R2

„ =i and ρ = iR1 R2 , (9.166)

R1

as in Section 9.5. Performing a T-duality on the ¬ber circle sends R1 ’ 1/R1

(for ± = 1), and as a result the moduli ¬elds of the resulting mirror torus

are

R2

„ = iR1 R2

˜ and ρ=i .

˜ (9.167)

R1

This shows that under the mirror map the complex-structure and K¨hler-

a

structure parameters have been interchanged, just as in the case of the

Calabi“Yau three-fold.

T 3 ¬brations

An approach to understanding mirror symmetry, which is based on T-

duality, was proposed by Strominger, Yau and Zaslow (SYZ). If mirror sym-

metry holds, then a necessary requirement is that the spectrum of BPS states

for the type IIA theory on M and type IIB on W must be the same. Ver-

ifying this would not constitute a complete proof, but it would give strong

support to the mirror-symmetry conjecture. That is often the best that can

be done for duality conjectures in string theory.

The BPS states to be compared arise from D-branes wrapping supersym-

metric cycles of the Calabi“Yau. In the case of the type IIA theory, Dp-

branes, with p = 0, 2, 4, 6, can wrap even-dimensional cycles of the Calabi“

Yau. However, since only BPS states can be compared reliably, only su-

persymmetric cycles should be considered. In the simplest case one only

considers the D0-brane, whose moduli space is the whole Calabi“Yau M ,

since the D0-brane can be located at any point in M . In the type IIB the-

ory the BPS spectrum of wrapped D-branes arises entirely from D3-branes

wrapping special Lagrangian three-cycles.

Since mirror symmetry relates the special Lagrangian three-cycle of W

to the whole Calabi“Yau manifold M , its properties are very constrained.

First, the D3-brane moduli space has to have three complex dimensions.

Three real moduli are provided by the transverse position of the D3-brane.

The remaining three moduli are obtained by assuming that mirror symmetry

is implemented by three T-dualities. D0-branes are mapped to D3-branes

under the action of three T-dualities. After performing the three T-dualities,

three ¬‚at U (1) gauge ¬elds appear in the directions of the D3-brane. These

are associated with the isometries of three circles which form a three-torus.

9.10 Heterotic string theory on Calabi“Yau three-folds 415

As a result, W is a T 3 ¬bration over a base B. By de¬nition, a Calabi“

Yau manifold is a T 3 ¬bration if it can be described by a three-dimensional

base space B, with a three-torus above each point of B assembled so as to

make a smooth Calabi“Yau manifold. A T 3 ¬bration is more general than a

T 3 ¬ber bundle in that isolated T 3 ¬bers are allowed to be singular, which

means that one or more of their cycles degenerate. Turning the argument

around, M must also be a T 3 ¬bration. Mirror symmetry is a ¬ber-wise

T-duality on all of the three directions of the T 3 . A simple example of a

¬ber bundle is depicted in Fig. 9.9.

Fig. 9.9. A Moebius strip is an example of a nontrivial ¬ber bundle. It is a line

segment ¬bered over a circle S 1 . Calabi“Yau three-folds that have a mirror are

conjectured to be T 3 ¬brations over a base B. In contrast to the simple example

of the Moebius strip, some of the T 3 ¬bers are allowed to be singular.

Since the number of T-dualities is odd, even forms and odd forms are in-

terchanged. As a result, the (1, 1) and (2, 1) cohomologies are interchanged,

as is expected from mirror symmetry. Moreover, there exists a holomor-

phic three-form on W , which implies that W is Calabi“Yau. The three

T-dualities, of course, also interchange type IIA and type IIB.

The argument given above probably contains the essence of the proof of

mirror symmetry. A note of caution is required though. We already pointed

out that there are Calabi“Yau manifolds whose mirrors are not Calabi“Yau,

so a complete proof would need to account for that. The T-duality rules and

the condition that a supersymmetric three-cycle has to be special Lagrangian

are statements that hold to leading order in ± , while the full description of

the mirror W requires, in general, a whole series of ± corrections.

9.10 Heterotic string theory on Calabi“Yau three-folds

As was discussed earlier, the fact that dH is an exact four-form implies that

1

tr(R §R) and tr(F §F ) = 30 Tr(F §F ) must belong to the same cohomology

class. The curvature two-form R takes values in the Lie algebra of the

416 String geometry

holonomy group, which is SU (3) in the case of Calabi“Yau compacti¬cation.

Specializing to the case of the E8 —E8 heterotic string theory, F takes values

in the E8 — E8 Lie algebra. The characteristic class tr(R § R) is nontrivial,

and so it is necessary that gauge ¬elds take nontrivial background values in

the compact directions.

The easiest way “ but certainly not the only one “ to satisfy the cohomol-

ogy constraint is for the ¬eld strengths associated with an SU (3) subgroup

of the gauge group to take background values that are equal to those of

the curvature form while the other ¬eld strengths have zero background

value. More fundamentally, the Yang“Mills potentials A can be identi¬ed

with the potentials that give the curvature, namely the spin connections.

This method of satisfying the constraint is referred to as embedding the spin

connection in the gauge group. There are many ways of embedding SU (3)

inside E8 — E8 and not all of them would work. The embedding is restricted

by the requirement that the cohomology class of tr(F § F ) gives exactly

the class of tr(R § R) and not just some multiple of it. The embedding

that satis¬es this requirement is one in which the SU (3) goes entirely into

one E8 factor in such a way that its commutant is E6 . In other words,

E8 ⊃ E6 — SU (3). Thus, for this embedding, the unbroken gauge symmetry

of the e¬ective four-dimensional theory is E6 — E8 .

This speci¬c scenario is not realistic for a variety of reasons, but it does

have some intriguing features that one could hope to preserve in a better

set-up. For one thing, E6 is a group that has been proposed for grand uni-

¬cation. In that context, the gauge bosons belong to the adjoint 78 and

chiral fermions are assigned to the 27, which is a complex representation.

This representation might also be used for Higgs ¬elds. Clearly, these rep-

resentations give a lot of extra ¬elds beyond what is observed, so additional

measures are required to lift them to high mass or else eliminate them alto-

gether.

The presence of the second unbroken E8 also needs to be addressed. The

important observation is that all ¬elds that participate in standard-model

interactions must carry nontrivial standard-model quantum numbers. But

the massless ¬elds belonging to the adjoint of the second E8 are all E6 sin-

glets. Fields that belong to nontrivial representations of both E8 s ¬rst occur

for masses comparable to the string scale. Thus, if the string scale is com-

parable to the Planck scale, the existence of light ¬elds carrying nontrivial

quantum numbers of the second E8 could only be detected by gravitational-

strength interactions. These ¬elds comprise the hidden sector. A hidden

sector could actually be useful. Assuming that the hidden sector has a mass

gap, perhaps due to con¬nement, one intriguing possibility is that hidden-

9.10 Heterotic string theory on Calabi“Yau three-folds 417

sector particles comprise a component of the dark matter. It has also been

suggested that gaugino condensation in the hidden sector could be the origin

of supersymmetry breaking.

The adjoint of E8 , the 248, is reducible with respect to the E6 — SU (3)

subgroup, with the decomposition

248 = (78, 1) + (1, 8) + (27, 3) + (27, ¯).

3 (9.168)

The massless spectrum in four dimensions can now be determined. There

are massless vector supermultiplets in the adjoint of E6 — E8 , since this is

the unbroken gauge symmetry. In addition, there are h1,1 chiral supermul-

tiplets containing (complexi¬ed) K¨hler moduli and h2,1 chiral supermul-

a

tiplets containing complex-structure moduli. These chiral supermultiplets

are all singlets of the gauge group, since the ten-dimensional graviton is a

singlet.

Let us now explain the origin of chiral matter, which belongs to chiral

supermultiplets. It is easiest to focus on the origin of the scalars and invoke

supersymmetry to infer that the corresponding massless fermions must also

be present. For this purpose let us denote the components of the gauge

¬elds as follows:

AM = (Aµ , Aa , Aa ). (9.169)

¯