2 2

where the last line has used the six-dimensional identity

1

(J § J) = —J.

2

As a result, one obtains

1†

Nmnp = ’ 12 ·+ H, γmnp + 3iγ[m Jnp] ·+

1†

= ’ 12 ·+ ‚ ¦, γmnp + 3iγ[m Jnp] ·+

/

= 0.

This proves that the manifold is complex. 2

EXERCISE 10.11

Prove that „¦ in Eq. (10.222) is holomorphic.

SOLUTION

¯

A holomorphic three-form is a ‚ closed form of type (3, 0). In order to prove

¯

that „¦ is holomorphic, we compute ‚„¦. We start by computing its covariant

derivative.

The covariant derivative (de¬ned with respect to the Christo¬el connec-

tion) acting on the tensor „¦ is

= ‚k „¦abc ’ 3“p „¦bc]p = ‚k „¦abc ’ “p „¦abc .

k „¦abc

¯ ¯ ¯

¯ ¯

k[a kp

Using the de¬nition of the Christo¬el connection and expanding Eq. (10.220)

in components implies

1

“p = g p¯‚[k gq]p =

q

Hkp¯g p¯ = ‚k ¦.

q

¯¯ ¯q ¯

¯

kp 2

518 Flux compacti¬cations

As a result,

= ‚k „¦abc ’ ‚k ¦„¦abc .

k „¦abc

¯ ¯ ¯

On the other hand, using the de¬nition of „¦, one obtains

= ’‚k ¦„¦abc .

k „¦abc

¯ ¯

Indeed, to see this last relation, use

e’2¦ ·’ γabc ·’ = ’2‚k ¦„¦abc + 2e’2¦ ·’ γabc

T T

k „¦abc = k ·’ .

¯ ¯ ¯ ¯

k

Using Eq. (10.211), this is equal to

1

’2‚k ¦„¦abc + Hkn¯g n¯„¦abc = ’‚k ¦„¦abc .

p

¯ ¯p ¯

2

This implies that „¦ is holomorphic. 2

10.5 The strongly coupled heterotic string

This feature is generic and is not special to the type IIB theory. It also

applies to the heterotic theory. The subject of moduli stabilization in the

strongly coupled heterotic string is still relatively unexplored and an active

area of current research.

A natural way to describe the strongly coupled E8 — E8 heterotic string

theory is in terms of M-theory. This formulation, called heterotic M-theory,

was introduced in Chapter 8. Recall that it has a space-time geometry

10 — S 1 / 1

2 . The quotient space S / 2 can be regarded as a line interval

¡

that arises when the E8 — E8 heterotic string is strongly coupled, with a

length equal to gs s . The gauge ¬elds of the two E8 gauge groups live on

the two ten-dimensional boundaries of the resulting 11-dimensional space-

time. This section explores some phenomenological implications of ¬‚uxes in

heterotic M-theory and brie¬‚y describes moduli stabilization in the context

of the strongly coupled theory. For heterotic M-theory compacti¬ed on a

Calabi“Yau three-fold, the four-form ¬eld strength F4 does not vanish if

higher-order terms in κ2/3 are taken into account. The Yang“Mills ¬elds act

as magnetic sources in the Bianchi-identity for F4 and therefore an F4 of

order κ2/3 is required for consistency. As in the previous sections, a warped

geometry again plays a crucial role in heterotic M-theory compacti¬cations.

One rather intriguing result is that, in heterotic M-theory, Newton™s con-

stant is bounded from below by an expression that is close to the correct

10.5 The strongly coupled heterotic string 519

value. This is in contrast to the weakly coupled heterotic string theory,

where the value of Newton™s constant comes out too large. Let us describe

this in more detail.

Newton™s constant from the D = 10 heterotic string

As was shown in Chapter 8, the leading terms of the ten-dimensional e¬ective

action for the heterotic string in the string frame are

√ 4 1

d x ’Ge’2¦

10

R ’ 3 tr|F |2 + . . . .

Le¬ = (10.240)

±4 ±

If this theory is compacti¬ed on a Calabi“Yau manifold with volume V, the

resulting four-dimensional low-energy e¬ective action takes the form

√ 4 1

’Ge’2¦

d4 x V R ’ 3 tr|F |2 + . . . .

Le¬ = (10.241)

±4 ±

In the supergravity approximation, the volume of the Calabi“Yau manifold

is assumed to be large V > ± 3 . Thus, the value of Newton™s constant from

the previous formula is

e2¦ ± 4

G4 = . (10.242)

64πV

The value of the uni¬cation gauge coupling constant is

e2¦ ± 3

±U = . (10.243)

16πV

The previous two formulas lead to an expression for Newton™s constant in

terms of these variables

1

G4 = ±U ± . (10.244)

4

If one assumes that the string is weakly coupled, then e2¦ 1, and the

volume of the Calabi“Yau is bounded from above

±3

V . (10.245)

16π±U

In heterotic-string compacti¬cations of the type described in Chapter 9,