Many body topological invariants in topological phases with point group symmetry

A way to detect topological phases from a given short-range entangled state is discussed. Many body topological invariants are defined as partition functions of topological quantum field theory (TQFT) on space-time manifolds, for example, real projective spaces. It is expected that by translating TQFT partition functions to the operator formalism one can get a definition of many body topological invariants made from ground state wave functions and symmetry operations. We propose that a kind of non-local operator, the "partial point group transformation", on a short-range entangled state is a unified measure to detect topologically nontrivial phases with point group symmetry. In this talk, I introduce (i) the partial rotations on (2$+$1)d chiral superconductors, and (ii) the Z16 invariant from the partial inversion on (3$+$1)d superconductors. These partial point group transformations can be analytically calculated from the boundary theory. We confirmed that analytical results from the boundary theory match with direct numerical calculations on bulk.