  Recent work on a family of boson-fermion mappings has emphasized the
interplay of symmetry and duality: Phases related by a particle-vortex duality
of bosons (fermions) are related by time-reversal symmetry in their fermionic
(bosonic) formulation. We present exact mappings for a number of concrete
models that make this property explicit on the operator level. We illustrate
the approach with one- and two-dimensional quantum Ising models, and then
similarly explore the duality web of complex bosons and Dirac fermions in (2+1)
dimensions.
