Abstract
"\theta-angle monodromy" occurs when a theory possesses a landscape of
metastable vacua which reshuffle as one shifts a periodic coupling \theta by a
single period. "Axion monodromy" models arise when this parameter is promoted
to a dynamical pseudoscalar field. This paper studies the phenomenon in
two-dimensional gauge theories which possess a U(1) factor at low energies: the
massive Schwinger and gauged massive Thirring models, the U(N) 't Hooft model,
and the {\mathbb CP}^N model. In all of these models, the energy dependence of
a given metastable false vacuum deviates significantly from quadratic
dependence on \theta just as the branch becomes completely unstable (distinct
from some four-dimensional axion monodromy models). In the Schwinger, Thirring,
and 't Hooft models, the meson masses decrease as a function of \theta. In the
U(N) models, the landscape is enriched by sectors with nonabelian \theta terms.
In the {\mathbb CP}^N model, we compute the effective action and the size of
the mass gap is computed along a metastable branch.
