Axion dynamics in wormhole background

When axions are coupled to gravity, it is known that wormholes exist as gravitational instantons. These wormholes break global U(1) symmetry explicitly thus giving rise to the wormhole-induced low-energy effective Lagrangian. We derive systematically this effective Lagrangian to lower orders in the derivative expansion. The role of U(1) symmetry to the third-quantized field theory of topology change and the invisible-axion phenomenology in the wormhole background are also discussed.