We study the effects of cavity-mediated interactions on the symmetries and topological properties of a Su-Schrieffer-Heeger chain coupled to a single-mode photonic cavity. Considering an off-resonant regime we use the high-frequency expansion in order to obtain an effective fermionic Hamiltonian with cavity-mediated interactions. We characterize the effects of the cavity on topology in a finite-size chain by studying three different fermionic many-body topological markers using exact diagonalization: correlation functions between edges in a chain with open boundary conditions, and a winding number based on the single-particle Green's function and bulk electric polarization via the many-body formula by Resta for a chain with periodic boundary conditions. Importantly, we show that chiral symmetry is not broken by the cavity in the many-body ground state in the high-frequency limit. Our work highlights the importance of the cavity-mediated many-body interactions in understanding the symmetry and topology of the system.

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• Cavity quantum electrodynamics
• Topological insulators
• 1-dimensional systems
• Exact diagonalization
• Su-Schrieffer-Heeger model