  We study the nontrivial topology of the parameter space of general
$U(1)$-symmetric fermionic non-degenerately gapped system and its consequences
on the transport properties in arbitrary dimensions. By a nonlinear
parameter-gauge topological response theory, we find that such nontrivial
topology can impose quantization constraints on the charge transport in the
presence of background fluxes or, more generally, instantons in general
dimensions and our result generalizes the Thouless pump and its higher
dimensional generalizations. We also show that these nontrivial transport
properties are related to an unconventional quantum anomaly, which generalizes
$-1$-form anomalies. This anomaly imposes non-perturbative ingappabilities of
various types of spatial interfaces or time-dependent system evolution.
