abs/hep-th/9709214))

* {#Kashima00} Ken Kashima, _The M2-brane Solution of Heterotic M-theory with the Gauss-Bonnet $R^2$ terms_, Prog.Theor.Phys. 105 (2001) 301-321 ([arXiv:hep-th/0010286](https://arxiv.org/abs/hep-th/0010286))

Meanwhile [[AdS-CFT duality]] was recognized in

* {#Maldacena97} [[Juan Maldacena]], _The Large N limit of superconformal field theories and supergravity_, Adv. Theor. Math. Phys. 2:231, 1998, [hep-th/9711200](http://arxiv.org/abs/hep-th/9711200);

where a dual description of the [[worldvolume]] theory of M2-brane appears in section 3.2.  More on this is in

* {#KlebanovTorri10} [[Igor Klebanov]], Giuseppe Torri, _M2-branes and AdS/CFT_, Int.J.Mod.Phys.A25:332-350,2010 ([arXiv:0909.1580](http://arxiv.org/abs/0909.1580))

An account of the history as of 1999 is in

* {#Duff99} [[Mike Duff]], chapter II of _[[The World in Eleven Dimensions]]: Supergravity, Supermembranes and M-theory_, IoP 1999 ([publisher](https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720))

More recent review is in

* Georgios Linardopoulos, chapter 13 of _Classical Strings and Membranes in the AdS/CFT Correspondence_ ([pdf](http://users.uoa.gr/~glinardo/Thesis.pdf), [spire](https://inspirehep.net/record/1391031/))

A detailed discussion of this [[black brane]]-realization of the [[M2-brane]] and its relation to [[AdS-CFT]] is in

* [[Gianguido Dall'Agata]], Davide FabbKri, Christophe Fraser, [[Pietro Fré]], Piet Termonia, Mario Trigiante, _The $Osp(8|4)$ singleton action from the supermembrane_, Nucl.Phys.B542:157-194, 1999 ([arXiv:hep-th/9807115](http://arxiv.org/abs/hep-th/9807115))

The generalization of this to $\geq 1/2$ BPS sugra solutions of the form $AdS_4 \times X_7$ is due to

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* {#MedeirosFigueroa10} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

Discussion of $G7$-[[C-field flux quantization]] in the $G_4$-fluxed case:

 {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Prop. 4.31 of: _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of: _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

### In $AdS_4/CFT_3$

In [[AdS4-CFT3 duality]]:

Review:

* Kazuo Hosomichi, _M2-branes and AdS/CFT: A Review_ ([arXiv:2003.13914](https://arxiv.org/abs/2003.13914))

### History

Discussion of the history includes

* [[Mike Duff]], ([arXiv:1501.04098](http://arxiv.org/abs/1501.04098))

### Decoupled worldvolume theory

On the [[BLG model]]/[[ABJM model]] for the [[worldvolume]]-theory of [[coincident branes|coincident]] [[M2-branes]] see there.

Review:

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

### Dualities

The role of and the relation to [[duality in string theory]] of the membrane is discussed in the following articles.

Relation to [[T-duality]] is discussed in:

* J.G. Russo, _T-duality in M-theory and supermembranes_ ([arXiv:hep-th/9701188](http://arxiv.org/abs/hep-th/9701188))

* M.P. Garcia del Moral, J.M. Pena, A. Restuccia, _T-duality Invariance of the Supermembrane_ ([arXiv:1211.2434](http://arxiv.org/abs/1211.2434))

Relation to [[U-duality]] is discussed in:

* [[Martin Cederwall]], _M-branes on U-folds_ ([arXiv:0712.4287](http://arxiv.org/abs/0712.4287))

* M.P. Garcia del Moral, _Dualities as symmetries of the Supermembrane Theory_ ([arXiv](http://arxiv.org/abs/1211.6265))

* [pdf](http://streaming.ictp.trieste.it/preprints/P/97/044.pdf)

Discussion from the point of view of [[E11]]-[[U-duality]] and [[current algebra]] is in

* [[Hirotaka Sugawara]], _Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra_, International Journal of Modern Physics A, Volume 32, Issue 05, 20 February 2017 ([arXiv:1701.06894](https://arxiv.org/abs/1701.06894))

* [[Shotaro Shiba]], [[Hirotaka Sugawara]], _M2- and M5-branes in $E_{11}$ Current Algebra Formulation of M-theory_ ([arXiv:1709.07169](https://arxiv.org/abs/1709.07169))

### M2-M5 bound states
 {#ReferencesDyonic}

Discussion of [[M2-M5 brane bound states]], i.e.
[[dyon|dyonic]]$\,$[[black brane|black]] [[M2-branes]] ([[M5-branes]] [[wrapped brane|wrapped]] on a [[3-manifold]], see also at _[NS5-branes -- D2/D4/NS5-bound states](NS5-brane#ReferencesNS5D4D2BoundStates)_):

* {#ILPT95} J.M. Izquierdo, [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic Membranes_, Nucl. Phys. B460:560-578, 1996 ([arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177))

* [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic $p$-branes from self-dual $(p+1)$-branes_, Phys. Lett. B384:86-92, 1996 ([arXiv:hep-th/9605146](https://arxiv.org/abs/hep-th/9605146))

* [[Troels Harmark]], Section 3.1 of _Open Branes in Space-Time Non-Commutative Little String Theory_, Nucl.Phys. B593 (2001) 76-98 ([arXiv:hep-th/0007147](https://arxiv.org/abs/hep-th/0007147))

* [[Troels Harmark]],  N.A. Obers, Section 5.1 of _Phase Structure of Non-Commutative Field Theories and Spinning Brane Bound States_, JHEP 0003 (2000) 024 ([arXiv:hep-th/9911169](https://arxiv.org/abs/hep-th/9911169))

* [[George Papadopoulos]], [[Dimitrios Tsimpis]], _The holonomy of the supercovariant connection and Killing spinors_, JHEP 0307:018, 2003 ([arXiv:hep-th/0306117](https://arxiv.org/abs/hep-th/0306117))

* {#Petri18} [[Nicolò Petri]], slide 14 of _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4 of _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

* Jay Armas, Vasilis Niarchos, Niels A. Obers, _Thermal transitions of metastable M-branes_ ([arXiv:1904.13283](https://arxiv.org/abs/1904.13283))

Further [[M2/M5-brane bound states]] to [[giant gravitons]]:

* J. M. Camino, A. V. Ramallo, _M-Theory Giant Gravitons with C field_, Phys. Lett. B525:337-346, 2002 ([arXiv:hep-th/0110096](https://arxiv.org/abs/hep-th/0110096))

[[!redirects M2-branes]]

[[!redirects M2 brane]]
[[!redirects M2 branes]]

[[!redirects M-theory membrane]]
[[!redirects M-theory membranes]]


M2-M5 brane bound state

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[bound states]] of [[M2-branes]] with [[M5-branes]];

in particular [[dyon|dyonic]]$\,$[[black brane|black]] [[M2-branes]], i.e. [[M5-branes]] [[wrapped brane|wrapped]] on a [[3-manifold]] ([ILPT 95](#ILPT95))

## Properties

### Image in type II string theory

Under suitable [[duality between M-theory and type IIA string theory]] the M2/M5-brane bound states become [[Dp-D(p+2)-brane bound states]] ([Campos, Ferretti & Salomonson 2000](#CamposFerrettiSalomonson00); [Basu & Harvey 2005](#BasuHarvey05); for review and more see [Bagger, Lambert, Mukhi & Papageorgakis 2013, pp. 37](#BaggerLambertMukhiPapageorgakis13)).

### Relation to giant gravitons

for the relation to [[giant gravitons]]: ([Camino-Ramallo 01](#CaminoRamallo01))

\linebreak

[[!include M2-M5 brane bound states in the BMN matrix model -- subsection]]

\linebreak

## Related entries

[[!include brane bound states -- table]]

* [[M-theory on 8-manifolds]]

* [[M-brane]]

* [[wrapped brane]]

## References

### As black branes

As [[black brane]]-solutions of [[D=11 supergravity]]:

* {#ILPT95} [[José M. Izquierdo]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], *Dyonic Membranes*, Nucl. Phys. B **460** (1996) 560-578 [[arXiv:hep-th/9508177](https://arxiv.org/abs/hep-th/9508177), doi:10.1016/0550-3213(95)00606-0]

* [[Arkady Tseytlin]], §3.2 in: *Harmonic superpositions of M-branes*, Nucl. Phys. B **475** (1996) 149-163 [[arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035), doi:10.1016/0550-3213(96)00328-8]

* [[Michael Green]], [[Neil Lambert]], [[George Papadopoulos]], [[Paul Townsend]], _Dyonic $p$-branes from self-dual $(p+1)$-branes_, Phys. Lett. B **384** (1996) 86-92 [[arXiv:hep-th/9605146](https://arxiv.org/abs/hep-th/9605146)]

* [[Troels Harmark]], Section 3.1 of _Open Branes in Space-Time Non-Commutative Little String Theory_, Nucl. Phys. B **593** (2001) 76-98 ([arXiv:hep-th/0007147](https://arxiv.org/abs/hep-th/0007147))

* [[Troels Harmark]],  N.A. Obers, Section 5.1 of _Phase Structure of Non-Commutative Field Theories and Spinning Brane Bound States_, JHEP 0003 (2000) 024 ([arXiv:hep-th/9911169](https://arxiv.org/abs/hep-th/9911169))

* [[Kazuo Hosomichi]], *On Branes Ending on Branes in Supergravity*, JHEP 0006 (2000) 004 [[arXiv:hep-th/0002069](https://arxiv.org/abs/hep-th/0002069), [doi:10.1088/1126-6708/2000/06/004](https://doi.org/10.1088/1126-6708/2000/06/004)]

* [[Arvind Rajaraman]], *Supergravity Solutions for Localised Brane Intersections*, JHEP 0109:018 (2001) [[arXiv:hep-th/0007241](https://arxiv.org/abs/hep-th/0007241), [doi:10.1088/1126-6708/2001/09/018](https://doi.org/10.1088/1126-6708/2001/09/018)]

* [[George Papadopoulos]], [[Dimitrios Tsimpis]], _The holonomy of the supercovariant connection and Killing spinors_, JHEP 0307:018, 2003 ([arXiv:hep-th/0306117](https://arxiv.org/abs/hep-th/0306117))

* [[Kurt Lechner]], _Intersecting M2- and M5-branes_, Phys. Lett. B589 (2004) 147-154 ([arXiv:hep-th/0402078](https://arxiv.org/abs/hep-th/0402078), [doi:10.1016/j.physletb.2004.03.056](https://doi.org/10.1016/j.physletb.2004.03.056))

* [[Eric D'Hoker]], John Estes, Michael Gutperle, Darya Krym, _Exact Half-BPS Flux Solutions in M-theory I, Local Solutions_, JHEP 0808:028, 2008 ([arXiv:0806.0605](https://arxiv.org/abs/0806.0605))

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *M2-M5 blackfold funnels*, Journal of High Energy Physics **2012** 175 (2012) [[arXiv:1205.1535](https://arxiv.org/abs/1205.1535), doi:10.1007/JHEP06(2012)175]

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *Entropy of the self-dual string soliton*, Journal of High Energy Physics **2012** 134 (2012) [[arXiv:1206.2935](https://arxiv.org/abs/1206.2935), doi:10.1007/JHEP07(2012)134]

* [[Vasilis Niarchos]], [[Konstadinos Siampos]], *The black M2-M5 ring intersection spins* [[arXiv:1302.0854](https://arxiv.org/abs/1302.0854)]

* [[Vasilis Niarchos]], *Localizing the black M2-M5 intersection* (2012) [[pdf](https://hep.physics.uoc.gr/Slides/Fall_2012/VasilisNiarchos.pdf), [[Niarchos-BlackM2M5Branes.pdf:file]]]

* Leon Berdichevsky, Bat-el Dahan, _Local gravitational solutions dual to M2-branes intersecting and/or ending on M5-branes_, JHEP 08 (2013) 061 ([arXiv:1304.4389](https://arxiv.org/abs/1304.4389))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _BPS objects in $D=7$ supergravity and their M-theory origin_, JHEP 12 (2017) 041 ([arxiv:1707.06152](https://arxiv.org/abs/1707.06152))

* {#Petri18} [[Nicolò Petri]], slide 14 of: _Surface defects in massive IIA_, talk at [Recent Trends in String Theory and Related Topics](http://physics.ipm.ac.ir/conferences/stringtheory3/) 2018 ([pdf](http://physics.ipm.ac.ir/conferences/stringtheory3/note/N.Petri.pdf))

* Jay Armas, [[Vasilis Niarchos]], [[Niels A. Obers]], Section 2 of: _Thermal transitions of metastable M-branes_, J. High Energ. Phys. (2019) 2019: 128 [[arXiv:1904.13283](https://arxiv.org/abs/1904.13283)]

* [[Iosif Bena]], [[Anthony Houppe]], [[Dimitrios Toulikas]], [[Nicholas P. Warner]], *Maze Topiary in Supergravity* [[arXiv:2312.02286](https://arxiv.org/abs/2312.02286)]

### As polarized M2 branes

The Myers effect in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of ([[fuzzy 3-sphere|fuzzy]]) [[3-sphere]]-shape ([[M2-M5 brane bound states]]):

* [[Iosif Bena]], _The M-theory dual of a 3 dimensional theory with reduced supersymmetry_, Phys. Rev. D62:126006, 2000 ([arXiv:hep-th/0004142](https://arxiv.org/abs/hep-th/0004142))

* Masato Arai, [[Claus Montonen]], Shin Sasaki, _Vortices, Q-balls and Domain Walls on Dielectric M2-branes_, JHEP 0903:119, 2009 ([arXiv:0812.4437](https://arxiv.org/abs/0812.4437))

* [[Iosif Bena]], [[Mariana Graña]], Stanislav Kuperstein, Stefano Massai, _Tachyonic Anti-M2 Branes_, JHEP 1406:173, 2014 ([arXiv:1402.2294](https://arxiv.org/abs/1402.2294))

With emphasis on the role of the [[Page charge]]/[[Hopf WZ term]]:

* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))

Via the [[mass-deformed ABJM model]]:

* [[Jaume Gomis]], Diego Rodriguez-Gomez, [[Mark Van Raamsdonk]], [[Herman Verlinde]], _A Massive Study of M2-brane Proposals_, JHEP 0809:113, 2008 ([arXiv:0807.1074](https://arxiv.org/abs/0807.1074))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 6.4 of: _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

The corresponding D2-NS5 bound state under [[duality between M-theory and type IIA string theory]]:

* [[Iosif Bena]], Aleksey Nudelman, _Warping and vacua of $(S)YM_{3+1}$_, Phys. Rev. D62 (2000) 086008 ([arXiv:hep-th/0005163](https://arxiv.org/abs/hep-th/0005163))

### Reduction to D$p$-D$(p+2)$-brane bound states

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (eventually motivating the [[BLG model]]/[[ABJM model]]):

* {#CamposFerrettiSalomonson00} Vanicson L. Campos, Gabriele Ferretti, Per Salomonson, *The Non-Abelian Self Dual String on the Light Cone*, JHEP 0012 (2000) 011 [[arXiv:hep-th/0011271](https://www.arxiv.org/abs/hep-th/0011271)]

* {#BasuHarvey05} [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

Argument, using the [[ABJM model]], that the apparent [[fuzzy 3-sphere]] geometry of [[M2-M5 brane bound states]] effectively collapses to a [[fuzzy 2-sphere]] geometry of [[Dp-D(p+2)-brane intersection]] [[fuzzy funnels]]:

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], [[Sanjaye Ramgoolam]], _The fuzzy $S^2$ structure of M2-M5 systems in ABJM membrane theories_, JHEP 0905:123, 2009 ([arXiv:0903.3966](https://arxiv.org/abs/0903.3966))

* [[Sanjaye Ramgoolam]], _Fuzzy geometry of membranes in M-theory_, 2009 ([[RamgoolamFuzzyGeometry.pdf:file]])

* [[Horatiu Nastase]], [[Constantinos Papageorgakis]], _Bifundamental Fuzzy 2-Sphere and Fuzzy Killing Spinors_, SIGMA 6:058, 2010 ([arXiv:1003.5590](https://arxiv.org/abs/1003.5590))

### Relation to giant gravitons

Relation to [[giant gravitons]]:

* {#CaminoRamallo01} J. M. Camino, A. V. Ramallo, _M-Theory Giant Gravitons with C field_, Phys.Lett.B525:337-346,2002 ([arXiv:hep-th/0110096](https://arxiv.org/abs/hep-th/0110096))

* Shinji Hirano, Yuki Sato, _Giant graviton interactions and M2-branes ending on multiple M5-branes_, JHEP 05 (2018) 065 ([arXiv:1803.04172](https://arxiv.org/abs/1803.04172))

### KK-reduction  to JT gravity

Realization of [[JT-gravity]] as [[Kaluza-Klein reduction]] of [[D=6 supergravity]] on the [[worldvolume]] of [[D1-D5 brane bound states]] or [[M2-M5 brane bound states]]:

* Yue-Zhou Li, Shou-Long Li, H. Lu, _Exact Embeddings of JT Gravity in Strings and M-theory_, Eur. Phys. J. C (2018) 78: 791 ([arXiv:1804.09742](https://arxiv.org/abs/1804.09742))

* Iosif Bena, Pierre Heidmann, David Turton, _$AdS_2$ Holography: Mind the Cap_, JHEP 1812 (2018) 028 ([arXiv:1806.02834](https://arxiv.org/abs/1806.02834))

[[!redirects M2-M5 brane bound states]]

[[!redirects M2-M5-brane bound state]]
[[!redirects M2-M5-brane bound states]]

[[!redirects M2-M5 brane intersection]]
[[!redirects M2-M5 brane intersections]]
[[!redirects M2-M5-brane intersections]]

[[!redirects M2/M5 brane bound state]]
[[!redirects M2/M5 brane bound states]]

[[!redirects M2/M5-brane bound state]]
[[!redirects M2/M5-brane bound states]]

[[!redirects M2/M5 brane intersection]]
[[!redirects M2/M5 brane intersections]]


M2-M5 brane bound states in the BMN matrix model -- subsection

### M2-M5 brane bound states in the BMN matrix model
 {#M2M5BraneBoundStatesInTheBMNMatrixModel}

There is the suggestion  ([MSJVR 02](BMN+matrix+model#MSJVR02), checked in [AIST 17a](BMN+matrix+model#AIST17a), [AIST 17b](BMN+matrix+model#AIST17b)) that, in the [[BMN matrix model]], [[supersymmetry|supersymmetric]] [[M2-M5-brane bound states]] are identified with [[isomorphism classes]] of certain "limit sequences" of longitudinal-[[light cone]]-[[constant function|constant]] $N \times N$-[[matrix]]-[[field (physics)|fields]] constituting [[finite-dimensional vector space|finite-dimensional]] [[complex numbers|complex]] [[Lie algebra representations]] of [[su(2)]].

#### Traditional discussion
 {#TraditionalDiscussion}

Concretely, if

$$
  \underset{ i }{\oplus}
  \big(
    N^{(M2)}_i \cdot \mathbf{N}^{(M5)}_i
  \big)
  \;\;\in\;\;
  \mathfrak{su}(2)_{\mathbb{C}}Rep^{fin}
$$

denotes [[generalized the|the]] [[Lie algebra representation|representation]] containing

* $N^{(M2)}_i$ [[direct sum|direct summands]]

of [[generalized the|the]]

* $N^{(M5)}_i$-[[dimension|dimensional]] [[irrep]] $\mathbf{N}^{(M5)}_i \in \mathfrak{su}(2)_{\mathbb{C}}Rep$

(for $\{N^{(M2)}_i, N^{(M5)}_i\}_{i} \in (\mathbb{N} \times \mathbb{N})^I$ some [[finite set|finitely]] [[indexed set]] of [[pairs]] of [[natural numbers]])

with total [[dimension]]

$$
  N
    \;\coloneqq\;
  dim
  \big(
    \underset{ i }{\oplus}
    \big(
      N^{(M2)}_i \cdot \mathbf{N}^{(M5)}_i
    \big)
  \big)
$$

then:

1. a configuration of a [[finite number]] of [[coincident brane|stacks of coincident]] [[M5-branes]] corresponds to a [[sequence]] of such representations for which

   1. $N^{(M2)}_i \to \infty$ (this being the relevant [[large N limit]])

   1. for fixed $N^{(M5)}_i$ (being the number of M5-branes in the $i$th stack)

   1. and fixed [[ratios]] $N^{(M2)}_i/N$ (being the [[charge]]/[[light-cone momentum]] carried by the $i$th stack);

1. an [[M2-brane]] configuration corresponds to a [[sequence]] of such representations for which

   1. $N^{(M5)}_i \to \infty$ (this being the relevant [[large N limit]])

   1. for fixed $N^{(M2)}_i$ (being the number of M2-brane in the $i$th stack)

   1. and fixed [[ratios]] $N^{(M5)}_i/N$ (being the [[charge]]/[[light-cone momentum]] carried by the $i$th stack)

for all $i \in I$.

Hence, by extension, any other sequence of finite-dimensional $\mathfrak{su}(2)$-representations is a kind of mixture of these two cases, interpreted as an [[M2-M5 brane bound state]] of sorts.

#### Formalization via weight systems on chord diagrams
 {#FormalizationViaWeightSystems}

To make this precise, let

$$
  \mathfrak{su}(2)_{\mathbb{C}} MetMod_{/\sim}
  \;\in\;
  Set
$$

be the [[set]] of [[isomorphism classes]] of [[complex numbers|complex]] [[metric Lie representations]] (hence [[finite-dimensional vector space|finite-dimensional representations]]) of [[su(2)]] ([hence](su2#Complexification) of the [[special linear Lie algebra]] $\mathfrak{sl}(2,C)$) and write

$$
  Span
  \big(
    \mathfrak{su}(2)_{\mathbb{C}} MetMod_{/\sim}
  \big)
  \;\in\;
  Vect_{\mathbb{C}}
$$

for its [[linear span]] (the [[complex vector space]] of [[formal linear combinations]] of [[isomorphism classes]] of [[metric Lie representations]]).

Finally, write

$$
  \array{
    Span
    \big(
      \mathfrak{su}(2)_{\mathbb{C}}MetMod_{/\sim}
    \big)
    &
      \longrightarrow
    &
    \big(
      \mathcal{W}^{{}^{s}}
    \big)^{deg}
    \\
    c_1 \cdot V_1
    +
    c_2 \cdot V_2
    &\mapsto&
    c_1 \cdot w_{V_1}
    +
    c_2 \cdot w_{V_2}
  }
$$

for the [[linear map]] which sends a [[formal linear combination]] of representations to the [[weight system]] on [[Sullivan chord diagrams]] with $deg \in \mathbb{N}$ chords which is given by [[trace|tracing]] in the given representation.

Then a [[M2-M5-brane bound state]] as in the traditional discussion [above](#TraditionalDiscussion), but now formalized as an [[Lie algebra weight system|su(2)-weight system]]

$$
  \Psi_{
    \left\{
      N^{(M2)}_i, N^{(M5)}_i
    \right\}_{i}
  }
  \;\in\;
  \underset{
    deg \in \mathbb{N}
  }{\prod}
  \big(
    \mathcal{W}^{{}^{\mathrm{s}}}
  \big)^{deg}
$$

hence a [[weight system]] [[horizontal chord diagrams]] closed to [[Sullivan chord diagrams]], these now being the [[multi-trace observables]] on these) is

\[
  \label{M2M5BraneBoundStateInWeightSystems}
  \Psi_{
    \left\{
      N^{(M2)}_i, N^{(M5)}_i
    \right\}_{i}
  }
  \;\coloneqq\;
  \tfrac{
    4 \pi \, 2^{2\,deg}
  }{
    \sum_i N^{(M2)}_i
  }
  \underset{i}{\sum}
  \left(
    N^{(M2)}_i
    \cdot
    \tfrac{1}{
      \left(
        \sqrt{
          \left(
            N^{(M5)}_i
          \right)^2 - 1
        }
      \right)^{1 + 2 deg}
    }
    \mathbf{N}^{(M5)}_i
  \right)
\]

> (from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]])

{#M2M5BraneBoundStatesAsWeightSystemsGraphics} **Normalization and large $N$ limit.** The first power of the square root in (eq:M2M5BraneBoundStateInWeightSystems) reflects the [[volume]] measure on the [[fuzzy 2-sphere]] (by the formula [here](fuzzy+sphere#eq:FuzzyS2Integration)), while the power of $2\,deg$ (which is the number of operators in the [[multi-trace observable]] evaluating the [[weight system]]) gives the normalization ([here](fuzzy+sphere#eq:BasisFunctionsOnFuzzy2Sphere)) of the functions on the fuzzy 2-sphere.

Hence this normalization is such that the [[single-trace observables]] among the [[multi-trace observables]], hence those which come from [[round chord diagrams]], coincide on those [[M2-M5 brane bound states]]
$
  \Psi_{
    \left\{
      N^{(M2)}_i, N^{(M5)}_i
    \right\}_{i}
  }
$
for which $N^{(M2)}_i = \delta_i^{i_0} N^{(M2)}$,
hence those which have a single constitutent [[fuzzy 2-sphere]],  with the shape observables on single fuzzy 2-spheres discussed  [here](fuzzy+sphere#ShapeObservablesAsVassilievInvariants):

> (from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]])

Therefore, with this normalization, the [[limit of a sequence|limits]] $N^{(M2)} \to \infty$ and $N^{(M5)} \to \infty$ of (eq:M2M5BraneBoundStateInWeightSystems) should exist in [[weight systems]]. The former trivially so, the latter by the usual convergence of the [[fuzzy 2-sphere]] to the [[round sphere|round]] [[2-sphere]] in the [[large N limit]].

Notice that the [[multi trace observables]] on these states only see the relative [[radii]] of the constitutent [[fuzzy 2-spheres]]: If $N^{(M2)}_i = \delta_i^{i_0} N^{(M2)}$ then the $N^{(M2)}$-dependence of (eq:M2M5BraneBoundStateInWeightSystems) cancels out, reflecting the fact that then there is only a single constituent 2-sphere of which the observable sees only the radius fluctuations, not the absolute radius (proportional to $N^{(M2)}$).