ponents, so 1/16 of the original supersymmetry is preserved. This is less

9.12 Manifolds with G2 and Spin(7) holonomy 439

supersymmetry than the minimal amount for a Lorentz-invariant supersym-

metric theory in four dimensions. Witten has speculated that the existence

of such a three-dimensional theory might indicate the existence of a theory

in four dimensions with no supersymmetry that upon circle compacti¬ca-

tion develops an N = 1 supersymmetry in three dimensions. This is one

of many speculations that have been considered in attempts to explain why

the observed cosmological constant is so tiny.

Spin(7) is the subgroup of Spin(8) that leaves invariant the self-dual four-

form

„¦ = dy 1234 + dy 1256 + dy 1278 + dy 1357 ’ dy 1368 ’ dy 1458 ’ dy 1467’

dy 2358 ’ dy 2367 ’ dy 2457 + dy 2468 + dy 3456 + dy 3478 + dy 5678,

where

dy ijkl = dy i § dy j § dy k § dy l , (9.224)

and yi with i = 1, . . . , 8 are the coordinates of 8 . This 21-dimensional Lie

¡

group is compact and simply-connected.

The decomposition of the adjoint is 28 = 21 + 7. Spin(8) has three

eight-dimensional representations: the fundamental and two spinors, which

are sometimes denoted 8v , 8s and 8c . Because of the triality of Spin(8),

discussed in Chapter 5, it is possible to embed Spin(7) inside Spin(8) such

that one spinor decomposes as 8c = 7 + 1, while the 8v and 8s both reduce

to the spinor 8 of the Spin(7) subgroup. By choosing such an embedding,

the Spin(7) holonomy preserves 1/16 of the original supersymmetry corre-

sponding to the singlet in the decomposition of the two Spin(8) spinors.

Examples of compact Spin(7) manifolds can be obtained, as in the G2

case, as the blow-ups of orbifolds. The simplest example starts with an

orbifold T 8 / 4 . Spin(7) manifolds are not K¨hler in general. As in the G2

a

2

case, it is interesting to consider manifolds with singularities, which can lead

to strongly coupled gauge theories.

EXERCISES

EXERCISE 9.16

Verify that the calibration (9.212) is invariant under 14 linearly independent

combinations of the 21 rotation generators of 7 . ¡

440 String geometry

SOLUTION

An in¬nitesimal rotation has the form Rij = δij + aij , where aij is in¬nites-

imal, and aij = ’aji . This acts on the coordinates by y i = Rij y j . Now

plug this into the three-form (9.212) and keep only the linear terms in a.

Requiring the three-form to be invariant results in the equations

a14 + a36 + a27 = 0, a15 + a73 + a26 = 0,

a16 + a43 + a52 = 0, a17 + a35 + a42 = 0,

a76 + a54 + a32 = 0, a12 + a74 + a65 = 0,

a13 + a57 + a64 = 0.

These seven constraints leave 21 ’ 7 = 14 linearly independent rotations

under which the calibration is invariant. This construction ensures that

they generate a group. 2

Appendix: Some basic geometry and topology

This appendix summarizes some basic geometry and topology needed in this

chapter as well as other chapters of this book. This summary is very limited,

so we refer the reader to GSW as well as some excellent review articles for

a more detailed discussion. The mathematically inclined reader may prefer

to consult the math literature for a more rigorous approach.

Real manifolds

What is a manifold?

A real d-dimensional manifold is a space which locally looks like Euclidean

space d . More precisely, a real manifold of dimension d is de¬ned by

¡

introducing a covering with open sets on which local coordinate systems are

introduced. Each of these coordinate systems provides a homeomorphism

between the open set and a region in d . The manifold is constructed by

¡

pasting together the open sets. In regions where two open sets overlap,

the two sets of local coordinates are related by smooth transition functions.

Some simple examples of manifolds are as follows:

d d

• and are examples of noncompact manifolds.

¡ £

Appendix: Some basic geometry and topology 441

Fig. 9.13. This is not a one-dimensional manifold, because the intersection points

are singularities.

• The n-sphere n+1 (x2 ) = 1 is an example of a compact manifold. The

i

i=1

case n = 0 corresponds to two points at x = ±1, n = 1 is a circle and n = 2

is a sphere. In contrast to the one-dimensional noncompact manifold 1 , ¡

the compact manifold S 1 needs two open sets to be constructed.

• The space displayed in Fig. 9.13 is not a one-dimensional manifold since

there is no neighborhood of the cross over points that looks like 1 . ¡

Homology and cohomology

Many topological aspects of real manifolds can be studied with the help of

homology and cohomology groups. In the following let us assume that M is

a compact d-dimensional manifold with no boundary.

A p-form Ap is an antisymmetric tensor of rank p. The components of Ap

are

1

Ap = Aµ1 ···µp dxµ1 § · · · § dxµp , (9.225)

p!

where § denotes the wedge product (an antisymmetrized tensor product).

From a mathematician™s viewpoint, these p-forms are the natural quantities

to de¬ne on a manifold, since they are invariant under di¬eomorphisms and

therefore do not depend on the choice of coordinate system. The possible

values of p are p = 0, 1, . . . , d.

The exterior derivative d gives a linear map from the space of p-forms into

the space of (p + 1)-forms given by

1