Topological Invariants and Ground-State Wave Functions of Topological Insulators on a Torus

We define topological invariants in terms of the ground states wave functions on a torus. This approach leads to precisely defined formulas for the Hall conductance in four dimensions and the topological magneto-electric $\theta$ term in three dimensions, and their generalizations in higher dimensions. They are valid in the presence of arbitrary many-body interaction and disorder. These topological invariants systematically generalize the two-dimensional Niu-Thouless-Wu formula, and will be useful in numerical calculations of disordered topological insulators and strongly correlated topological insulators, especially fractional topological insulators.