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Cavity quantum electrodynamics (QED) studies the interaction between light and matter at the single quantum level and has played a central role in quantum science and technology. Combining the idea of cavity QED with moiré materials, we theoretically show that strong quantum light-matter interaction provides a way to control frustrated magnetism. Specifically, we develop a theory of moiré materials confined in a cavity consisting of thin polar van der Waals crystals. We show that nontrivial quantum geometry of moiré flat bands leads to electromagnetic vacuum dressing of electrons, which produces appreciable changes in single-electron energies and manifests itself as long-range electron hoppings. We apply our general formulation to a twisted transition metal dichalcogenide heterobilayer encapsulated by ultrathin hexagonal boron nitride layers and predict its phase diagram at different twist angles and light-matter coupling strengths. Our results indicate that the cavity confinement enables one to control magnetic frustration of moiré materials and might allow for realizing various exotic phases such as a quantum spin liquid.

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