Nonlinear parameter-gauge coupling approach to generalization of generalized Thouless pumps and $-1$-form anomaly

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.