In a quantum field theory with fermions, (−1)F is a unitary, Hermitian, involutive operator where F is the fermion number operator and is equal to the sum of the lepton number plus the baryon number, F=B+L, for all particles in the Standard Model. The action of this operator is to multiply bosonic states by 1 and fermionic states by −1. This is always a global internal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2π. This splits the Hilbert space into two superselection sectors. Bosonic operators commute with (−1)F whereas fermionic operators anticommute with it.
This operator really shows its utility in supersymmetric theories. It's trace is often a useful computation.