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While the phase structure of the $U(1)\times U(1)$-symmetric Higgs theory isstill under debate, a version of this theory with an additional Chern-Simonsterm 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 fermionsfeaturing two gauge fields. Here we elaborate on several aspects of thisduality, focusing on the critical current correlators and on the nature of thecritical point as reflected by the bosonization duality. The currentcorrelators associated to the $U(1)\times U(1)$ symmetry and the topologicalcurrent are shown to coincide up to a universal prefactor, which we find to bethe same for both $U(1)$ and $U(1)\times U(1)$ topological Higgs theories. Theestablished duality offers in addition another way to substantiate the claimabout the existence of a critical point in the bosonic Chern-Simons $U(1)\timesU(1)$ Higgs model: a Schwinger-Dyson analysis of the fermionic dual model showsthat no dynamical mass generation occurs. The same cannot be said for thetheory without the Chern-Simons term in the action.

Bosonization duality in 2+1 dimensions and critical current correlation functions in Chern-Simons $U(1)\times U(1)$ Abelian Higgs model