  We study aspects of Berry phase in gapped many-body quantum systems by means
of effective field theory. Once the parameters are promoted to
spacetime-dependent background fields, such adiabatic phases are described by
Wess-Zumino-Witten (WZW) and similar terms. In the presence of symmetries,
there are also quantized invariants capturing generalized Thouless pumps.
Consideration of these terms provides constraints on the phase diagram of
many-body systems, implying the existence of gapless points in the phase
diagram which are stable for topological reasons. We describe such diabolical
points, realized by free fermions and gauge theories in various dimensions,
which act as sources of "higher Berry curvature" and are protected by the
quantization of the corresponding WZW terms or Thouless pump terms. These are
analogous to Weyl nodes in a semimetal band structure. We argue that in the
presence of a boundary, there are boundary diabolical points---parameter values
where the boundary gap closes---which occupy arcs ending at the bulk diabolical
points. Thus the boundary has an "anomaly in the space of couplings" in the
sense of C\'ordova et al. Consideration of the topological effective action for
the parameters also provides some new checks on conjectured infrared dualities
and deconfined quantum criticality in 2+1d.
