  Motivated by the three-dimensional topological field theory / two-dimensional
conformal field theory (CFT) correspondence, we study a broad class of
one-dimensional quantum mechanical models, known as anyonic chains, that can
give rise to an enormously rich (and largely unexplored) space of
two-dimensional critical theories in the thermodynamic limit. One remarkable
feature of these systems is the appearance of non-local microscopic
"topological symmetries" that descend to topological defects of the resulting
CFTs. We derive various model-independent properties of these theories and of
this topological symmetry / topological defect correspondence. For example, by
studying precursors of certain twist and defect fields directly in the anyonic
chains, we argue that (under mild assumptions) the two-dimensional CFTs
correspond to particular modular invariants with respect to their maximal
chiral algebras and that the topological defects descending from topological
symmetries commute with these maximal chiral algebras. Using this map, we apply
properties of defect Hilbert spaces to show how topological symmetries give a
handle on the set of allowed relevant deformations of these theories.
Throughout, we give a unified perspective that treats the constraints from
discrete symmetries on the same footing as the constraints from topological
ones.
