where

ξ = ∆1/4 ·. (10.28)

Since ξ is a nonvanishing covariantly constant complex spinor with de¬-

nite chirality, the second expression in Eq. (10.27) states that the internal

manifold M is conformal to a Calabi“Yau four-fold.

Conditions on the ¬‚uxes

The mathematical properties of Calabi“Yau four-folds are similar to those

of three-folds, as discussed in Chapter 9. The covariantly constant spinor

appearing in Eq. (10.27) can be used to de¬ne the almost complex structure

of the internal manifold

Ja b = ’iξ † γa b ξ, (10.29)

which has the same properties as for the Calabi“Yau three-fold case, as you

are asked to verify in Problems 10.2, 10.3. Recall that the Dirac algebra for

a K¨hler manifold

a

¯ ¯ ¯

{γ a , γ b } = 0, {γ a , γ b } = 0,

¯

{γ a , γ b } = 2g ab , (10.30)

can be interpreted as an algebra of raising and lowering operators. This

is useful for evaluating the solution of Eq. (10.27). To see this rewrite

Eq. (10.29) as

Ja¯ = iga¯ = ’iξ † γa¯ξ = ’iξ † (γa γ¯ ’ ga¯)ξ. (10.31)

b b b b b

This implies

0 = ξ † γa γ¯ξ = (γa ξ)† (γ¯ξ). (10.32)

¯

b b

10.1 Flux compacti¬cations and Calabi“Yau four-folds 467

By setting a = ¯ the previous equation implies that ξ is a highest-weight

¯ b

state that is annihilated by γa ,

¯

γa ξ = γ a ξ = 0, (10.33)

¯

for all indices on the Calabi“Yau four-fold. Using this result, Exercise 10.3

shows that the ¬rst condition in Eq. (10.27) implies the vanishing of the

following ¬‚ux components:

F 4,0 = F 0,4 = F 1,3 = F 3,1 = 0, (10.34)

and that the only nonvanishing component is F ∈ H (2,2) , which must satisfy

the primitivity condition

¯

Fa¯ d g cd = 0. (10.35)

bc ¯

Since ξ has a de¬nite chirality, F is self-dual on the Calabi“Yau four-fold,

as is explained in Exercise 10.2. The self-duality implies that Eq. (10.35)

can be written in the following form:5

F 2,2 § J = 0. (10.36)

As a result of the above analysis, supersymmetry is unbroken if F lies in the

primitive (2, 2) cohomology, that is,

(2,2)

F ∈ Hprimitive (M ). (10.37)

In the following the general de¬nition of primitive forms is given and their

relevance in building the complete de Rham cohomology is discussed.

Primitive cohomology

Any harmonic (p, q)-form of a K¨hler manifold can be expressed entirely in

a

terms of primitive forms, a representation known as the Lefschetz decompo-

sition. This construction closely resembles the Fock-space construction of

angular momentum states |j, m using raising and lowering operators J± .

Chapter 9 discussed the Hodge decomposition of the de Rham cohomology

of a compact K¨hler manifold. The Lefschetz decomposition is compatible

a

with the Hodge decomposition, as is shown below.

On a compact K¨hler manifold M of complex dimension d (and real di-

a

mension 2d) with K¨hler form J, one can de¬ne an SU (2) action on harmonic

a

5 Problem 10.5 asks you to verify that the primitivity condition is modi¬ed when the complex

spinor on the internal manifold is nonchiral.

468 Flux compacti¬cations

forms (and hence the de Rham cohomology) by

: G ’ 1 (d ’ n)G,

J3 2

J’ : G ’ J § G, (10.38)

1 p1 p2 G p3 § · · · § dxpn ,

J+ : G ’ 2(n’2)! J p1 p2 ...pn dx

where G is a harmonic n-form. Notice that J’ lowers the J3 eigenvalue by

one and as a result acts as a lowering operator while J+ increases the value

of J3 by one and thus acts as a raising operator. Problem 10.6 asks you to

verify that these operators satisfy an SU (2) algebra.

As in the case of the angular momentum algebra, the space of harmonic

forms can be classi¬ed according to their J3 and J 2 eigenvalues, with basis

states denoted by

|j, m m = ’j, ’j + 1, . . . , j ’ 1, j.

with (10.39)

Primitive forms are de¬ned as highest-weight states that are annihilated by

J+ , that is,

J+ Gprimitive = 0, (10.40)

and may be denoted by |j, j . All other states (or harmonic forms) can then

be obtained by acting with lowering operators J’ on primitive forms. A

primitive n-form also satis¬es

d’n

2j+1

J’ Gprimitive = 0 where j= . (10.41)

2

Notice that the primitive forms in the middle-dimensional cohomology (that

is, with n = d) correspond to j = 0. So they are singlets |0, 0 that are

annihilated by both the raising and lowering operators

J+ G = 0 or J’ G = 0. (10.42)

These two formulas correspond to Eqs (10.35) and (10.36), respectively.

This discussion makes it clear that primitive forms can be used to construct

any harmonic form and hence representatives of every de Rham cohomology

class. Schematically, the Lefschetz decomposition is6

k n’2k

H n (M ) = J’ Hprimitive (M ). (10.43)

k

6 It would be more precise to write Harmn (M ) instead of H n (M ).

10.1 Flux compacti¬cations and Calabi“Yau four-folds 469

The Lefschetz decomposition is compatible with the Hodge decomposition,

so that we can also write

(p’k,q’k)

H (p,q) (M ) = k

J’ Hprimitive (M ). (10.44)

k

In this way any harmonic (p, q)-form can be written in terms of primitive

forms. If M is a Calabi“Yau four-fold, it follows from Eq. (10.41) that