Exactly Solvable Floquet Dynamics for Conformal Field Theories in Dimensions Greater than Two

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.