  Conformal field theories (CFTs) with a globally conserved U(1) charge Q can
be deformed into compressible phases by modifying their Hamiltonian, H, by a
chemical potential H -> H - \mu Q. We study 2+1 dimensional CFTs upon which an
explicit S duality mapping can be performed. We find that this construction
leads naturally to compressible phases which are superfluids, solids, or
non-Fermi liquids which are more appropriately called `Bose metals' in the
present context. The Bose metal preserves all symmetries and has Fermi surfaces
of gauge-charged fermions, even in cases where the parent CFT can be expressed
solely by bosonic degrees of freedom. Monopole operators are identified as
order parameters of the solid, and the product of their magnetic charge and Q
determines the area of the unit cell. We discuss implications for holographic
theories on asymptotically AdS4 spacetimes: S duality and monopole/dyon fields
play important roles in this connection.
