This paper studies the symmetry rules that govern theories with many axions — light, periodic fields that appear in particle physics and string theory and are sometimes called the “axiverse.” The authors classify both ordinary (invertible) and more exotic (non-invertible) generalized symmetries that act on these fields. They then ask how those symmetries can be broken, including by effects coming from gravity, and what that means for the possible potentials and interactions of axions.

To do this the authors write down a fairly general four-dimensional effective field theory with an arbitrary number of axions and abelian (U(1)) gauge fields. They build explicit topological operators that realize the invertible and non-invertible symmetries, and they explain how those operators fit together with standard higher-form symmetries (symmetries associated with objects extended in space, not just pointlike charges). The setup also allows for certain curvature-squared couplings (Gauss–Bonnet and Pontryagin terms) and can be extended to simple supersymmetric models called N=1 axiverses.

At a conceptual level the paper emphasizes that non-invertible axion shift symmetries can forbid the formation of an axion potential and so protect axions from acquiring mass. These non-invertible symmetries are generated by topological defects that do not have inverses under fusion, so their conservation is more subtle than ordinary symmetries. The authors connect these field-theory considerations to quantum-gravity ideas about wormholes. In the picture they use, certain wormhole saddle configurations have imaginary axion profiles that interpolate between boundary conditions at the two ends. The appearance of these “imaginary wormholes” signals that the effective description must be modified and, in particular, that the axion shift symmetries can be broken.