Euclidean Axion Wormholes Have Multiple Negative Modes

We show that Euclidean axion wormholes in theories of gravity coupled to a single axion have several independent inhomogeneous perturbations that lower the Euclidean action. Our analysis relies on a judiciously chosen gauge-invariant variable which makes the negative mode structure about axion wormholes transparent. Perturbations lowering the action are concentrated in the neck region and exist for wormholes in flat space and in AdS. Their presence means axion wormholes are not relevant saddle points of the functional integral in quantum gravity. This resolves the paradoxes associated with these solutions from the viewpoint of AdS/CFT.