  The topology of an object describes global properties that are insensitive to
local perturbations. Classic examples include string knots and the genus
(number of handles) of a surface: no manipulation of a closed string short of
cutting it changes its "knottedness"; and no deformation of a closed surface,
short of puncturing it, changes how many handles it has. Topology has recently
become an intense focus of condensed matter physics, where it arises in the
context of the quantum Hall effect [1] and topological insulators [2]. In each
case, topology is defined through invariants of the material's bulk [3-5], but
experimentally measured through chiral/helical properties of the material's
edges. In this work we measure topological invariants of a quantum Hall
material through local response of the bulk: treating the material as a
many-port circulator enables direct measurement of the Chern number as the
spatial winding of the circulator phase; excess density accumulation near
spatial curvature quantifies the curvature-analog of charge known as mean
orbital spin, while the moment of inertia of this excess density reflects the
chiral central charge. We observe that the topological invariants converge to
their global values when probed over a few magnetic lengths lB, consistent with
intuition that the bulk/edge distinction exists only for samples larger than a
few lB. By performing these experiments in photonic Landau levels of a twisted
resonator [6], we apply quantum-optics tools to topological matter. Combined
with developments in Rydberg-mediated interactions between resonator photons
[7], this work augurs an era of precision characterization of topological
matter in strongly correlated fluids of light.
