Mesoscopic charging effects in the $\sigma$-model for granular metals

We derive the $\sigma$-model for granular metals. It is valid for any relation between the temperature, $T$, and the grain mean level spacing. The continuum limit of the $\sigma$-model describes non-perturbative charging effects in homogeneous disordered metals. The $\sigma$-model action contains a novel term which is crucial for studying Coulomb blockade effects. Topological properties of the $\sigma$-model encoding charge discreteness are studied. For $T$ below the escape rate from the grain, $\mathrm{\Gamma }$, Coulomb blockade effects are described by solitons of the $Q$-matrix which are delocalized in real space. At $T>\mathrm{\Gamma }$ the solitons transform into phase instantons localized at a single tunneling contact.