  The concepts of geometric phase and wave-particle duality are interlinked to
several fundamental phenomena in quantum physics, but their mutual relationship
still forms an uncharted open problem. Here we address this question by
studying the geometric phase of a photon in double-slit interference. We
especially discover a general complementarity relation for the photon that
connects the geometric phase it exhibits in the observation plane and the
which-path information it encases at the two slits. The relation can be seen as
quantifying wave-particle duality of the photon via the geometric phase, thus
corroborating a foundational link between two ubiquitous notions in quantum
physics research.
