  Controlling the topological properties of quantum matter is a major goal of
condensed matter physics. A major effort in this direction has been devoted to
using classical light in the form of Floquet drives to manipulate and induce
states with non-trivial topology. A different route can be achieved with cavity
photons. Here we consider a prototypical model for topological phase
transition, the one-dimensional Su-Schrieffer-Heeger (SSH) model, coupled to a
single mode cavity. We show that quantum light can affect the topological
properties of the system, including the finite-length energy spectrum hosting
edge modes and the topological phase diagram. In particular we show that
depending on the lattice geometry and the strength of light-matter coupling one
can either turn a trivial phase into a topological one or viceversa using
quantum cavity fields. Furthermore, we compute the polariton spectrum of the
coupled electron-photon system, and we note that the lower polariton branch
disappears at the topological transition point. This phenomenon can be used to
probe the phase transition in the SSH model.
