On anomalies and fermionic unitary operators

We point out that fermionic unitary operators which anticommute among themselves appear in various situations in quantum field theories with anomalies in the Hamiltonian formalism. To illustrate, we give multiple derivations of the fact that position-dependent $U(1)$ transformations of two-dimensional theories with $U(1)$ symmetry of odd level are fermionic when the winding number is odd. We then relate this mechanism to the anomalies of the discrete ${\mathbb{Z}}_N\subset U(1)$ symmetry, whose description also crucially uses unitary operators which are fermionic. We also show that position-dependent $SU(2)$ transformations of four-dimensional theories with $SU(2)$ symmetry with Witten anomaly are fermionic and anticommute among themselves when the winding number is odd.