01 10

σ1 = , σ2 = and σ3 = . (10.10)

0 ’1

10 i0

Moreover, γm are the 16 — 16 gamma matrices of M and γ9 is the eight-

2

dimensional chirality operator that satis¬es γ9 = 1 and anticommutes with

the other eight γm ™s. It is both possible and convenient to choose a repre-

sentation in which the γm and γ9 are real symmetric matrices. In a tangent-

space basis one can choose the eight 16 — 16 Dirac matrices on the internal

space M to be

σ2 — σ2 — 1 — σ1,3 , σ2 — σ1,3 — σ2 — 1,

(10.11)

σ2 — 1 — σ1,3 — σ2 , σ1,3 — 1 — 1 — 1 .

Then one can de¬ne a ninth symmetric matrix that anticommutes with all

of these eight as

γ9 = γ1 . . . γ8 = σ2 — σ2 — σ2 — σ2 , (10.12)

from which the chirality projection operators

P± = (1 ± γ9 )/2 (10.13)

are constructed.

Decomposition of the spinor

The 11-dimensional Majorana spinor µ decomposes into a sum of two terms

of the form

µ(x, y) = ζ(x) — ·(y) + ζ — (x) — · — (y), (10.14)

where ζ is a two-component anticommuting spinor in three dimensions, while

· is a commuting 16-component spinor in eight dimensions. A theory with

N = 2 supersymmetry in three dimensions has two linearly independent

Majorana“Weyl spinors ·1 , ·2 on M , which have been combined into a

complex spinor in Eq. (10.14). In general, these two spinors do not need to

have the same chirality. However, for Calabi“Yau four-folds the spinor on

the internal manifold is complex and Weyl

(γ9 ’ 1)· = 0.

· = ·1 + i·2 with (10.15)

This sign choice for the eigenvalue of γ9 , which is just a convention, is

called positive chirality. The two real spinors ·1 , ·2 correspond to the two

singlets in the decomposition of the 8c representation of Spin(8) to SU (4),

the holonomy group of a Calabi“Yau four-fold,

8c ’ 6 • 1 • 1. (10.16)

464 Flux compacti¬cations

Fig. 10.1. This ¬gure illustrates the Poincar´“Hopf index theorem. A continuous

e

vector ¬eld on a sphere must have at least two zeros, which in this case are located

at the north and south poles, since the Euler characteristic is 2. On the other hand,

a vector ¬eld on a torus can be nonvanishing everywhere since χ = 0.

Nonchiral spinors

If ·1 and ·2 have opposite chirality the complex spinor · = ·1 + i·2 is

nonchiral. The two spinors of opposite chirality de¬ne a vector ¬eld on the

internal manifold

†

va = ·1 γa ·2 . (10.17)

Requiring this vector to be nonvanishing leads to an interesting class of solu-

tions. Indeed, the Poincar´“Hopf index theorem of algebraic topology states

e

that the number of zeros of a continuous vector ¬eld must be at least equal to

the absolute value of the Euler characteristic χ of the background geometry.

As a result, a nowhere vanishing vector ¬eld only exists for manifolds with

χ = 0. An example of this theorem is illustrated in Fig. 10.1. Flux back-

grounds representing M5-branes ¬lling the three-dimensional space-time and

wrapping supersymmetric three-cycles on the internal space are examples of

this type of geometries. Moreover, once the spinor is nonchiral, compacti-

¬cations to AdS3 spaces become possible. Compacti¬cations to AdS space

are considered in Chapter 12, so the discussion in this chapter is restricted

to spinors of positive chirality. It will turn out that AdS3 is not a solution

in this case.

Solving the supersymmetry constraints

The constraints that follow from Eq. (10.4) are in¬‚uenced by the warp-

factor dependence of the metric. As was pointed out in Chapter 8, there

is a relation between the covariant derivatives of a spinor with respect to

a pair of metrics that di¬er by a conformal transformation. In particular,

in the present case, the internal and external components of the metric are

rescaled with a di¬erent power of the warp factor and the vielbeins are given

10.1 Flux compacti¬cations and Calabi“Yau four-folds 465

by Eµ = ∆’1/2 e± and Em = ∆1/4 e± . This leads to

± ±

µ m

1

’ 4 ∆’7/4 (γµ — γ9 γ m ) ‚m ∆µ,

’

µµ µµ

(10.18)

1 ’1 n ) µ.

’ — γm

mµ mµ + 8 ∆ ‚n ∆ (1

For compacti¬cations to maximally symmetric three-dimensional space-time,

Poincar´ invariance restricts the possible nonvanishing components of F4 to

e

Fmnpq (y) and Fµνρm = µµνρ fm (y), (10.19)

where µµνρ is the completely antisymmetric Levi“Civita tensor of M3 . Once

the gamma matrices are decomposed as in Eq (10.8), the nonvanishing ¬‚ux

components take the form

F(4) = ∆’1 (1 — F) + ∆5/4 (1 — γ9 f ),

(4)

= ∆3/4 (γµ — f ), (10.20)

Fµ

(4)

= ’∆3/2 fm (y) (1 — γ9 ) + ∆’3/4 (1 — Fm ),

Fm

where F, Fm and f are de¬ned like their ten-dimensional counterparts, but

the tensor ¬elds are now contracted with eight-dimensional Dirac matrices

1 1

Fmnpq γ mnpq , Fmnpq γ npq f = γ m fm .

F= Fm = and

24 6

(10.21)

The gravitino supersymmetry transformation Eq. (10.4) has external and

internal components depending on the value of the index M .

External component of the gravitino equation

Let us analyze the external component δΨµ = 0 ¬rst. In three-dimensional

Minkowski space-time a covariantly constant spinor, satisfying

µζ = 0, (10.22)

can be found. As a result, the δΨµ = 0 equation becomes

1

‚ ∆’3/2 · + f · + ∆’9/4 F· = 0,

/ (10.23)

2

which by projecting on the two chiralities using the projection operators P±

leads to

F· = 0, (10.24)

and

fm (y) = ’‚m ∆’3/2 . (10.25)

466 Flux compacti¬cations

The last of these equations provides a relation between the external ¬‚ux

component and the warp factor.

Internal component of the gravitino equation

After decomposing the gamma matrices and ¬‚uxes using Eqs (10.8) and

(10.20), respectively, the internal component of the supersymmetry trans-

formation δΨm = 0 takes the form

1 1

+ ∆’1 ‚m ∆ · ’ ∆’3/4 Fm · = 0.

m· (10.26)

4 4

This equation leads to