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Topological materials have recently become a distinct focus in condensed matter physics, appearing famously in the quantum Hall effect and topological insulators. In these materials, certain essential properties are governed by ‘topological invariants,’ quantities insensitive to local perturbation or manipulation. As such, understanding and measuring topological invariants play a central role in investigations of topological materials.

Synthetic materials in which the constituent particles are photons trapped in an optical resonator offer an exciting platform on which to study topological materials. Recent efforts have realized broad control over the single-photon Hamiltonian, including a strong synthetic magnetic field for photons, and strong photon-photon interactions. In this talk, I will present how a nonplanar resonator can harbor a quantum Hall system on the surface of a cone. I will then discuss measurements of three distinct topological indices, offering insight onto their physical meaning and application. We measure the Chern number via real-space local projectors: non-reciprocal products of transmission amplitudes reveal an Aharanov-Bohm phase associated with a non-zero Chern number. We access two additional topological invariants, the mean orbital spin and chiral central charge, via the variation of the local density of states near a singularity of spatial curvature, revealing a complex interplay between geometry and topology. I will then transition to ongoing work introducing Rydberg-mediated interactions between individual photons and the resulting creation of a fractional quantum Hall system, paying close attention to the role topological invariants play in characterizing interacting topological phases of matter.