  Driven quantum systems exhibit a large variety of interesting and sometimes
exotic phenomena. Of particular interest are driven conformal field theories
(CFTs) which describe quantum many-body systems at criticality. In this paper,
we develop both a spacetime and a quantum information geometry perspective on
driven 2d CFTs. We show that for a large class of driving protocols the
theories admit an alternative but equivalent formulation in terms of a CFT
defined on a spacetime with a time-dependent metric. We prove this equivalence
both in the operator formulation as well as in the path integral description of
the theory. A complementary quantum information geometric perspective for
driven 2d CFTs employs the so-called Bogoliubov-Kubo-Mori (BKM) metric, which
is the counterpart of the Fisher metric of classical information theory, and
which is obtained from a perturbative expansion of relative entropy. We compute
the BKM metric for the universal sector of Virasoro excitations of a thermal
state, which captures a large class of driving protocols, and find it to be a
useful tool to classify and characterize different types of driving. For
M\"obius driving by the SL(2,R) subgroup, the BKM metric becomes the hyperbolic
metric on the unit disk. We show how the non-trivial dynamics of Floquet driven
CFTs is encoded in the BKM geometry via M\"obius transformations. This allows
us to identify ergodic and non-ergodic regimes in the driving. We also explain
how holographic driven CFTs are dual to driven BTZ black holes with evolving
horizons. The deformation of the black hole horizon towards and away from the
asymptotic boundary provides a holographic understanding of heating and cooling
in Floquet CFTs.
