  We propose a system that exploits the fundamental features of topological
photonics and synthetic dimensions to force many semiconductor laser resonators
to synchronize, mutually lock, and under suitable modulation emit a train of
transform-limited mode-locked pulses. These lasers exploit the Floquet
topological edge states in a 1D array of ring resonators, which corresponds to
a 2D topological system with one spatial dimension and one synthetic frequency
dimension. We show that the lasing state of the multi-element laser system
possesses the distinct characteristics of spatial topological edge states while
exhibiting topologically protected transport. The topological synthetic-space
edge mode imposes a constant-phase difference between the multi-frequency modes
on the edges, and together with modulation of the individual elements forces
the ensemble of resonators to mode-lock and emit short pulses, robust to
disorder in the multi-resonator system. Our results offer a proof-of-concept
mechanism to actively mode-lock a laser diode array of many lasing elements,
which is otherwise extremely difficult due to the presence of many spatial
modes of the array. The topological synthetic-space concepts proposed here
offer an avenue to overcome this major technological challenge, and open new
opportunities in laser physics.
