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We study generalizations of the Berry phase for quantum lattice systems inarbitrary dimensions. For a smooth family of gapped ground states in ddimensions, we define a closed (d+2)-form on the parameter space whichgeneralizes the curvature of the Berry connection. Its cohomology class is atopological invariant of the family. When the family is equivariant under theaction of a compact Lie group G, topological invariants take values in theequivariant cohomology of the parameter space. These invariants unify andgeneralize the Hall conductance and the Thouless pump. A key role in theseconstructions is played by a certain differential graded Frechet-Lie algebraattached to any quantum lattice system. As a by-product, we describeambiguities in charge densities and conserved currents for arbitrary latticesystems with rapidly decaying interactions.

Local Noether theorem for quantum lattice systems and topological invariants of gapped states