  We study the topological phase transitions occurring in three-dimensional
(3D) multicomponent lattice Abelian-Higgs (LAH) models, in which an
$N$-component scalar field is minimally coupled with a noncompact Abelian gauge
field, with a global SU($N$) symmetry. Their phase diagram presents a
high-temperature Coulomb (C) phase, and two low-temperature molecular (M) and
Higgs (H) phases, both characterized by the spontaneous breaking of the SU($N$)
symmetry. The molecular-Higgs (MH) and Coulomb-Higgs (CH) transitions are
topological transitions, separating a phase with gapless gauge modes and
confined charges from a phase with gapped gauge modes and deconfined charged
excitations. These transitions are not described by effective
Landau-Ginzburg-Wilson theories, due to the active role of the gauge modes. We
show that the MH and CH transitions belong to different charged universality
classes. The CH transitions are associated with the $N$-dependent charged fixed
point of the renormalization-group (RG) flow of the 3D Abelian-Higgs field
theory (AHFT). On the other hand, the universality class of the MH transitions
is independent of $N$ and coincides with that controlling the continuous
transitions of the one-component ($N=1$) LAH model. In particular, we verify
that the gauge critical behavior always corresponds to that observed in the 3D
inverted XY model, and that the correlations of an extended charged
gauge-invariant operator (in the Lorenz gauge, this operator corresponds to the
scalar field, thus it is local, justifing the use of the RG framework) have an
$N$-independent critical universal behavior. This scenario is supported by
numerical results for $N=1,\,2,\,25$. The MH critical behavior does not
apparently have an interpretation in terms of the RG flow of the AHFT, as
determined perturbatively close to four dimensions or with standard large-$N$
methods.
