  We discuss the effective twisted superpotentials of 2d $\mathcal{N}=(2,2)$
theories arising upon the reduction of 4d $\mathcal{N}=2$ gauge theories on the
$\Omega$-deformed cigar-like geometry. We explain field-theoretic origins of
the gaps in the spectrum in the corresponding quantum mechanical (QM) systems.
We find local 2d descriptions of the physics near these gaps by resumming the
non-perturbative part of the twisted superpotential and discuss arising
wall-crossing phenomena. The interpretation of the associated phenomena in the
classical Liouville theory and in the scattering of two heavy states in $AdS_3$
gravity is suggested. Some comments concerning a possible interpretation of the
band structure in QM in terms of the Schwinger monopole-pair production in 4d
are presented.
