  We show how Yang-Mills theory on $S^3\times R$ can exhibit a spectrum with
continuous bands if coupled either to a topological 3-form gauge field, or to a
dynamical axion with heavy Peccei-Quinn scale. The basic mechanism consists in
associating winding histories to a bosonic zero mode whose role is to convert a
circle in configuration space into a helix. The zero mode is, respectively, the
holonomy of the 3-form field or the axion momentum. In these models different
theta sectors coexist but are not mixed by local operators. Our analysis sheds
light on, and extends Seiberg's proposal for modifying the topological sums in
quantum field theories. It refutes a recent claim that $B+L$ violation at LHC
is unsuppressed.
