In fact, for exactly this reason, only the bosonic parts of fermionic super-

symmetry transformations were presented in Chapter 8. If the expectation

values for the fermions still vanish after performing a supersymmetry vari-

ation, then one obtains a solution of the bosonic equations of motion that

preserves supersymmetry for the type of backgrounds considered here. In

fact, as is shown in Exercise 9.4, a solution to the supersymmetry constraints

is always a solution to the equations of motion, while the converse is not

necessarily true. Here we are applying this result for theories with local

supersymmetry. This can be done if we impose the Bianchi identity sat-

is¬ed by the three-form H as an additional constraint. In order to obtain

unbroken N = 1 supersymmetry, Eq. (9.44) needs to hold for four linearly

376 String geometry

independent choices of µ forming a four-component Majorana spinor (or

equivalently a two-component Weyl spinor and its complex conjugate).

The supergravity approximation to heterotic string theory was described

in Section 8.1. In particular, the bosonic part of the ten-dimensional action

was presented. The full supergravity approximation also contains terms in-

volving fermionic ¬elds, which are incorporated in such a way that the theory

has N = 1 local supersymmetry (16 fermionic symmetries). As described in

Section 8.1, the bosonic terms of the supersymmetry transformations of the

fermionic ¬elds can be written in the form11

1

’ 4 HM µ,

δΨM = Mµ

= ’ 2 ‚ ¦µ + 1 Hµ,

1

(9.45)

δ» / 4

1

= ’ 2 Fµ,

δχ

in the string frame. In addition, the three-form ¬eld strength H satis¬es

±

[tr(R § R) ’ tr(F § F )] .

dH = (9.46)

4

The left-hand side is exact. Therefore, the cohomology classes of tr(R § R)

and tr(F § F ) have to be the same. In compacti¬cations with branes, this

condition can be modi¬ed by additional contributions.

Since the H-¬‚ux is assumed to vanish, the supersymmetry transformation

of the gravitino simpli¬es,

δΨM = M µ. (9.47)

For an unbroken supersymmetry this must vanish, and therefore there should

exist a nontrivial solution to the Killing spinor equation

Mµ = 0. (9.48)

This equation means that µ is a covariantly constant spinor.

N = 1 supersymmetry implies that one such spinor should exist. Since

the manifold M10 is a direct product, the covariantly constant spinor µ can

be decomposed into a product structure

µ(x, y) = ζ(x) — ·(y), (9.49)

or a sum of such terms. The properties of these spinors and the form of the

decomposition are discussed in the next section. In making such decompo-

sitions of anticommuting (Grassmann-odd) spinors, it is always understood

˜

˜

11 The notation introduced in Section 8.1 is simpli¬ed here according to H(3) ’ H and H3 ’ H.

Also, the fermionic variables that had tildes there are written here without tildes.

9.4 Calabi“Yau compacti¬cations of the heterotic string 377

that the space-time components ζ(x) are anticommuting (Grassmann odd),

while the internal components ·(y) are commuting (Grassmann even).

Properties of the external space

Let us consider the external components of Eq. (9.48) for which the index

takes value M = µ. The existence of a covariantly constant spinor ζ(x) on

M4 , satisfying

µζ = 0, (9.50)

implies that the curvature scalar R in Eq. (9.42) vanishes, and hence M4 is

Minkowski space-time. This follows from

1

Rµνρσ “ρσ ζ = 0

[ µ, ν] ζ = (9.51)

4

and the assumption of maximal symmetry (9.42). The details are shown in

Exercises 9.6 and 9.7. Then ζ is actually constant, not just covariantly con-

stant, and it is the in¬nitesimal transformation parameter of an unbroken

global supersymmetry in four dimensions. This is a nontrivial result inas-

much as unbroken supersymmetry does not necessarily imply a vanishing

cosmological constant by itself. AdS spaces can also be supersymmetrical,

a fact that plays a crucial role in Chapter 12. However, this result does

not solve the cosmological constant problem. The question that needs to be

answered in order to make contact with the real world is whether the cos-

mological constant can vanish, or at least be extremely small, when super-

symmetry is broken. The present result has nothing to say about this, since

it is derived by requiring unbroken supersymmetry. To summarize: super-

symmetry constrains the external space to be four-dimensional Minkowski

space.

Properties of the internal manifold

Let us now consider the restrictions coming from the internal components

M = m of Eq. (9.48). The existence of a spinor that satis¬es

m· = 0, (9.52)

and therefore is covariantly constant on M , leads to the integrability condi-

tion

1

[ m , n ] · = Rmnpq “pq · = 0. (9.53)

4

378 String geometry

This implies that the metric on the internal manifold M is Ricci-¬‚at (see

Exercises 9.6 and 9.7)

Rmn = 0. (9.54)

However, in contrast to the external space-time, where maximal symmetry

is assumed, it does not mean that M is ¬‚at, since the Riemann tensor can

still be nonzero.

Holonomy and unbroken supersymmetry

For an orientable six-dimensional spin manifold,12 the main case of inter-

est here, parallel transport of a spinor · around a closed curve generically

gives a rotation by a Spin(6) = SU (4) matrix. This is the generic holonomy

group.13 A real spinor on such a manifold has eight components, but the

eight components can be decomposed into two irreducible SU (4) represen-

tations

8 = 4 • ¯,

4 (9.55)

where the 4 and ¯ represent spinors of opposite chirality, which are complex

4

conjugates of one another. Thus, a spinor of de¬nite chirality has four

complex components.

A spinor that is covariantly constant remains unchanged after being par-

allel transported around a closed curve. The existence of such a spinor is

required if some supersymmetry is to remain unbroken; see Eq. (9.48). The

largest subgroup of SU (4) for which a spinor of de¬nite chirality can be

invariant is SU (3). The reason is that the 4 has an SU (3) decomposition

4 = 3 • 1, (9.56)

and the singlet is invariant under SU (3) transformations. Since the condi-

tion for N = 1 unbroken supersymmetry in four dimensions is equivalent to

the existence of a covariantly constant spinor on the internal six-dimensional

manifold, it follows that the manifold should have SU (3) holonomy.

The supersymmetry charge of the heterotic string in ten dimensions is a

Majorana“Weyl spinor with 16 real components, which form an irreducible

representation of Spin(9, 1). Group theoretically, this decomposes with re-

spect to an SL(2, ) — SU (4) subgroup as14

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