Revealing the topological nature of entangled orbital angular momentum states of light.

Topology has emerged as a fundamental property of many systems yet mostly limited to low dimensions. Here, we reveal the hidden topology in entangled states carrying orbital angular momentum (OAM), in arbitrary dimensions. For two-dimensional systems, we demonstrate multiple skyrmion topologies and their equivalence to 't Hooft-Polyakov magnetic monopoles, experimentally connecting them to the Higgs field. In higher dimensions, we use non-Abelian gauge fields of SU(d) Yang-Mills theory to predict a rich tapestry of topological maps and their invariants, which we confirm experimentally for dimensionality up to seven, showing an underlying topology of 48 dimensions and a topological spectrum spanning over 17000 invariants. In addition to inducing robustness to perturbation, the topological spectrum enables probing them, by observing their emergent signatures in its non-topological spaces. The only degree of freedom we use to construct the topology is OAM, breaking away from the optical paradigm of polarisation-based spin-textured fields and forgoing the need for quantum state engineering. Our theoretical framework can be extrapolated to any dimension and degree of freedom, opening a distinct path for finding topologies in light.