  Topological Physics relies on the specific structure of the eigenstates of
Hamiltonians. Their geometry is encoded in the quantum geometric tensor
containing both the celebrated Berry curvature, crucial for topological matter,
and the quantum metric. The latter is at the heart of a growing number of
physical phenomena such as superfluidity in flat bands, orbital magnetic
susceptibility, exciton Lamb shift, and non-adiabatic corrections to the
anomalous Hall effect. Here, we report the first direct measurement of both
Berry curvature and quantum metric in a two-dimensional continuous medium. The
studied platform is a planar microcavity of extremely high finesse, in the
strong coupling regime. It hosts mixed exciton-photon modes
(exciton-polaritons) subject to photonic spin-orbit-coupling which makes emerge
Dirac cones and exciton Zeeman splitting breaking time-reversal symmetry. The
monopolar and half-skyrmion pseudospin textures are measured by
polarisation-resolved photoluminescence. The associated quantum geometry of the
bands is straightforwardly extracted from these measurements. Our results
unveil the intrinsic chirality of photonic modes which is at the basis of
topological photonics. This technique can be extended to measure Bloch band
geometries in artificial lattices. The use of exciton-polaritons (interacting
photons) opens wide perspectives for future studies of quantum fluid physics in
topological systems.
