  While the phase structure of the $U(1)\times U(1)$-symmetric Higgs theory is
still under debate, a version of this theory with an additional Chern-Simons
term was recently shown to undergo a second-order phase transition [V. Shyta,
J. van den Brink, and F. S. Nogueira, Phys. Rev. Lett. 127, 045701 (2021)].
This theory is dual to a topological field theory of massless fermions
featuring two gauge fields. Here we elaborate on several aspects of this
duality, focusing on the critical current correlators and on the nature of the
critical point as reflected by the bosonization duality. The current
correlators associated to the $U(1)\times U(1)$ symmetry and the topological
current are shown to coincide up to a universal prefactor, which we find to be
the same for both $U(1)$ and $U(1)\times U(1)$ topological Higgs theories. The
established duality offers in addition another way to substantiate the claim
about the existence of a critical point in the bosonic Chern-Simons $U(1)\times
U(1)$ Higgs model: a Schwinger-Dyson analysis of the fermionic dual model shows
that no dynamical mass generation occurs. The same cannot be said for the
theory without the Chern-Simons term in the action.
