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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 ofthe gaps in the spectrum in the corresponding quantum mechanical (QM) systems.We find local 2d descriptions of the physics near these gaps by resumming thenon-perturbative part of the twisted superpotential and discuss arisingwall-crossing phenomena. The interpretation of the associated phenomena in theclassical Liouville theory and in the scattering of two heavy states in $AdS_3$gravity is suggested. Some comments concerning a possible interpretation of theband structure in QM in terms of the Schwinger monopole-pair production in 4dare presented.

Bands and gaps in Nekrasov partition function