4

¯

2

V = eκ4 K Gab Da W D¯W ’ 3κ2 |W |2 .

4

b

For small κ4 ,

¯

V = Gab ‚a W ‚¯W + O(κ2 ).

4

b

As expected, one ¬nds the global supersymmetry formula plus corrections

proportional to Newton™s constant. 2

10.4 Fluxes, torsion and heterotic strings

This section explores compacti¬cations of the weakly coupled heterotic string

in the presence of a nonzero three-form ¬eld H.20 A nonvanishing H ¬‚ux has

two implications for the background geometry. First, the background geom-

etry becomes a warped product, like that discussed in the previous sections.

The second consequence of nonvanishing H is that its contributions to the

various equations can be given a geometric interpretation as torsion of the

internal manifold. If the gauge ¬elds are not excited, heterotic supergravity

is a truncation of either type II supergravity theory. Therefore, some of the

analysis in this section applies to those cases and vice versa.

Warped geometry

As in the previous sections, when H ¬‚ux is included, the space-time is no

longer a direct-product space of the form M10 = M4 — M . (For simplicity, in

the following we assume that the external space-time is four-dimensional.)

Analysis of the heterotic supersymmetry transformation laws will show that

a warp factor e2D(y) must be included in the metric in order to provide

a consistent solution. In the Einstein frame, let us write the background

metric for the warped compacti¬cation in the form

ds2 = e2D(y) (gµν (x)dxµ dxν + gmn (y)dy m dy n ) (10.194)

6D

4D

As before, x denotes the coordinates of the external space, y the internal

coordinates, the indices µ, ν label the coordinates of the external space and

m, n label the coordinates of the internal space.

The function D(y) depends only on the internal coordinates. It will be

shown that supersymmetry can be satis¬ed when there is nonzero H ¬‚ux

provided that

D(y) = ¦(y), (10.195)

20 The index on H3 is suppressed.

10.4 Fluxes, torsion and heterotic strings 509

where ¦ is the dilaton ¬eld. In the case without H ¬‚ux, the dilaton is

constant, so the geometry is a direct product in the Einstein frame. When

‚m ¦ = 0, it becomes a warped product. This warp factor is exactly the

one that converts the Einstein frame to the string frame. So the geometry

actually is a direct product with respect to the string-frame metric even

when there is nonzero H ¬‚ux. Since the internal space is compact and

the dilaton ¬eld ¦(y) is nonsingular (in the absence of NS5-branes), the

dilaton is bounded. Therefore, shifting by a constant can make the coupling

arbitrarily weak, so that perturbation theory is justi¬ed.

Torsion

The use of a connection with torsion is natural, since the three-form H is

part of the supergravity multiplet. The torsion two-form T ± is de¬ned in

terms of the frame and spin-connection one-forms by21

T ± = de± + ω ± γ § eγ , (10.196)

which can be written in terms of connection coe¬cients “r according to

mn

T ± = “r e±r dxm § dxn , (10.197)

mn

Since torsion is a tensor, it has intrinsic geometric meaning. A connection

is torsion-free if it is symmetric in its lower indices.

In de¬ning the geometry one is free to choose what torsion tensor to

include in the connection as one pleases. A connection, which is not a

tensor, can always be rede¬ned by a tensor, and in this way the torsion

is changed. In particular, one can choose to use the Christo¬el connection,

which has no torsion. The use of a connection with torsion has the geometric

consequences described below. However, you are never required to use such

a connection. In ¬‚ux compacti¬cations of the heterotic string there is a

natural choice, since by incorporating the three-form ¬‚ux in the connection,

in the way described below, one is able to de¬ne a covariantly constant

spinor.

Geometrically, torsion measures the failure of in¬nitesimal parallelograms,

de¬ned by the parallel transport of a pair of vectors, to close. Parallel

transport for the case in which the torsion vanishes is illustrated in Fig. 10.9

and a case in which it does not vanish is illustrated in Fig. 10.10.

As a simple example consider the Euclidean metric ds2 = dx2 + dy 2 on

the two-dimensional plane 2 . If parallel transport is de¬ned in the usual

¡

21 There are other meanings of the word torsion that should not be confused with the one intro-

duced here.

510 Flux compacti¬cations

Fig. 10.9. The vectors V and W are parallel transported to V and W using a

torsion-free connection. The resulting parallelogram closes.

sense of elementary geometry, the Christo¬el connection vanishes in carte-

sian coordinates. However, any connection compatible with the ¬‚at metric

is allowed. This means one can choose any connection that respects angles

and distances or equivalently which leaves the metric covariantly constant.

In the present case this means that one can choose any spin connection one-

form taking values in the Lie algebra of the two-dimensional rotation group,

so

ω±β = hµ±β , (10.198)

where h can be any one-form. Parallel transport of a vector now leads to

a (would-be) parallelogram that fails to close, as indicated in Fig. 10.10.

Mathematically, this means that V W ’ W V = [V, W ].

Fig. 10.10. The vectors V and W are parallel transported to V and W using a

connection that has torsion. The resulting parallelogram fails to close.

10.4 Fluxes, torsion and heterotic strings 511

Conditions for unbroken supersymmetry

The goal of this subsection is to derive the supersymmetry constraints

for compacti¬cations of the heterotic string to maximally symmetric four-

dimensional space-time allowing for nonzero H ¬‚ux. As was explained in

Section 9.4, a supersymmetric con¬guration is one for which a spinor µ exists

that satis¬es

’ 1 HM µ = 0,

δΨM = Mµ 4

= ’ 1 ‚ ¦µ + 1 Hµ = 0, (10.199)

δ» 2/ 4

1

= ’ 2 Fµ = 0,

δχ

in the notation of Section 8.1. A very convenient fact is that these formulas

are written in the string frame. Therefore, the warp factor is already taken

into account, and they can be analyzed using a space-time that is a direct

product of external and internal spaces, just as in Chapter 9. As before, ¦