  Nonlinearly realized Abelian global symmetries can be reformulated as local
shift symmetries gauged by three-form gauge fields. The anomalous symmetries of
the Standard Model (such as Peccei-Quinn or $B+L$) can be dualized to local
symmetries gauged by the Chern-Simons three-forms of the Standard Model gauge
group. In this description the strong CP problem can be reformulated as the
problem of a massless three-form field in QCD, which creates an arbitrary
CP-violating constant four-form electric field in the vacuum. Both the axion as
well as the massless quark solutions amount to simply Higgsing the three-form
gauge field, hence screening the electric field in the vacuum. This language
gives an alternative way for visualizing the physics of the axion solution as
well as the degree of its vulnerability due to gravitational corrections. Any
physics that can jeopardize the axion solution must take the QCD three-form out
of the Higgs phase. This can only happen if the physics in question provides an
additional massless three-form. The axion then Higgses one combination of the
three-forms and the QCD electric field gets partially unscreened, reintroducing
the strong CP problem. Gravity provides such a candidate in form of the
Chern-Simons spin connection three-form, which could un-Higgs the QCD
three-form in the absence of additional chiral symmetries. We also discuss
analogous effects for the baryon number symmetry.
