factor has 18 — 2 real moduli or 18 complex moduli. In fact, mathematicians

knew before the discovery of F-theory that this is the moduli space of ellip-

tically ¬bered K3 manifolds. Thus, F-theory compacti¬ed on an elliptically

¬bered K3 (with section) is conjectured to be dual to the heterotic string

theory compacti¬ed on T 2 .

This duality can be related to the others, and so it constitutes one more

link in a consistent web of dualities. For example, if one compacti¬es on

another circle, and uses the duality between type IIB on a circle and M-

theory on a torus, this torus becomes identi¬ed with the F-theory ¬ber

torus, which now has ¬nite area. Then one recovers the duality between M-

theory on K3 and the heterotic string on T 3 for the special case of elliptically

¬bered K3 s.

The F-theory construction described above is the simplest example of a

large class of possibilities. More generally, F-theory on an elliptically ¬bered

Calabi“Yau n-fold (with section) gives a solution for (12 ’ 2n)-dimensional

Minkowski space-time. For example, using elliptically ¬bered Calabi“Yau

four-folds one can obtain four-dimensional F-theory vacua with N = 1 su-

persymmetry. It is an interesting challenge to identify duality relations

between such constructions and other ones that can give N = 1, such as the

heterotic string compacti¬ed on a Calabi“Yau three-fold.

9.12 Manifolds with G2 and Spin(7) holonomy

Since the emergence of string dualities and the discovery of M-theory, special-

holonomy manifolds have received considerable attention. Manifolds of

SU (3) holonomy have already been discussed at length. 7-manifolds with

G2 holonomy and 8-manifolds with Spin(7) holonomy are also of interest

for a number of reasons. They constitute the exceptional-holonomy man-

ifolds. We refer to them simply as G2 manifolds and Spin(7) manifolds,

respectively.

G2 manifolds

Suppose that M-theory compacti¬ed to four dimensions on a 7-manifold M7 ,

M11 = M4 — M7 , (9.210)

gives rise to N = 1 supersymmetry in four dimensions. An analysis of the

supersymmetry constraints, along the lines studied for Calabi“Yau three-

434 String geometry

folds, constrains M7 to have G2 holonomy. In such a compacti¬cation to

¬‚at D = 4 Minkowski space-time, there should exist one spinor (with four

independent components) satisfying

δψM = Mµ = 0. (9.211)

The background geometry is then 3,1 — M7 , where M7 has G2 holonomy,

¡

and µ is the covariantly constant spinor of the G2 manifold tensored with a

constant spinor of 3,1 . As in the case of Calabi“Yau three-folds, Eq. (9.211)

¡

implies that M7 is Ricci ¬‚at. Of course, it cannot be K¨hler, or even complex,

a

since it has an odd dimension. Let us now examine why Eq. (9.211) implies

that M7 has G2 holonomy.

The exceptional group G2

G2 can be de¬ned as the subgroup of the SO(7) rotation group that preserves

the form

• = dy 123 + dy 145 + dy 167 + dy 246 ’ dy 257 ’ dy 347 ’ dy 356 , (9.212)

where

dy ijk = dy i § dy j § dy k , (9.213)

and y i are the coordinates of 7 . G2 is the smallest of the ¬ve exceptional

¡

simple Lie groups (G2 , F4 , E6 , E7 , E8 ), and it has dimension 14 and rank 2.

Its Dynkin diagram is given in Fig. 9.11. Let us describe its embedding in

Spin(7), the covering group of SO(7), by giving the decomposition of three

representations of Spin(7), the vector 7, the spinor 8 and the adjoint 21:

• Adjoint representation: decomposes under G2 as 21 = 14 + 7.

• The vector representation is irreducible 7 = 7.

• The spinor representation decomposes as 8 = 7 + 1.

G2

Fig. 9.11. The G2 Dynkin diagram.

The singlet in the spinor representation precisely corresponds to the co-

variantly constant spinor in Eq. (9.211) and this decomposition is the reason

why G2 compacti¬cations preserve 1/8 of the original supersymmetry, lead-

ing to an N = 1 theory in four dimensions in the case of M-theory. While

9.12 Manifolds with G2 and Spin(7) holonomy 435

Calabi“Yau three-folds are characterized by the existence of a nowhere van-

ishing covariantly constant holomorphic three-form, a G2 manifold is charac-

terized by a covariantly constant real three-form ¦, known as the associative

calibration

1

¦ = ¦abc ea § eb § ec , (9.214)

6

where ea are the seven-beins of the manifold. The Hodge dual four-form ¦

is known as the coassociative calibration.

A simple compact example

Smooth G2 manifolds were ¬rst constructed by resolving the singularities

of orbifolds. A simple example is the orbifold T 7 /“, where T 7 is the ¬‚at

seven-torus and “ is a ¬nite group of isometries preserving the calibration

Eq. (9.212) generated by

± : (y 1 , . . . , y 7 ) ’ (y 1 , y 2 , y 3 , ’y 4 , ’y 5 , ’y 6 , ’y 7 ), (9.215)

β : (y 1 , . . . , y 7 ) ’ (y 1 , ’y 2 , ’y 3 , y 4 , y 5 , 1/2 ’ y 6 , ’y 7 ), (9.216)

γ : (y 1 , . . . , y 7 ) ’ (’y 1 , y 2 , ’y 3 , y 4 , 1/2 ’ y 5 , y 6 , 1/2 ’ y 7 ). (9.217)

In a homework problem you are asked to verify that ±, β, γ have the following

properties: (1) they preserve the calibration, (2) ±2 = β 2 = γ 2 = 1, (3) the

three generators commute. The group “ is isomorphic to 3 . The ¬xed

2

3 , while (β, γ)

points of ± (and similarly for β and γ) are 16 copies of T

act freely on the ¬xed-point set of ± (similarly for the ¬xed-point set of β

and γ). The singularities of this orbifold can be blown up in a similar way

discussed in Section 9.1 for K3, that is, by cutting out a ball B 4 / 2 around

each singularity and replacing it with an Eguchi“Hanson space. The result

is a smooth G2 manifold.

Supersymmetric cycles in G2 manifolds

As in the case of Calabi“Yau three-folds, supersymmetric cycles in G2 man-

ifolds play a crucial role in describing nonperturbative e¬ects. Supersym-

metric three-cycles can be de¬ned for G2 manifolds in a similar manner as

for Calabi“Yau three-folds in Section 9.8. A supersymmetric three-cycle is

a con¬guration that solves the equation