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

Abstract

Abstract. By normalizing a component of the space of commuting pairs of elements ina reductive Lie group G, and the corresponding space for the Langlands dual group,we construct pairs of hyperk¨ahler orbifolds which satisfy the conditions to be mirrorpartners in the sense of Strominger-Yau-Zaslow. The same holds true for commutingquadruples in a compact Lie group. The Hodge numbers of the mirror partners, ormore precisely their orbifold E-polynomials, are shown to agree, as predicted by mirrorsymmetry. 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 linearsigma-models in superstring theory [10, 21]. The targets were Calabi-Yau 3-folds, so mirrorsymmetry predicted that these should come in pairs, M and Mˆ , satisfying h p,q (M) =H p,3−q (Mˆ).Although many examples were known, the physics did not immediately provide anygeneral construction of a mathematical nature for the mirror. Since then, however, twomathematical 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.Ofthe two, Batyrev’s construction has theadvantageofbeing precise, andmoreamenableto explicit calculations. One can prove, for example, that the Hodge numbers of the Batyrevmirror satisfy the desired relationship. On the other hand, it is deeply rooted in toricgeometry. This has led skeptics to suggest that mirror symmetry is an intrinsically toricphenomenon, despite work [3, 40] extending Batyrev’s point of view some ways beyond thetoric 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-modelswhose targets are dual tori. Remarkably, although it is supposed to transform one projectivevariety into another, the construction is not algebraic, or even K¨ahler, in nature. Rather, itis symplectic: one must find a foliation of M by special Lagrangian tori, and replace eachtorus with its dual.This bold idea has already led to some interesting work on the existence of families ofLagrangiantoriin Calabi-Yau3-folds [22, 23,41], which isessentially aproblem in symplectictopology. But it is not yet sufficiently advanced that the mirror can be constructed in anyprecise sense, nor any of its invariants computed beyond the Euler characteristic. It is noteven known how to construct families of tori which are special Lagrangian (as opposed tojust Lagrangian). And the further questions of what complex structure to place on the dualfamily, and how to deal with singular fibers, remain mysterious.