dz § d¯z

R = ’2 .

(1 + z z )2

¯

Using this result compute the Chern class and the Chern number (that is,

the integral of the Chern class over S 2 ) for the tangent bundle of S 2 .

PROBLEM 9.4

Show that the K¨hler potential for P n given in Eq. (9.31) undergoes a

a £

K¨hler transformation when one changes from one set of local coordinates

a

to another one. Construct the Fubini“Study metric.

PROBLEM 9.5

Show that K = ’ log( J) is the K¨hler potential for the K¨hler-structure

a a

modulus of a two-dimensional torus.

PROBLEM 9.6

Consider a two-dimensional torus characterized by two complex parame-

ters „ and ρ (that is, an angle θ is also allowed). Show that T-duality

interchanges the complex-structure and K¨hler parameters, as mentioned in

a

Section 9.5, and that the spectrum is invariant under this interchange.

PROBLEM 9.7

Verify the Lichnerowicz equation discussed in Section 9.5:

k

+ 2Rmp nq δgpq = 0.

k δgmn

Hint: use Rmn = 0 and the gauge condition in Eq. (9.91).

454 String geometry

PROBLEM 9.8

Use (9.92) to show that (9.93) and (9.96) are harmonic.

PROBLEM 9.9

Check the result for the K¨hler potential Eq. (9.117).

a

PROBLEM 9.10

Show that Eq. (9.132) agrees with Eq. (9.129).

PROBLEM 9.11

Compute the scalar curvature of the conifold metric in Eq. (9.143), and

show that it diverges at X 1 = 0. Thus, the conifold singularity is a real

singularity in the moduli space.

PROBLEM 9.12

Show that the operators in Eq. (9.146) are projection operators.

PROBLEM 9.13

Consider the E8 — E8 heterotic string compacti¬ed on a six-dimensional

orbifold

T2 — T2 — T2

,

4

where 4 acts on the complex coordinates (z1 , z2 , z3 ) of the three tori, as

(z1 , z2 , z3 ) ’ (iz1 , iz2 , ’z3 ). Identify the spin connection with the gauge

connection of one of the E8 s to ¬nd the spectrum of massless modes and

gauge symmetries in four dimensions.

PROBLEM 9.14

Verify that Eq. (9.172) vanishes if J is the K¨hler form.

a

PROBLEM 9.15

As mentioned in Section 9.11, compacti¬cation of the type IIB theory on

K3 leads to a chiral theory with N = 2 supersymmetry in six dimensions.

Since this theory is chiral, it potentially contains gravitational anomalies.

Using the explicit form of the anomaly characteristic classes discussed in

Chapter 5, show that anomaly cancellation requires that the massless sector

contain 21 matter multiplets (called tensor multiplets) in addition to the

supergravity multiplet.

Homework Problems 455

PROBLEM 9.16

Consider the second term in the action (9.189) restricted to two dimensions

described by a complex variable z. Form the equation of motion of the ¬eld

„ and show that it is satis¬ed by any holomorphic function „ (z).

PROBLEM 9.17

Consider a Calabi“Yau three-fold given as an elliptically ¬bered manifold

over P 1 — P 1

£ £

y 2 = x3 + f (z1 , z2 )x + g(z1 , z2 ),

P 1 s and f, g are polynomials in f in (z1 , z2 ).

where z1 , z2 represent the two £

(i) What is the degree of the polynomials f and g? Hint: write down

the holomorphic three-form and insist that it has no zeros or poles

at in¬nity.

(ii) Compute the number of independent complex structure deformations

of this Calabi“Yau. What do you obtain for the Hodge number h2,1 ?

(iii) How many K¨hler deformations do you ¬nd, and what does this imply

a

for h1,1 ?

PROBLEM 9.18

Verify properties (i)“(iii) for the G2 orbifold T 7 /“ de¬ned in Section 9.12.

Show that the blow-up of each ¬xed point gives 12 copies of T 3 .

PROBLEM 9.19

Verify that the solution to the constraint equation for a supersymmetric

three-cycle in a G2 manifold Eq. (9.218) is given by Eq. (9.219). Repeat the

calculation for the supersymmetric four-cycle.

PROBLEM 9.20

Show that the direct product of the multi-center Taub“NUT metric dis-

cussed in Section 8.3 with ¬‚at 3 corresponds to a 7-manifold with G2

¡

holonomy.

PROBLEM 9.21

Find the conditions, analogous to those in Exercise 9.16, de¬ning the Spin(7)

action that leaves invariant the four-form (9.224). Verify that there are the

correct number of conditions.

10

Flux compacti¬cations