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 nonperturbative charging effects in homogeneous disordered metals. The $\sigma$-model action contains a 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, $\Gamma ,$ 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.