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Odd-angular momentum exciton states are dark to light in monolayers oftransition-metal dichalcogenides and can be addressed only by two-photonprobes. Besides, $2p$ excitons states are expected to show a fine splittingthat arises from the peculiar electronic band structure of the material,characterized by a finite Berry curvature around each valley. In this work westudy in detail the coupling of photons to $p$ exciton states and we find thatthe Rashba spin-orbit interaction or a Skyrmion in the transition-metaldichalcogenide substrate can be exploited to engineer finite optical selectionrules. The basis mechanism relies on a matching of the exciton angular momentumwith the winding of the Rashba spin-orbit interaction or the Skyrmiontopological charge. In a photonic cavity, the coupling is enhanced due tophoton confinement and the resulting polaritonic branches acquire a mixing with$2p^\pm$ excitons, thus providing a useful tool to detect excitonfine-splitting and and enable novel uses of odd-parity dark excitons.

Brightening odd-parity excitons in transition-metal dichalcogenides: Rashba spin-orbit interaction, skyrmions, and cavity polaritons