On duality in |$\mathcal{N}=2$| supersymmetric Liouville theory.

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 Kähler 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 I will review his unpublished but insightful idea and present our attempts to extend it to more general cases. [ABSTRACT FROM AUTHOR]