  Bosonization is one of the most significant frameworks to analyze fermionic
systems. In this work, we propose a new bosonization of Dirac fermion coupled
with $U(1)$ background gauge field consistent with gauge invariance, global
chiral anomaly matching and fermion-boson operator correspondence, either of
which is not satisfied by previously developed bosonizations. The bilinear
Dirac-mass term condensation paradox and its generalized form are resolved by
our bosonization. This new bosonization approach to interacting systems also
correctly captures the conformal characters of a significant class of critical
lattice models, such as conformal dimensions and conformal anomaly of XXZ spin
chain with twisted boundary condition. Our work clarifies the equivalence of
fermionic flux insertion and bosonic background charge insertion for
two-dimensional conformal field theory.
