Embedding a monolayer of a transition metal dichalcogenide in a high-$Q$ optical cavity results in the formation of distinct exciton polariton modes. The polaritons are affected by the strong exciton-phonon interaction in the monolayer. We use a time convolutionless master equation to calculate the phonon influence on the spectra of the polaritons. We discuss the nontrivial dependence of the line shapes of both branches on temperature and detuning. The peculiar polariton dispersion relation results in a linewidth of the lower polariton being largely independent of the coupling to acoustic phonons. For the upper polariton, acoustic phonons lead to a low-energy shoulder of the resonance in the linear response. Furthermore, we analyze the influence of inhomogeneous broadening being the dominant contribution to the lower polariton linewidth at low temperatures. Our results point towards interesting phonon features in polariton spectra in transition metal dichalcogenides.

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• Electron-phonon coupling
• Exciton polariton
• Photonics
• Transition metal dichalcogenides
• Approximation methods for many-body systems
• Density matrix methods