  We introduce the concept of Floquet topological magnons --- a mechanism by
which a synthetic tunable Dzyaloshinskii-Moriya interaction (DMI) can be
generated in quantum magnets using circularly polarized electric (laser) field.
The resulting effect is that Dirac magnons and nodal magnons in two-dimensional
(2D) and three-dimensional (3D) quantum magnets can be tuned to magnon Chern
insulators and Weyl magnons respectively under circularly polarized laser
field. The Floquet formalism also yields a tunable intrinsic DMI in insulating
quantum magnets without an inversion center. We demonstrate that the Floquet
topological magnons possess a finite thermal Hall conductivity tunable by the
laser field.
