16 = (2, 4) • (¯ ¯

2, 4). (9.57)

12 A spin manifold is a manifold on which spinors can be de¬ned, that is, it admits spinors.

13 More information about holonomy and spinors is given in the appendix.

14 The other 16-dimensional spinor, which is not a supersymmetry of the heterotic string, then

has the decomposition 16 = (2, ¯ + (¯ 4).

4) 2,

9.4 Calabi“Yau compacti¬cations of the heterotic string 379

Here SL(2, ) is the four-dimensional Lorentz group, so 2 and ¯ correspond

2

£

to positive- and negative-chirality Weyl spinors. On a manifold of SU (3)

holonomy only the singlet pieces of the 4 and the ¯ in Eq. (9.56) lead to

4

covariantly constant spinors. Denoting them by ¬elds ·± (y), the covariantly

constant spinor µ can be decomposed into a sum of two terms

µ(x, y) = ζ+ — ·+ (y) + ζ’ — ·’ (y), (9.58)

where ζ± are two-component constant Weyl spinors on M4 . Note that

— —

·’ = ·+ and ζ ’ = ζ+ , (9.59)

since µ is assumed to be in a Majorana basis.

A representation of the gamma matrices that is convenient for this 10 =

4 + 6 split is

“µ = γ µ — 1 “ m = γ5 — γm ,

and (9.60)

where γµ and γm are the gamma matrices of M4 and M , respectively, and

γ5 is the usual four-dimensional chirality operator

γ5 = ’iγ0 γ1 γ2 γ3 , (9.61)

2

which satis¬es γ5 = 1 and anticommutes with the other four γµ s.

Internal Dirac matrices

The 8 — 8 Dirac matrices on the internal space M can be chosen to be an-

tisymmetric. A possible choice of the six antisymmetric matrices satisfying

{γi , γj } = 2δij is

σ2 — 1 — σ1,3 σ1,3 — σ2 — 1 1 — σ1,3 — σ2 . (9.62)

One can then de¬ne a seventh antisymmetric matrix that anticommutes

with all of these six as γ7 = iγ1 . . . γ6 or

γ7 = σ2 — σ2 — σ2 . (9.63)

The chirality projection operators are

P± = (1 ± γ7 )/2. (9.64)

In terms of the matrices de¬ned above, one de¬nes matrices γm = ei γi in a

m

real basis or γa and γa in a complex basis.

¯

380 String geometry

K¨hler form and complex structure

a

Now let us consider possible fermion bilinears constructed from ·+ and ·’ .

Since these spinors are covariantly constant they can be normalized accord-

ing to

† †

·+ ·+ = ·’ ·’ = 1. (9.65)

Next, de¬ne the tensor

† †

Jm n = i·+ γm n ·+ = ’i·’ γm n ·’ , (9.66)

which by using the Fierz transformation formula (given in the appendix of

Chapter 10) satis¬es

Jm n Jn p = ’δm p . (9.67)

As a result, the manifold is almost complex, and J is the almost complex

structure.

Since the spinors ·± and the metric are covariantly constant, the almost

complex structure is also covariantly constant, that is

p

m Jn = 0. (9.68)

This implies that the almost complex structure satis¬es the condition that

it is a complex structure, since it satis¬es

N p mn = 0, (9.69)

where N p mn is the Nijenhuis tensor (see the appendix and Exercise A.4).

So one can introduce local complex coordinates z a and z a in terms of which

¯

¯ ¯ ¯

Ja b = iδa b , Ja b = ’iδa b Ja b = Ja b = 0.

and (9.70)

¯ ¯ ¯

Note that

gmn = Jm k Jn l gkn , (9.71)

which together with Eq. (9.70) implies that the metric is hermitian with

respect to the almost complex structure. Moreover, Eq. (9.71) implies that

the quantity

Jmn = Jm k gkn , (9.72)

is antisymmetric and as a result de¬nes a two-form

1

J = Jmn dxm § dxn . (9.73)

2

The components of J are related to the metric according to

Ja¯ = iga¯. (9.74)

b b

9.4 Calabi“Yau compacti¬cations of the heterotic string 381

One important property of J is that it is closed, since

¯

dJ = ‚J + ‚J = i‚a gb¯dz a § dz b § dz c + i‚a gb¯dz a § dz b § dz c = 0. (9.75)

¯ ¯ ¯

¯c

c

To see this, note that the metric is covariantly constant and take into account

that we are using a torsion-free connection. As a result, the background is

K¨hler, and J is the K¨hler form.

a a

Holomorphic three-form

Let us now consider possible fermion bilinears, starting with ones that are

bilinear in ·’ . Remembering that · is Grassmann even, one can see that

T T

the bilinears ·’ γa ·’ and ·’ γab ·’ vanish by symmetry. Also, the bilinear

T

·’ ·’ vanishes by chirality. The only nonzero possibility, consistent with

both chirality and symmetry, is

T

„¦abc = ·’ γabc ·’ . (9.76)

This can be used to de¬ne a nowhere-vanishing (3, 0)-form

1

„¦ = „¦abc dz a § dz b § dz c . (9.77)

6

• Let us now show that „¦ is closed. Since · and the metric are covariantly

constant, it satis¬es d „¦abc = 0. The connection terms vanish for a

¯

¯

K¨hler manifold, and therefore one deduces that ‚„¦ = 0. It is obvious

a

that ‚„¦ = 0, since there are only three holomorphic dimensions. Thus,

¯ ¯

„¦ is closed, d„¦ = (‚ + ‚)„¦ = 0. The fact that ‚„¦ = 0 implies that the

coe¬cients „¦abc are holomorphic.

• On the other hand, „¦ is not exact. This can be understood as a conse-

quence of the fact „¦ § „¦ is proportional to the volume form, which has a

nonzero integral over M (see Exercise 9.8). Therefore, „¦ § „¦ is not exact,