  The dualities that map hard-to-solve, interacting theories to free,
non-interacting ones often trigger a deeper understanding of the systems to
which they apply. However, simplifying assumptions such as Lorentz invariance,
low dimensionality, or the absence of axial gauge fields, limit their
application to a broad class of systems, including topological semimetals. Here
we derive several axial field theory dualities in 2+1 and 3+1 dimensions by
developing an axial slave-rotor approach capable of accounting for the axial
anomaly. Our 2+1-dimensional duality suggests the existence of a dual, critical
surface theory for strained three-dimensional non-symmorphic topological
insulators. Our 3+1-dimensional duality maps free Dirac fermions to Dirac
fermions coupled to emergent U(1) and Kalb-Ramond vector and axial gauge
fields. Upon fixing an axial field configuration that breaks Lorentz
invariance, this duality maps free to interacting Weyl semimetals, thereby
suggesting that the quantization of the non-linear circular photogalvanic
effect can be robust to certain interactions. Our work emphasizes how axial and
Lorentz-breaking dualities improve our understanding of topological matter.
