  We study the response of quantum many-body systems to coupling some of their
degrees of freedom to external gauge fields. This serves to understand the
current Green functions and transport properties of interacting many-body
systems. Our analysis leads to a "gauge theory of states of matter"
complementary to the well known Landau theory of order parameters. We
illustrate the power of our approach by deriving and interpreting the
gauge-invariant effective actions of (topological) superconductors, 2D electron
gases exhibiting the quantized Hall- and spin-Hall effect, 3D topological
insulators, as well as axion electrodynamics. We also use the theory to
elucidate the structure of surface modes in these systems.
