Black hole attractors and the entropy function in four- and five-dimensional N=2 supergravity

Extremal black holes in theories of gravity coupled to abelian
gauge fields and neutral scalars, such as those arising in the
low-energy description of compactifications of string theory on
Calabi-Yau manifolds, exhibit the attractor phenomenon: on the
event horizon the scalars settle to values determined by the
charges carried by the black hole and independent of the values at
infinity. It is so, because on the horizon the energy contained in
vector fields acts as an effective potential (the black hole
potential), driving the scalars towards its minima. For spherically
symmetric black holes in theories where gauge potentials appear in
the Lagrangian solely through field strengths, the attractor
phenomenon can be alternatively described by a variational
principle based on the so-called entropy function, defined as the
Legendre transform with respect to electric fields of the
Lagrangian density integrated over the horizon. Stationarity
conditions for the entropy function then take the form of attractor
equations relating the horizon values of the scalars to the black
hole charges, while the stationary value itself yields the entropy
of the black hole. In this study we examine the relationship
between the entropy function and the black hole potential in
four-dimensional N=2 supergravity and demonstrate that in the
absence of higher-order corrections to the Lagrangian these two
notions are equivalent. We also exemplify their practical
application by finding a supersymmetric and a non-supersymmetric
solution to the attractor equations for a conifold prepotential.
Exploiting a connection between four- and five-dimensional black
holes we then extend the definition of the entropy function to a
class of rotating black holes in five-dimensional N=2 supergravity
with cubic prepotentials, to which the original formulation did not
apply because of broken spherical symmetry and explicit dependence
of the Lagrangian on the gauge potentials in the Chern-Simons term.
We also display two types of solutions to the respective attractor
equations. The link between four- and five-dimensional black holes
allows us further to derive five-dimensional first-order
differential flow equations governing the profile of the fields
from infinity to the horizon and construct non-supersymmetric
solutions in four dimensions by dimensional reduction. Finally,
four-dimensional extremal black holes in N=2 supergravity can be
also viewed as certain two-dimensional string compactifications
with fluxes. Motivated by this fact the recently proposed entropic
principle postulates as a probability measure on the space of these
string compactifications the exponentiated entropy of the
corresponding black holes. Invoking the conifold example we find
that the entropic principle would favor compactifications that
result in infrared-free gauge theories.