  In-plane polarized light experiences a non-trivial topological evolution as
it propagates resonantly in a M\"obius ring resonator. The resultant geometric
phase varies continuously when changing the light ellipticity, which leads to
constructive interference for a non-integer number of wavelengths, and
therefore to the occurrence of an arbitrary fractional number of optical modes.
The geometric phase in M\"obius-ring resonators is topologically robust and
implies excellent intrinsic fault-tolerance.
