  We study charge transport of quantum critical points described by conformal
field theories in 2+1 spacetime dimensions. The transport is described by an
effective field theory on an asymptotically anti-de Sitter spacetime, expanded
to fourth order in spatial and temporal gradients. The presence of a horizon at
non-zero temperatures implies that this theory has quasi-normal modes with
complex frequencies. The quasi-normal modes determine the poles and zeros of
the conductivity in the complex frequency plane, and so fully determine its
behavior on the real frequency axis, at frequencies both smaller and larger
than the absolute temperature. We describe the role of particle-vortex or
S-duality on the conductivity, specifically how it maps poles to zeros and vice
versa. These analyses motivate two sum rules obeyed by the quantum critical
conductivity: the holographic computations are the first to satisfy both sum
rules, while earlier Boltzmann-theory computations satisfy only one of them.
Finally, we compare our results with the analytic structure of the O(N) model
in the large-N limit, and other CFTs.
