  We analyze the properties of strongly coupled excitons and photons in systems
made of semiconducting two-dimensional transition-metal dichalcogenides
embedded in optical cavities. Through a detailed microscopic analysis of the
coupling we unveil novel, highly tunable features of the spectrum, that result
in polariton splitting and a breaking of light-matter selection rules. The
dynamics of the composite polaritons is influenced by the Berry phase arising
both from their constituents and from the confinement-enhanced coupling. We
find that light-matter coupling emerges as a mechanism that enhances the Berry
phase of polaritons well beyond that of its elementary constituents, paving the
way to achieve a polariton Hall effect.
