String Theory and M-Theory

11.5 The attractor mechanism 589
The vector multiplets contain the complex-structure moduli, while the hypermultiplets
contain the Kähler moduli and the dilaton. The following
discussion focuses on the fields contained in vector multiplets, since the entropy
does not depend on the hypermultiplets, at least in the supergravity
approximation, as will become clear.
Brief review of special geometry
An N = 2 vector multiplet contains a complex scalar, a gauge field and
a pair of Majorana (or Weyl) fermions. The moduli space describing the
scalars is h2,1-dimensional and is a special-Kähler manifold.
potential for the complex-structure moduli space is
The Kähler
K = − log i Ω ∧ Ω , (11.110)
where Ω is the holomorphic three-form of the Calabi–Yau manifold, as usual.
In this set up a black hole can be realized by wrapping a set of D3-branes on
a special Lagrangian three-cycle C. In order to describe this, let us introduce
the Poincaré dual three-form to C, which we denote by Γ.
This black hole carries electric and magnetic charges with respect to the
h 2,1 U(1) gauge fields originating from the ten-dimensional type IIB self-dual
five-form F5 as well as the graviphoton belonging to the N = 2 supergravity
multiplet. In order to describe the charges, let us introduce a basis of threecycles
A I , BJ (with I, J = 1, . . . , h 2,1 + 1), which can be chosen such that
the intersection numbers are
A I ∩ BJ = −BJ ∩ A I = δ I J and A I ∩ A J = BI ∩ BJ = 0. (11.111)
The Poincaré dual three-forms are denoted αI and βI. The group of transformations
that preserves these properties is the symplectic modular group
Sp(2h2,1 + 2; ). The symplectic coordinates introduced in Chapter 9 are
X I = e K/2
Ω and FI = e K/2
Ω. (11.112)
A I
Recall that the definition of Ω can be rescaled by a factor that is independent
of the manifold coordinates and that this corresponds to a rescaling of the
homogeneous coordinates X I . This freedom has been used to include the
factors of e K/2 , which will be convenient later.
The electric and magnetic charges, qI and p I , that result in four dimensions
are encoded in the homology class C = p I BI − qIA I or the equivalent
cohomology class Γ = p I αI − qIβ I . Thus, in terms of a canonical homology
M
BI