String Theory and M-Theory

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 field
τ and show that it is satisfied by any holomorphic function τ(z).
PROBLEM 9.17
Consider a Calabi–Yau three-fold given as an elliptically fibered manifold
over £ P 1 × £ P 1
y 2 = x 3 + f(z1, z2)x + g(z1, z2),
where z1, z2 represent the two £ P 1 s and f, g are polynomials in f in (z1, z2).
(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 infinity.
(ii) Compute the number of independent complex structure deformations
of this Calabi–Yau. What do you obtain for the Hodge number h 2,1 ?
(iii) How many Kähler deformations do you find, and what does this imply
for h 1,1 ?
PROBLEM 9.18
Verify properties (i)–(iii) for the G2 orbifold T 7 /Γ defined in Section 9.12.
Show that the blow-up of each fixed 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 discussed
in Section 8.3 with flat ¡ 3 corresponds to a 7-manifold with G2
holonomy.
PROBLEM 9.21
Find the conditions, analogous to those in Exercise 9.16, defining the Spin(7)
action that leaves invariant the four-form (9.224). Verify that there are the
correct number of conditions.