  Appearance of quantized vortices in a superfluid and a Bose-Einstein
condensate (BEC) stems from their nontrivial response to broken time-reversal
symmetry (TRS). Here we show that breaking of the TRS by, for example, rotation
or an external synthetic magnetic field in a transversely-driven BEC coupled to
a single mode of an optical cavity modifies drastically Dicke-superradiance and
self-ordering phenomena in this system. In particular, photon scattering from
the pump laser into the cavity is amplified by the rotational motion of the
BEC, leading to so-called 'rotational superradiance' - in a loose analogy to
black-hole physics - with distinct critical scaling properties. Another notable
finding is that cavity photons mediate long-range, periodic attractive
interactions among the vortices, which compete with pair-wise logarithmic
repulsive vortex interactions and deform the Abrikosov triangular vortex
lattice favoring a stripe-like pattern. Remarkably, the rotation of the BEC and
topological properties of the vortex lattice can be monitored nondestructively
through the cavity output field.
