  We study a charged dilatonic black hole in AdS_5, derived from a lagrangian
involving a gauge field whose kinetic term is modified by the exponential of a
neutral scalar. This black hole has two properties which one might reasonably
demand of the dual of a Fermi liquid: Its entropy is proportional to
temperature at low temperature, and its extremal limit supports normal modes of
massless, charged bulk fermions. The black hole we study has a simple analytic
form because it can be embedded in type IIB string theory as the near-horizon
limit of D3-branes with equal spins in two of the three independent transverse
planes. Two further properties can be deduced from this embedding: There is a
thermodynamic instability, reminiscent of ferromagnetism, at low temperatures;
and there is an AdS_3 factor in the extremal near-horizon geometry which
accounts for the linear dependence of entropy on temperature. Altogether, it is
plausible that the dilatonic black hole we study, or a relative of it with
similar behavior in the infrared, is the dual of a Fermi liquid; however, the
particular embedding in string theory that we consider is unlikely to have such
a dual description, unless through some unexpected boson-fermion equivalence at
large N.
