dimensional space-time. As explained above, the corresponding fermions

are chiral. The subscript a labels a quantity that transforms as a 3 of

the holonomy SU (3). However, the embedding of the spin connection in

the gauge group means that this SU (3) is identi¬ed with the SU (3) in

the decomposition of the gauge group. Therefore, the components of Aa

belonging to the (27, ¯ term in the decomposition can be written in the form

3)

Aa,¯¯, where s labels the components of the 27 and ¯ labels the components

¯ b

sb

of the ¯ This can be regarded as a (1, 1)-form taking values in the 27.

3.

However, a (1, 1)-form has h1,1 zero modes. Thus, we conclude that there

are h1,1 massless chiral supermultiplets belonging to the 27 of E6 . The next

case to consider is Aa,sb . To recast this as a di¬erential form, one uses the

inverse K¨hler metric and the antiholomorphic (0, 3)-form to de¬ne

a

Aad¯s = Aa,sb g b¯„¦cd¯.

c

(9.170)

¯e ¯ ¯e

This is a 27-valued (1, 2)-form. It then follows that there are h2,1 massless

chiral supermultiplets belonging to the 27 of E6 .

As an exercise in group theory, let us explore how the reasoning above

is modi¬ed if the background gauge ¬elds take values in SU (4) or SU (5)

418 String geometry

rather than SU (3). In the ¬rst case, the appropriate embedding would be

E8 ⊃ SO(10) — SU (4), so that the unbroken gauge symmetry would be

SO(10) — E8 , and the decomposition of the adjoint would be

248 = (45, 1) + (1, 15) + (10, 6) + (16, 4) + (16, ¯

4). (9.171)

This could lead to a supersymmetric SO(10) grand-uni¬ed theory with gen-

erations of chiral matter in the 16, antigenerations in the 16 and Higgs

¬elds in the 10. This is certainly an intriguing possibility. In the SU (5)

case, the embedding would be E8 ⊃ SU (5) — SU (5), so that the unbroken

gauge symmetry would be SU (5) — E8 . This could lead to a massless ¬eld

content suitable for a supersymmetric SU (5) grand-uni¬ed theory.

As a matter of fact, there are more complicated constructions in which

these possibilities are realized. For the gauge ¬elds to take values in SU (4)

or SU (5), rather than SU (3), requires more complicated ways of solving

the topological constraints than simply embedding the holonomy group in

the gauge group. The existence of solutions is guaranteed by a theorem

of Uhlenbeck and Yau, though the details are beyond the scope of this

book. For these more general embeddings there is no longer a simple relation

between the Hodge numbers and the number of generations.

Starting from a Calabi“Yau compacti¬cation scenario that leads to a su-

persymmetric grand-uni¬ed theory, there are still a number of other issues

that need to be addressed. These include breaking the gauge symmetry to

the standard-model gauge symmetry and breaking the residual supersym-

metry. If the Calabi“Yau space is not simply connected, as happens for

certain quotient-space constructions, there is an elegant possibility. Wilson

lines Wi = exp( γi A) can be introduced along the noncontractible loops γi

without changing the ¬eld strengths. The unbroken gauge symmetry is then

reduced to the subgroup that commutes with these Wilson lines. This can

break the gauge group to SU (3) — SU (2) — U (1)n , where n = 3 for the E6

case, n = 2 for the SO(10) case and n = 1 for the SU (5) case. Experimen-

talists are on the lookout for heavy Z bosons, which would correspond to

extra U (1) factors.

9.11 K3 compacti¬cations and more string dualities

Compacti¬cations of string theory that lead to a four-dimensional space-

time are of interest for making contact with the real world. However, it is

also possible to construct other consistent compacti¬cations, which can also

be of theoretical interest. This section considers a particularly interesting

class of four-dimensional compact manifolds, namely Calabi“Yau two-folds.

9.11 K3 compacti¬cations and more string dualities 419

As discussed earlier, the only Calabi“Yau two-fold with SU (2) holonomy is

the K3 manifold. It can be used to compactify superstring theories to six

dimensions, M-theory to seven dimensions or F-theory to eight dimensions.

Compacti¬cation of M-theory on K3

M-theory has a consistent vacuum of the form M7 —K3, where M7 represents

seven-dimensional Minkowski space-time. The compacti¬cation breaks half

of the supersymmetries, so the resulting vacuum has 16 unbroken super-

symmetries. The moduli of the seven-dimensional theory have two potential

sources. One source is the moduli-space of K3 manifolds, itself, which is

manifested as zero modes of the metric tensor on K3. The other source is

from zero modes of antisymmetric-tensor gauge ¬elds. However, the only

such ¬eld in M-theory is a three-form, and the third cohomology of K3 is

trivial. Therefore, the three-form does not contribute any moduli in seven

dimensions, and the moduli space of the compacti¬ed theory is precisely the

moduli space of K3 manifolds.

Moduli space of K3

Let us count the moduli of K3. K¨hler-structure deformations are given

a

by closed (1, 1)-forms,29 so their number in the case of K3 is h1,1 = 20.

Complex-structure deformations in the case of K3 correspond to coe¬cients

for the variations

¯

δgab ∼ „¦ac g cd ωbd + (a ” b), (9.172)

¯

where „¦ is the holomorphic two-form and ωbd is a closed (1, 1)-form. This

¯

variation vanishes if ω is the K¨hler form, as you are asked to verify in a

a

homework problem. Thus, there are 38 real (19 complex) complex-structure

moduli. Combined with the 20 K¨hler moduli this gives a 58-dimensional

a

moduli space of K3 manifolds.

This moduli space is itself an orbifold. The result, worked out by mathe-

maticians, is + — M19,3 , where

¡

M19,3 = M0 /O(19, 3; ) (9.173)

19,3

and

O(19, 3; )

M0 = . (9.174)

¡

19,3

O(19, ) — O(3, )

¡ ¡

The + factor corresponds to the overall volume modulus, and the factor

¡

M19,3 describes a space of dimension 19 — 3 = 57, as required. In contrast

29 This is true for any Calabi“Yau n-fold.

420 String geometry

to the case of Calabi“Yau three-folds, the dependence on K¨hler moduli and

a

complex-structure moduli does not factorize. The singularities of the moduli