  By example of the nonlinear Kerr-mode driven by a laser, we show that
hysteresis phenomena in systems featuring a driven-dissipative phase transition
(DPT) can be accurately described in terms of just two collective, dissipative
Liouvillian eigenmodes. The key quantities are just two components of a
nonabelian geometric connection, even though a single parameter is driven. This
powerful geometric approach considerably simplifies the description of
driven-dissipative phase transitions, extending the range of computationally
accessible parameter regimes, and providing a new starting point for both
experimental studies and analytical insights.
