p,q

J § · · · § J §Fprimitive = 0. (10.45)

5’p’q times

In the case of a Calabi“Yau four-fold, it is a useful fact that the Hodge

operator has the eigenvalue (’1)p on the primitive (p, 4 ’ p) cohomology

(see Exercise 10.4). This is of relevance in Section 10.3.

Tadpole-cancellation condition

We have learned that unbroken supersymmetry requires that the internal

¬‚ux components Fmnpq (y) are given by a primitive (2, 2)-form, Eq. (10.37),

and the external ¬‚ux components fm (y) are determined in terms of the warp

factor by Eq. (10.25). The equation that determines the warp factor follows

from the equation of motion of the four-form ¬eld strength. Using self-

duality, it would make the energy density |F4 |2 proportional to the Laplacian

of log ∆, which gives zero when integrated over the internal manifold. If

this were the whole story, one would be forced to conclude that the ¬‚ux

vanishes, so that one is left with ordinary Calabi“Yau compacti¬cation of

the type discussed in Chapter 9. However, quantum gravity corrections to

11-dimensional supergravity must be taken into account, and then nonzero

¬‚ux is required for consistency. Let us explain how this works.

The action for 11-dimensional supergravity receives quantum corrections,

denoted δS, coming from an eight-form X8 that is quartic in the Riemann

tensor

δS = ’TM2 A3 § X8 , (10.46)

M

where

1 1 1

trR4 ’ (trR2 )2 .

X8 = (10.47)

4 192

(2π) 768

This correction term was ¬rst derived by considering a one-loop scattering

amplitude in type IIA string theory involving four gravitons Gµν and one

two-form tensor ¬eld Bµν . In the type IIA theory the correction takes a

similar form as in M-theory, with the three-form A3 replaced by the NS“NS

470 Flux compacti¬cations

Bµν

Gµν Gµν

Gµν Gµν

Fig. 10.2. The higher-order interaction in Eq. (10.46) can be determined by calcu-

lating a one-loop diagram in type IIA string theory, involving four gravitons and

one NS“NS two-form ¬eld, whose result can then be lifted to M-theory.

two-form B2 . Since the result does not depend on the dilaton, it can be

lifted to M-theory.

The δS term is also required for the cancellation of anomalies on bound-

aries of the 11-dimensional space-time, such as those that are present in the

strongly coupled E8 — E8 theory, which is also know as heterotic M-theory.

This was discussed in Chapter 5. Together with the original A3 § F4 § F4

term it gives the complete Chern“Simons part of the theory, so it is not just

the leading term in some expansion. In fact, it is the only higher-derivative

term that can contribute to the problem at hand in the large-volume limit.

The ¬eld strength satis¬es the Bianchi identity

dF = 0. (10.48)

Furthermore, the δS term contributes to the 11-dimensional equation of

motion of the four-form ¬eld strength. Combining Eqs (10.3) and (10.46),

the result is

1

d F4 = ’ F4 § F4 ’ 2κ2 TM2 X8 . (10.49)

11

2

Using Eq. (10.25) this gives an equation for the warp factor

1 4

F § F + κ2 TM2 X8 .

d d log ∆ = (10.50)

8

3 11

3

Integrating this expression over the internal manifold leads to the tadpole-

cancellation condition, as follows. The integral of the left-hand side vanishes,

since it is exact, and (for the time being) it is assumed that no explicit delta

function singularities are present. In other words, it is assumed that no

10.1 Flux compacti¬cations and Calabi“Yau four-folds 471

space-¬lling M2-branes are present. To obtain the result of the X8 integra-

tion, it is convenient to express the anomaly characteristic class X8 in terms

of the ¬rst and second Pontryagin forms of the internal manifold

1 1 1 1 1

’ trR2 ’ trR4 + (trR2 )2 .

P1 = and P2 =

(2π)2 (2π)4

2 4 8

(10.51)

This gives

1 2

(P1 ’ 4P2 ).

X8 = (10.52)

192

For complex manifolds the Pontryagin classes are related to the Chern classes

by

P1 = c2 ’ 2c2 P2 = c2 ’ 2c1 c3 + 2c4 .

and (10.53)

1 2

Thus

14

(c1 ’ 4c2 c2 + 8c1 c3 ’ 8c4 ).

X8 = (10.54)

1

192

Calabi“Yau manifolds have vanishing ¬rst Chern class, so the only nontrivial

contribution comes from the fourth Chern class. This in turn is related to

the Euler characteristic χ, so

1 χ

X8 = ’ c4 = ’ . (10.55)

24 24

M M

Thus, Eq. (10.50) leads to the tadpole-cancellation condition

1 χ

F §F = . (10.56)

4κ2 TM2 24

M

11

Fluxes without sources

Using the last equation, it is possible to estimate the order of magnitude of

the internal ¬‚ux components. Expressing κ2 and the M2-brane tension in

11

terms of the 11-dimensional Planck length p yields

4κ2 TM2 = 2(2π p )6 . (10.57)

11

As a result, the order of magnitude of the ¬‚uxes is

3

p

√

Fmnpq O , (10.58)

v

where v is the volume of the Calabi“Yau four-fold. Comparing this result

with Eq. (10.50) shows that the warp factor satis¬es log ∆ ∼ 6 /v 3/4 , or if

p

472 Flux compacti¬cations

Fig. 10.3. According to Maxwell™s theory, an electric current in a solenoid gener-