z c (t± + δt± ) = z c (t± ) + M± δt± ,

c

9.7 Type IIA and type IIB on Calabi“Yau three-folds 399

which implies that

‚(dz c ) c c

‚M± d ‚M± d ¯

c

= dM± = dz + ¯ d¯ .

z

‚t± ‚z d ‚zd

¯

Therefore, the (2, 1)-form is equal to

c

1 ‚M± a ¯

b

zd

¯ dz § dz § d¯ .

„¦abc

2 ‚zd

¯

We want to show that this is equal to

1 e ‚gd¯ ¯

¯e

χ± = ’ „¦abc g c¯ dz a § dz b § d¯d .

z

±

4 ‚t

Therefore, we need to show that

c

‚M± 1 e ‚gd¯¯e

= ’ g c¯ .

¯¯ ‚t±

2

‚zd

¯

Di¬erentiating the hermitian metric ds2 = 2ga¯dz a d¯b in the same way that

z

b

we did the holomorphic three-form gives the desired result

c

‚gd¯ ‚M±

¯e

= ’2gc¯ .

e

¯¯

‚t± ‚zd

2

9.7 Type IIA and type IIB on Calabi“Yau three-folds

The compacti¬cation of type IIA or type IIB superstring theory on a Calabi“

Yau three-fold M leads to a four-dimensional theory with N = 2 supersym-

metry. The metric perturbations and other scalar zero modes lead to moduli

¬elds that belong to N = 2 supermultiplets. These supermultiplets can be

either vector multiplets or hypermultiplets, since these are the only massless

N = 2 supermultiplets that contain scalar ¬elds.

D = 4, N = 2 supermultiplets

Massless four-dimensional supermultiplets have a structure that is easily

derived from the superalgebra by an analysis that corresponds to the mass-

less analog of that presented in Exercise 8.2. The physical states are labeled

by their helicities, which are Lorentz-invariant quantities for massless states.

For N -extended supersymmetry the multiplet is determined by the maximal

helicity with the rest of the states having multiplicities given by binomial

400 String geometry

coe¬cients. When the multiplet is not TCP self-conjugate, one must also

adjoin the conjugate multiplet.21

In the case of N = 2 this implies that the supermultiplet with maximal

helicity 2 also has two helicity 3/2 states, and one helicity 1 state. Adding

the TCP conjugate multiplet (with the opposite helicities) gives the N = 2

supergravity multiplet, which contains one graviton, two gravitinos and one

graviphoton. If the maximal helicity is 1, and one again adds the TCP con-

jugate, the same reasoning gives the N = 2 vector multiplet, which contains

one vector, two gauginos and two scalars. Finally, the multiplet with max-

imal helicity 1/2, called a hypermultiplet contains two spin 1/2 ¬elds and

four scalars. In each of these three cases there is a total of four bosonic and

four fermionic degrees of freedom.

Type IIA

When the type IIA theory is compacti¬ed on a Calabi“Yau three-fold M , the

resulting four-dimensional theory contains h1,1 abelian vector multiplets and

h2,1 + 1 hypermultiplets. The scalar ¬elds in these multiplets parametrize

the moduli spaces. There is no mixing between the two sets of moduli, so

the moduli space can be expressed in the product form

M1,1 (M ) — M2,1 (M ). (9.138)

Each vector multiplet contains two real scalar ¬elds, so the dimension of

M1,1 (M ) is 2h1,1 . In fact, this space has a naturally induced geometry that

promotes it into a special-K¨hler manifold (with a holomorphic prepoten-

a

tial). Each hypermultiplet contains four real scalar ¬elds, so the dimen-

sion of M2,1 (M ) is 4(h2,1 + 1). This manifold turns out to be of a special

type called quaternionic K¨hler.22 These geometric properties are inevitable

a

consequences of the structure of the interaction of vector multiplets and hy-

permultiplets with N = 2 supergravity. The massless ¬eld content of the

compacti¬ed type IIA theory is explored in Exercise 9.12.

Type IIB

Compacti¬cation of the type IIB theory on a Calabi“Yau three-fold W yields

h2,1 abelian vector multiplets and h1,1 + 1 hypermultiplets. The correspond-

21 The only self-conjugate multiplets in four dimensions are the N = 4 vector multiplet and the

N = 8 supergravity multiplet.

22 Note that quaternionic K¨hler manifolds are not K¨hler. The de¬nition is given in the appendix.

a a

9.7 Type IIA and type IIB on Calabi“Yau three-folds 401

ing moduli space takes the form

M1,1 (W ) — M2,1 (W ). (9.139)

In this case the situation is the opposite to type IIA, in that M2,1 (W )

is special K¨hler and M1,1 (W ) is quaternionic K¨hler. The massless ¬eld

a a

content of the compacti¬ed type IIB theory is explored in Exercise 9.13.

For each of the type II theories the dilaton belongs to the universal hyper-

multiplet, which explains the extra hypermultiplet in each case. This scalar

is complex because there is a second scalar, an axion a, which is the four-

dimensional dual of the two-form Bµν (dB = da). The complex-structure

moduli, being associated with complex (2, 1)-forms, are naturally complex.

The h1,1 K¨hler moduli are complex due to the B-¬eld contribution in the

a

complexi¬ed K¨hler form (Ja¯ + iBa¯) as in the case of the heterotic string.

a b b

EXERCISES

EXERCISE 9.11

Explain the origin of the massless scalar ¬elds in ¬ve dimensions that are

obtained by compactifying M-theory on a Calabi“Yau three-fold.

SOLUTION

The massless ¬elds in 11 dimensions are

{GM N , AM N P , ΨM }.

Let us decompose the indices of these ¬elds in a SU (3) covariant way, M =

(µ, i, ¯ The ¬elds belong to the following ¬ve-dimensional supermultiplets:

i).