# nLab M-theory on G2-manifolds

Contents

## Surveys, textbooks and lecture notes

#### Gravity

gravity, supergravity

# Contents

## Idea

The Kaluza-Klein reduction of 11-dimensional supergravity on G2 manifolds (notably Freund-Rubin compactifications and variants) yields an 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.

KK-compactifications of higher dimensional supergravity with minimal ($N=1$) supersymmetry:

perspectiveKK-compactification with $N=1$ supersymmetry
M-theoryM-theory on G2-manifolds
F-theoryF-theory on CY4-manifolds
heterotic string theoryheterotic string theory on CY3-manifolds

In order for this to yield 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 gauge group of the effective theory by “geometric engineering of quantum field theory” (Acharya 98, Atiyah-Witten 01, section 6), see below. Specifically for discussion of string phenomenology obtaining or approximating the standard model of particle physics by this procedure see at G2-MSSM.

## Details

### Vacuum solutions

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). (If one includes non-vanishing $C$-field strength one finds “weak $G_2$-holonomy” instead, see below).

Notice that vanishing gravitino field strength (i.e. covariant derivative) means that the torsion of the super-vielbein is in each tangent space the canonical torsion of the super Minkowski spacetime. This 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

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 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), Acharya 02, (32) (59) (74), Grigorian-Yau 08, (4.57), Acharya-Bobkov 08, (4)).

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

The KK-compactification of vacuum 11-dimensional supergravity on a smooth G2-manifold $Y$ results in a effective N=1 D=4 super Yang-Mills theory with 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). 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 (Acharya 98, Acharya 00, review includes Acharya 02, section 3, BBS 07, p. 422, 436, Ibáñez-Uranga 12, section 6.3.3, Wijnholt 14, part III (from which the graphics below is grabbed));

2. at a (non-orbifold) conifold singularity chiral fermions appear (Witten 01, p. 3, Atiyah-Witten 01, Acharya-Witten 01, Berglund-Brandhuber 02, Bourjaily-Espahbodi 08).

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

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, Witten 01, p. 1, based on Cvetic-Shiu-Uranga 01).

2. The duality between M-theory compactified on K3 and heterotic string theory on a 3-torus (Acharya-Witten 01). 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). This comes down (Intriligator-Seiberg 96) 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 of a super Yang-Mills theory.

3. 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)

M2-branes may 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), for more see at M-theory lift of gauge enhancement on D6-branes.

4. 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 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

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-vanishning $G_4$-flux (field strength of the supergravity C-field). In other words, non-vanishing flux always breaks the supersymmetry.

e.g. (Acharya-Spence 00) see the Introduction of (Beasley-Witten 02)

$\,$

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

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 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 metric tensor) saying that both spaces are Einstein manifolds (BSS 02, (5.4)). 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)), saying that $\phi \propto \star F_7$ exhibits weak G2-holonomy with weakness parameter given by the component of the C-field on $X_4$.

### 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, review is in Acharya-Gukov 04, section 5.3.

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 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 $X_{1,\Gamma}$)

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

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. 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.

### Relation to intersecting D-brane models

relation to intersecting D-brane models: see there

## References

### General

The KK-compactification of 11d supergravity of fibers of special holonomy was originally considered in

Specifically string phenomenology for the case of compactification on G2-manifolds (or rather orbifolds ) goes back to

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

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

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

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

Further discussion of membrane instantons in this context includes

Survey and further discussion includes

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

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

• Bobby Acharya, M Theory, $G_2$-manifolds and Four Dimensional Physics, Classical and Quantum Gravity Volume 19 Number 22, 2002 (pdf)

• Bobby Acharya, Sergei Gukov, M theory and Singularities of Exceptional Holonomy Manifolds, Phys.Rept.392:121-189,2004 (arXiv:hep-th/0409191)

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

• Adam B. Barrett, M-Theory on Manifolds with $G_2$ Holonomy (arXiv:hep-th/0612096)

• James Halverson, David Morrison, The Landscape of M-theory Compactifications on Seven-Manifolds with $G_2$ Holonomy (arXiv:1412.4123)

• Aaron Kennon, $G_2$-Manifolds and M-Theory Compactifications (arXiv:1810.12659)

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

Discussion of duality with F-theory on CY4-manifolds includes

Discussion of duality with heterotic string theory on CY3-manifolds:

• Andreas Braun, Sakura Schaefer-Nameki, Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality, JHEP04(2018)126 (arXiv:1708.07215)

The moduli space is discussed in

### Phenomenology

Popular exposition of the G2-MSSM phenomenology is in

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

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

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

and specifically for the G2-MSSM in

the strong CP problem is discussed in

and realization of GUTs in

• Edward Witten, Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting, (arXiv: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)

The phenomenology of compactifications on compact twisted connected sum G2-manifolds (Kovalev 00) is in

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)

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)

• Andreas Braun, Sebastjan Cizel, Max Hubner, Sakura Schafer-Nameki, Higgs Bundles for M-theory on $G_2$-Manifolds (arXiv:1812.06072)