  Axions, periodic scalar fields coupled to gauge fields through the instanton
density, have a rich variety of higher-form global symmetries. These include a
two-form global symmetry, which measures the charge of axion strings. As we
review, these symmetries typically combine into a higher-group, a kind of
non-abelian structure where symmetries that act on operators of different
dimensions, such as points, lines, and strings, are mixed. We use this
structure to derive model independent constraints on renormalization group
flows that realize theories of axions at long distances. These give universal
inequalities on the energy scales where various infrared symmetries emerge. For
example, we show that in any UV completion of axion-Yang-Mills, the energy
scale at which axion strings can decay is always larger than the mass scale of
charged particles.
