Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a nonlocal operation, "partial" reflection or transpose. As an application of our scheme, we study the Z_{8} and Z_{16} classification of topological superconductors in one and three dimensions.