h1,1 vector multiplets : Aµj k , Gj k , fermions

¯ ¯

h2,1 hypermultiplets : Aij k , Gjk , fermions.

¯

A ¬ve-dimensional duality transformation allows one to replace Aµνρ by a

real scalar ¬eld. The gravity multiplet has eight bosonic and eight fermionic

degrees of freedom. The other supermultiplets each have four bosonic and

402 String geometry

four fermionic degrees of freedom. The total number of massless scalar ¬elds

is

4h2,1 + h1,1 + 3.

2

EXERCISE 9.12

Consider the type IIA theory compacti¬ed on a Calabi“Yau three-fold. Ex-

plain the ten-dimensional origin of the massless ¬elds in four dimensions.

SOLUTION

The massless ¬elds in ten dimensions are

(+) (’)

{GM N , BM N , ¦, CM , CM N P , ΨM , ΨM , Ψ(+) , Ψ(’) },

(+) (’)

where ΨM , ΨM are the two Majorana“Weyl gravitinos of opposite chiral-

ity, while Ψ(+) , Ψ(’) are the two Majorana“Weyl dilatinos. Writing indices

in a SU (3) covariant way, M = (µ, i, ¯ we can arrange the ¬elds in N = 2

i),

supermultiplets:

gravity multiplet : Gµν , Ψµ , Ψµ , Cµ

h1,1 vector multiplets : Cµi¯, Gi¯, Bi¯, fermions

j j j

h2,1 hypermultiplets : Cij k , Gij , fermions

¯

universal hypermultiplet : Cijk , ¦, Bµν , fermions.

Bµν is dual to a scalar ¬eld. Since the ¬elds Cij k , Gij , Cijk are complex, the

¯

number of the massless scalar ¬elds is 2h1,1 + 4h2,1 + 4. There are h1,1 + 1

massless vector ¬elds. 2

EXERCISE 9.13

Consider the type IIB theory compacti¬ed on a Calabi“Yau three-fold. Ex-

plain the ten-dimensional origin of the massless ¬elds in four dimensions.

SOLUTION

The massless ¬elds in ten dimensions are

(+) (+)

{GM N , BM N , ¦, C, CM N , CM N P Q , ΨM , ΨM , Ψ(’) , Ψ(’) }.

9.8 Nonperturbative e¬ects in Calabi“Yau compacti¬cations 403

Let us use the same SU (3) covariant notation as in the previous exercise.

Compacti¬cation on a Calabi“Yau three-fold again gives N = 2, D = 4

supersymmetry. The ¬elds belong to the following supermultiplets:

gravity multiplet : Gµν , Ψµ , Ψµ , Cµijk

h2,1 vector supermultiplets : Cµij k , Gij , fermions

¯

h1,1 hypermultiplets : Cµνi¯, Gi¯, Bi¯, Ci¯, fermions

j j j j

universal hypermultiplet : ¦, C, Bµν , Cµν , fermions.

Now taking into account that Gij is complex and that the four-form C has a

self-duality constraint on its ¬eld strength, the total number of the massless

scalar ¬elds is 2h2,1 + 4(h1,1 + 1). The total number of massless vector ¬elds

is h2,1 + 1. 2

9.8 Nonperturbative e¬ects in Calabi“Yau compacti¬cations

Until now we have discussed perturbative aspects of Calabi“Yau compact-

i¬cation that were understood prior to the second superstring revolution.

This section and the following ones discuss some nonperturbative aspects

of Calabi“Yau compacti¬cations that were discovered during and after the

second superstring revolution.

The conifold singularity

In addition to their nonuniqueness, one of the main problems with Calabi“

Yau compacti¬cations is that their moduli spaces contain singularities, that

is, points in which the classical description breaks down. By analyzing a

particular example of such a singularity, the conifold singularity, it became

clear that the classical low-energy e¬ective action description breaks down.

Nonperturbative e¬ects due to branes wrapping vanishing (or degenerating)

cycles have to be taken into account.

To be concrete, let us consider the type IIB theory compacti¬ed on a

Calabi“Yau three-fold. As we have seen in the previous section, the moduli

space M2,1 (M ) can be described in terms of homogeneous special coordi-

nates X I . A conifold singularity appears when one of the coordinates, say

X1 = „¦, (9.140)

A1

vanishes. The period of „¦ over A1 goes to zero, and therefore A1 is called

404 String geometry

a vanishing cycle. At these points the metric on moduli space generically

becomes singular. Indeed, the subspace X 1 = 0 has complex codimension 1,

which is just a point if h2,1 = 1, and so it can be encircled by a closed loop.

Upon continuation around such a loop the basis of three-cycles comes back

to itself only up to an Sp(2; ) monodromy transformation. In general, the

monodromy is

X1 ’ X1 F 1 ’ F1 + X 1 .

and (9.141)

This implies that near the conifold singularity

11

F1 (X 1 ) = const + X log X 1 . (9.142)

2πi

In the simplest case one can assume that the other periods transform triv-

ially. This result implies that near the conifold singularity the K¨hler po-

a

tential in Eq. (9.117) takes the form

K2,1 ∼ log(|X 1 |2 log |X 1 |2 ). (9.143)

2

It follows that the metric G1¯ = ‚1 K 1 is singular at X 1 = 0. This is

1

‚X ‚X

a real singularity, and not merely a coordinate singularity, since the scalar

curvature diverges, as you are asked to verify in a homework problem.

The singularity of the moduli space occurs for the following reason. The

Calabi“Yau compacti¬cation is a description in terms of the low-energy e¬ec-

tive action in which the massive ¬elds have been integrated out. At the coni-

fold singularity certain massive states become massless, and an inconsistency

appears when such ¬elds have been integrated out. The particular states

that become massless at the singularity arise from D3-branes wrapping cer-

tain three-cycles, called special Lagrangian cycles, which are explained in

the next section. Near the conifold singularity these states becomes light,

and it is no longer consistent to exclude them from the low-energy e¬ective

action.