Mirror symmetry, Langlands duality, and commuting elements of Lie groups

By normalizing a component of the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkähler orbifolds which satisfy the conditions to be mirror partners in the sense of Strominger-Yau-Zaslow. The same holds true for commuting quadruples in a compact Lie group. The Hodge numbers of the mirror partners, or more precisely their orbifold E -polynomials, are shown to agree, as predicted by mirror symmetry. These polynomials are explicitly calculated when G is a quotient of SL(n). Mirror symmetry made its first appearance in 1990 as an equivalence between two linear sigma-models in superstring theory [10, 21]. The targets were Calabi-Yau 3-folds, so mirror symmetry predicted that these should come in pairs, M and M̂ , satisfying h(M) = H(M̂). Although many examples were known, the physics did not immediately provide any general construction of a mathematical nature for the mirror. Since then, however, two mathematical constructions have emerged: that of Batyrev [1] and Batyrev-Borisov [2], generalizing the original idea of Greene-Plesser [21], and that of Strominger-Yau-Zaslow [46] with which this paper is concerned. Of the two, Batyrev’s construction has the advantage of being precise, and more amenable to explicit calculations. One can prove, for example, that the Hodge numbers of the Batyrev mirror satisfy the desired relationship. On the other hand, it is deeply rooted in toric geometry. This has led skeptics to suggest that mirror symmetry is an intrinsically toric phenomenon, despite work [3, 40] extending Batyrev’s point of view some ways beyond the toric setting. The construction proposed by Strominger-Yau-Zaslow in 1996 has quite a different flavor. It is directly inspired by a physical duality, the so-called T -duality between sigma-models whose targets are dual tori. Remarkably, although it is supposed to transform one projective variety into another, the construction is not algebraic, or even Kähler, in nature. Rather, it is symplectic: one must find a foliation of M by special Lagrangian tori, and replace each torus with its dual. This bold idea has already led to some interesting work on the existence of families of Lagrangian tori in Calabi-Yau 3-folds [22, 23, 41], which is essentially a problem in symplectic topology. But it is not yet sufficiently advanced that the mirror can be constructed in any precise sense, nor any of its invariants computed beyond the Euler characteristic. It is not even known how to construct families of tori which are special Lagrangian (as opposed to just Lagrangian). And the further questions of what complex structure to place on the dual family, and how to deal with singular fibers, remain mysterious. Partially supported by NSF grant DMS–9808529.