  We construct charged black branes in type IIA flux compactifications that are
dual to (2+1)-dimensional field theories at finite density. The internal space
is a general Calabi-Yau manifold with fluxes, with internal dimensions much
smaller than the AdS radius. Gauge fields descend from the 3-form RR potential
evaluated on harmonic forms of the Calabi-Yau, and Kaluza-Klein modes decouple.
Black branes are described by a four-dimensional effective field theory that
includes only a few light fields and is valid over a parametrically large range
of scales. This effective theory determines the low energy dynamics, stability
and thermodynamic properties. Tools from flux compactifications are also used
to construct holographic CFTs with no relevant scalar operators, that can lead
to symmetric phases of condensed matter systems stable to very low
temperatures. The general formalism is illustrated with simple examples such as
toroidal compactifications and manifolds with a single size modulus. We
initiate the classification of holographic phases of matter described by flux
compactifications, which include generalized Reissner-Nordstrom branes,
nonsupersymmetric $AdS_2 \times R^2$ and hyperscaling violating solutions.
