String Theory and M-Theory

9.8 Nonperturbative effects in Calabi–Yau compactifications 409
in which a Euclidean M2-brane wraps the circle and a holomorphic two-cycle
of the Calabi–Yau.
EXERCISES
EXERCISE 9.14
Show that the submanifold X = X is a supersymmetric three-cycle inside
the Calabi–Yau three-fold given by a quintic hypersurface in £ P 4 .
SOLUTION
To prove the above statement, we should first check that the pullback of
the Kähler form is zero. This is trivial in this case, because X → X under
the transformation J → −J. On the other hand, the pullback of J onto the
fixed surface X = X should give J → J, so the pullback of J is zero.
Now let us consider the second condition, and compute the pullback of
the holomorphic three-form. The equation for a quintic hypersurface in £ P 4
discussed in Section 9.3 is
5
m=1
(X m ) 5 = 0.
Defining inhomogeneous coordinates Y k = X k /X 5 , with k = 1, 2, 3, 4, on
the open set X 5 = 0, the holomorphic three-form can be written as
The norm of Ω is
Ω = dY 1 ∧ dY 2 ∧ dY 3
(Y 4 ) 4 .
Ω 2 = 1
6 ΩabcΩ abc =
1
ˆg|Y 4 ,
| 8
where ˆg = det g a ¯ b . Using Eqs (9.104) and (9.129), as well as Exercise 9.8,
one has
e −K2,1
which implies that
= i
Ω ∧ Ω = V Ω 2 = 1
8 e−K1,1Ω
2
Ω 2 = 8e 2K ,