  Similarly to the bosonic Liouville theory, the $\mathcal{N}=2$ supersymmetric
Liouville theory was conjectured to be equipped with the duality that exchanges
the superpotential and the K\"ahler potential. The conjectured duality,
however, seems to suffer from a mismatch of the preserved symmetries. More than
fifteen years ago, when I was a student, my supervisor Tohru Eguchi gave a
beautiful resolution of the puzzle when the supersymmetry is enhanced to
$\mathcal{N}=4$ based on his insight into the underlying geometric structure of
the $A_1$ singularity. I will review his unpublished but insightful idea and
present our attempts to extend it to more general cases.
