  We present an effective field theory approach to the topological response of
Floquet systems with symmetry group $G$. This is achieved by introducing a
background $G$ gauge field in the Schwinger-Keldysh formalism, which is
suitable for far from equilibrium systems. We carry out this program for chiral
topological Floquet systems (anomalous Floquet-Anderson insulators) in two
spatial dimensions, and the group cohomology models of topological Floquet
unitaries. These response actions serve as many-body topological invariants for
topological Floquet unitaries. The effective action approach also leads us to
propose novel topological response functions.
