The Witten-Kontsevich KdV tau function of topological gravity has a generalization to an arbitrary DrinfeldSokolov hierarchy associated to a simple complex Lie algebra. Using the Feigin-Frenkel isomorphism we describe the affine opers describing such generalized Witten-Kontsevich functions in terms of Segal-Sugawara operators associated to the Langlands dual Lie algebra. In the case where the Lie algebra is simply laced there is a second role these Segal-Sugawara operators play: Their action, in the basic representation of the affine algebra associated to the Lie algebra, singles out the Witten-Kontsevich tau function within the phase space. We show that these two Langlands dual roles of Segal-Sugawara operators correspond to a duality between the first and last operator for a complete set of Segal-Sugawara operators.

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Witten-Kontsevich 点的 Feigin-Frenkel 图像

拓扑引力的 Witten-Kontsevich KdV tau 函数可推广到与简单复李代数相关联的任意 DrinfeldSokolov 层次结构。使用 Feigin-Frenkel 同构，我们根据与 Langlands 对偶李代数相关的 Segal-Sugawara 算子来描述描述这种广义 Witten-Kontsevich 函数的仿射运算。在李代数被简单地关联的情况下，这些 Segal-Sugawara 算子扮演了第二个角色：他们的行为，在与李代数相关联的仿射代数的基本表示中，挑选出相位内的 Witten-Kontsevich tau 函数空间。我们表明 Segal-Sugawara 算子的这两个 Langlands 双重角色对应于完整的 Segal-Sugawara 算子集的第一个和最后一个算子之间的对偶性。