Geometric phase in Taub-NUT spacetime

Constructing the Hamiltonian in the $\eta$-representation, we explore the geometric phase in the Taub-NUT spacetime, which is spherically symmetric and stationary. The geometric phase around a non-rotating Taub-NUT spacetime reveals both the gravitational analog of Aharonov-Bohm effect and Pancharatnam-Berry phase, similar to the rotating Kerr background. On the other hand, only the latter emerges in the spherically symmetric Schwarzschild geometry as well as in the axisymmetric magnetized Schwarzschild geometry. With this result, we argue that the Aharonov-Bohm effect and Pancharatnam-Berry phase both can emerge in the stationary spacetime, whereas only the latter emerges in the static spacetime. We outline plausible measurements of these effects/phases, mostly for primordial black holes.