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 I will review his unpublished but insightful idea and present our attempts to extend it to more general cases.

中文翻译：

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$\mathcal{N}=2$超对称刘维尔理论中的对偶性

与玻色子刘维尔理论类似，$\mathcal{N}=2$ 超对称刘维尔理论被推测配备了交换超势和 K\"ahler 势的对偶性。然而，这种猜想的对偶性似乎受到了影响保留对称性的不匹配。十五多年前，当我还是一名学生时，我的导师 Tohru Eguchi 根据他对超对称性增强到 $\mathcal{N}=4$ $A_1$ 奇点的基本几何结构。我将回顾他未发表但有见地的想法，并介绍我们将其扩展到更一般情况的尝试。