three-form ¬eld strength satisfying the Bianchi identity

±

[tr(R § R) ’ tr(F § F )] .

dH = (10.200)

4

Poincar´ invariance of the external space-time requires some components to

e

vanish

Hµνρ = Hµνp = Hµnp = 0 and Fµν = Fµn = 0. (10.201)

The nonvanishing ¬elds can depend on the coordinates of the internal man-

ifold only.

One class of consistent solutions of Eq. (10.199) has a vanishing three-

form and a constant dilaton. These solutions are the conventional Calabi“

Yau compacti¬cations described in Chapter 9. Now let us consider solutions

with

Hmnp = 0 and ‚m ¦ = 0. (10.202)

The supersymmetry transformation of the gravitino can be rewritten con-

veniently in terms of a covariant derivative with torsion. To understand

this, recall that

1 AB

M µ = ‚M µ + ωM AB “ µ. (10.203)

4

This result is written for tangent-space indices A, B and base-space indices

512 Flux compacti¬cations

M, N, P of the ten-dimensional space-time. In the ten-dimensional theory,

the supersymmetry variation of the gravitino can be written as

1

’ HM AB “AB )µ,

Mµ =( (10.204)

M

8

where M is the torsion-free connection, since this combination appears in

the supersymmetry transformation of the gravitino ¬eld. Here the derivative

M is de¬ned with respect to a connection with torsion. The three-form

¬‚ux shifts the spin connection according to

1

ω A = ω A B ’ HM AB dxM .

˜B (10.205)

2

Using Eq. (10.196) one sees that the three-form ¬‚ux represents an additional

contribution to the torsion one-form

1

˜

T A = T A + H AM N dxM § dxN . (10.206)

2

˜

We will choose T A = 0 so that T A is given by the three-form ¬‚ux.

The supersymmetry parameter and gamma matrices decompose into in-

ternal and external pieces

µ(x, y) = ζ+ (x) — ·+ (y) + ζ’ (x) — ·’ (y), (10.207)

where ζ± are Weyl spinors on M4 and ·± are Weyl spinors on M that satisfy

ζ’ = ζ+ and · ’ = ·+ . (10.208)

The gamma matrices split as

“µ = γ µ — 1 “ m = γ5 — γm .

and (10.209)

The conditions (10.199) have several components. From the external com-

ponent of the gravitino transformation one obtains

δψµ = µ ζ+ = 0, (10.210)

which implies that R = 0. Here R is the scalar curvature of the external

space, which by maximal symmetry is a constant. Even though solutions

with a negative cosmological constant, that is, AdS compacti¬cations, can

be compatible with supersymmetry, only Minkowski-space compacti¬cations

are possible in the present set-up. This part of the analysis is una¬ected by

the H ¬‚ux and is the same as in Chapter 9.

The internal component of the gravitino supersymmetry condition re-

quires the existence of H-covariant spinors ·± with

1

’ Hmnp γ np )·± = 0

m ·± =( (10.211)

m

8

10.4 Fluxes, torsion and heterotic strings 513

for a supersymmetric solution. Eq. (10.211) implies that the scalar quantity

† †

·+ ·+ is a constant, and so once again it can be normalized so that ·+ ·+ = 1.

In terms of this spinor, one can de¬ne the tensor

† †

Jm n = i·+ γm n ·+ = ’i·’ γm n ·’ . (10.212)

Moreover, using Fierz transformations, it is possible to show that

Jm n Jn p = ’δm p . (10.213)

Thus, the background geometry is almost complex, and J is an almost com-

plex structure. This implies that the metric has the property

gmn = Jm k Jn l gkl , (10.214)

and that the quantity

Jmn = Jm k gkn (10.215)

is antisymmetric. As a result, it can be used to de¬ne a two-form

1

J = Jmn dxm § dxn , (10.216)

2

which is sometimes called the fundamental form. It should not be confused

with the K¨hler form.

a

The tensor Jn p is covariantly constant with respect to the connection

with torsion,

1 1

p p

’ Hsm p Jn s ’ H s mn Js p = 0.

m Jn = m Jn (10.217)

2 2

Again, it is understood that uses the Christo¬el connection. Using this

result, it follows that the Nijenhuis tensor, de¬ned in the appendix of chapter

9, vanishes (see Exercise 10.10). As a result, J is a complex structure, and

the manifold is complex. 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 (10.218)

¯ ¯ ¯

The metric is hermitian with respect to J, since combining Eqs (10.214)

and (10.218) implies that the metric has only mixed components ga¯. The

b

fundamental form J is then related to the metric by

Ja¯ = iga¯. (10.219)

b b

Inserting the relation between the fundamental form and the metric into

Eq. (10.217) gives

¯

H = i(‚ ’ ‚)J. (10.220)

514 Flux compacti¬cations

By de¬nition dJ = 0 for a K¨hler manifold. As a result, backgrounds with

a

nonvanishing H are non-K¨hler.

a

Let us consider the implications of the dilatino equation in Eq. (10.199).

Evaluating it in complex coordinates and using γ a ·+ = γ a ·’ = 0, one ¬nds

¯

that

1

‚a ¦ = ’ Hab¯g b¯ c

(10.221)

c

2

and the complex-conjugate relation. This relation implies the existence of

a unique nowhere-vanishing holomorphic three-form „¦. This three-form is

given by

„¦ = e’2¦ ·’ γabc ·’ dz a § dz b § dz c .

T

(10.222)