1 ’ µ±βγ ‚± X M ‚β X N ‚γ X P “M N P

P’ = = 0, (9.218)

2 6

where now the spinor lives in seven dimensions. Here ±, β, . . . are indices

on the cycle while M, N, . . . are D = 11 indices. By a similar calculation

436 String geometry

to that in Exercise 9.15, one can verify that the de¬ning equation for a

supersymmetric three-cycle is

‚[± X a ‚β X b ‚γ] X c ¦abc = µ±βγ . (9.219)

This means that the pullback of the three-form onto the cycle is proportional

to the volume form. A G2 manifold can also have supersymmetric four-

cycles, which solve the equation

1 i ±βγσ

‚± X M ‚β X N ‚γ X Q ‚σ X P “M N P Q

1’

P’ = µ = 0. (9.220)

2 4!

The solution has the same form as Eq. (9.219) with the associative calibra-

tion replaced by the dual coassociative calibration ¦. Both type of cycles

break 1/2 of the original supersymmetry.

Obviously, there is interest in the phenomenological implications of M-

theory compacti¬cations on G2 manifolds, because these give N = 1 theo-

ries in four dimensions. Let us mention a few topics in this active area of

research.

G2 manifolds and strongly coupled gauge theories

Compacti¬cation of M-theory on a smooth G2 manifold does not lead to

chiral matter or nonabelian gauge symmetry. The reason is that M-theory

is a nonchiral theory and compacti¬cation on a smooth manifold cannot lead

to a chiral theory. A chiral theory can only be obtained if singularities or

other defects, where chiral fermions live, are included. Singularities arise,

for example, when a supersymmetric cycle shrinks to zero size.

M-theory compacti¬cation on a G2 manifold with a conical singularity

leads to interesting strongly coupled gauge theories, which have been in-

vestigated in some detail. The local structure of a conical singularity is

described by a metric of the form

ds2 = dr2 + r2 d„¦2 . (9.221)

n’1

Here r denotes a radial coordinate and d„¦2 is the metric of some compact

n’1

manifold Y . In general, this metric describes an n-dimensional space X

that has a singularity at r = 0 unless d„¦2 is the metric of the unit sphere,

n’1

n’1 . An example is a lens space S 3 /

S N +1 , which corresponds to an AN

singularity.

Singularities can give rise to nonabelian gauge groups in the low-energy

e¬ective action. Recall from Chapter 8 that M-theory compacti¬ed on K3

is dual to the heterotic string on T 3 , and that there is enhanced gauge

symmetry at the singularities of K3, which have an ADE classi¬cation.

9.12 Manifolds with G2 and Spin(7) holonomy 437

Invoking this duality for ¬bered manifolds, there should be a duality between

compacti¬cation of heterotic theories on Calabi“Yau manifolds with a T 3

¬bration and M-theory on G2 manifolds with a K3 ¬bration.

In order to obtain four-dimensional theories with nonabelian gauge sym-

metry, one strategy is to embed ADE singularities in G2 manifolds. In

general, the singularities of four-dimensional manifolds can be described as

2 /“, where “ is a subgroup of the holonomy group SO(4). The points that

£

are left invariant by “ then correspond to the singularities. The holonomy

group of K3 is SU (2), and as a result “ has to be a subgroup of SU (2) to

give unbroken supersymmetry. The ¬nite subgroups of SU (2) also have an

ADE classi¬cation consisting of two in¬nite series (An , n = 1, 2, . . . and Dk ,

k = 4, 5 . . . ) and three exceptional subgroups (E6 , E7 and E8 ). So for exam-

ple, the generators for the two in¬nite series can be represented according

to

e2πi/n 0

’2πi/n , (9.222)

0 e

for the An series. Meanwhile Dk has two generators given by

eπi/(k’2) 0 0i

and . (9.223)

e’πi/(k’2) i0

0

In the heterotic/M-theory duality discussed in Section 9.11, the heterotic

string gets an enhanced symmetry group whenever the K3 becomes sin-

gular. In general, M-theory compacti¬ed on a background of the form

4 /“ 6,1 gives rise to a Yang“Mills theory with the correspond-

ADE —

¡ ¡

ing ADE gauge group, near the singularity. Embedding four-dimensional

singular spaces into G2 manifolds, M-theory compacti¬cation can therefore

give rise to nonabelian gauge groups in four dimensions.

G2 manifolds and intersecting D6-brane models

Another area where G2 manifolds play an important role is intersecting D6-

brane models.32 Recall that Section 8.3 showed that N parallel D6-branes

in the type IIA theory are interpreted in M-theory as a multi-center Taub“

NUT metric times a ¬‚at seven-dimensional Minkowski space-time. Half

of the supersymmetry is preserved by a stack of parallel branes. If they

are not parallel, the amount of supersymmetry preserved depends on types

of rotations that relate the branes. Any con¬guration preserving at least

one supersymmetry is described by a special-holonomy manifold from the

M-theory perspective. If the position of the branes is such that they can

32 This is one of the constructions used in attempts to obtain realistic models.

438 String geometry

be interpreted in M-theory as a seven-manifold on which one covariantly

constant real spinor can be de¬ned times ¬‚at four-dimensional Minkowski

space-time, then this is a G2 holonomy con¬guration.

For parallel D6-branes, the 7-manifold with G2 holonomy is a direct prod-

uct of the multi-center Taub“NUT metric times 3 , as you are asked to

¡

verify in a homework problem. As discussed in Chapter 8, certain type IIA

¬elds, such as the dilaton and the U (1) gauge ¬eld, lift to pure geometry in

11 dimensions. From the M-theory perspective, strings stretched between

two D6-branes have an interpretation as membranes wrapping one of the

n(n + 1)/2 holomorphic embeddings of S 2 in multi-center Taub“NUT, as

shown in Fig. 9.12. When two D6-branes come close to each other, these

strings become massless, resulting in nonabelian gauge symmetry. Without

entering into the details, let us mention that chiral matter can be realized

when D6-branes intersect at appropriate angles, because the GSO projection

removes massless fermions of one chirality. This leads to interesting models

with some realistic features.

Fig. 9.12. Strings stretched between two D6-branes can be interpreted as mem-

branes wrapping a holomorphically embedded S 2 in a multi-center Taub“NUT ge-

ometry.

Spin(7) manifolds

Eight-dimensional manifolds of Spin(7) holonomy are of interest in the study

of string dualities including connections to strongly coupled gauge theories.

Compacti¬cation of M-theory on a Spin(7) manifold gives a theory with