  We study Tomonaga-Luttinger liquids thrown out of equilibrium by marginal
deformations in the form of interaction modulations. This is modeled by
quenching or periodically driving the Luttinger parameter or, equivalently, the
compactification radius of the free boson conformal field theory between two
different values. We obtain exact analytical results for the evolution of the
Loschmidt echo and observables such as the particle and energy densities.
Starting from generic initial states, the quench dynamics are shown to exhibit
revivals and temporal orthogonalities. For the periodic drive, we show
stability or instability of time-evolved physical quantities dependent on the
drive parameters. We also compare the corresponding marginally deformed thermal
density matrices by non-perturbatively evaluating their R\'{e}nyi divergence as
a Euclidean quench. All the dynamics are shown to be crucially dependent on the
ratio of the Luttinger parameters, which corresponds to the Zamolodchikov
distance in the space of marginal deformations. Our setup is equivalently
interpreted as the dynamics of the bosonic string upon instantaneous changes of
the target-space radius.
