  We find classes of driven conformal field theories (CFT) in d + 1 dimensions
with d > 1, whose quench and Floquet dynamics can be computed exactly. The
setup is suitable for studying periodic drives, consisting of square pulse
protocols for which Hamiltonian evolution takes place with different
deformations of the original CFT Hamiltonian in successive time intervals.
These deformations are realized by specific combinations of conformal
generators with a deformation parameter $\beta$; the $\beta < 1$ ($\beta > 1$)
Hamiltonians can be unitarily related to the standard (Luscher-Mack) CFT
Hamiltonian. The resulting time evolution can be then calculated by conformal
transformations. For $d\leq 3$ we show that the transformations can be obtained
in a quaternion formalism. Evolution with such a single Hamiltonian yields
qualitatively different time dependences of observables depending on the value
of $\beta$, ranging from exponential decays characteristic of heating to
oscillations and power law decays. This manifests in the behavior of the
fidelity, unequal-time correlator, and the energy density at the end of a
single cycle of a square pulse protocol with different hamiltonians in
successive time intervals. When the Hamiltonians in a cycle involve generators
of a single SU(1, 1) subalgebra we calculate the Floquet Hamiltonian. We show
that one can get dynamical phase transitions by varying the time period of a
cycle, where the system can go from a non-heating phase which is oscillatory as
a function of the time period to a heating phase with an exponentially damped
behavior. Our methods can be generalized to other discrete and continuous
protocols. We also point out that our results are expected to hold for a
broader class of QFTs that possesses an SL(2, C) symmetry with fields that
transform as quasi-primaries under this. As an example, we briefly comment on
celestial CFTs in this context.
