String Theory and M-Theory

9.8 Nonperturbative effects in Calabi–Yau compactifications 407
The meaning of this equation is that the pullback of the holomorphic
(3, 0)-form Ω of the Calabi–Yau manifold to the membrane world volume
is proportional to the membrane volume element. The complex-conjugate
equation implies the same thing for the (0, 3)-antiholomorphic form Ω.
The phase ϕ is a constant that simply reflects an arbitrariness in the
definition of Ω. The factor e K , where K is given by
K = 1
2 (K1,1 − K 2,1 ), (9.151)
is a convenient normalization factor. The term K 2,1 is a function of the
complex moduli belonging to h 2,1 hypermultiplets. K 1,1 is a function of
the real moduli belonging to h 1,1 vector supermultiplets.
The supersymmetric three-cycle conditions (9.149) and (9.150) define a
special Lagrangian submanifold. When these conditions are satisfied, there
exists a nonzero covariantly constant spinor of the form ε = P+η. Thus,
the conclusion is that a Euclidean M2-brane wrapping a special Lagrangian
submanifold of the Calabi–Yau three-fold gives a supersymmetric instanton
contribution to the five-dimensional low-energy effective theory.
The conditions (9.149) and (9.150) imply that the membrane has minimized
its volume. In order to derive a bound for the volume of the membrane
consider
where Σ is the membrane world volume. Since
the inequality becomes
Σ
2V ≥ e −K
ε † P †
− P−ε d 3 σ ≥ 0, (9.152)
P †
− P− = P−P− = P−, (9.153)
e iϕ
Σ
Ω + e −iϕ
Σ
Ω , (9.154)
where ϕ is a phase which can be adjusted so that we obtain
V ≥ e −K
Ω
. (9.155)
The bound is saturated whenever the membrane wraps a supersymmetric
cycle C, in which case
V = e −K
Ω
. (9.156)
Type IIA or type IIB superstring theory, compactified on a Calabi–Yau
three-fold, also has supersymmetric cycles, which can be determined in a
Σ
C