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

2000/09/08 by Michael Thaddeus, Thaddeus, Michael
Mathematics · Physics and Astronomy · #14H60 #14J32 (Primary) #14J60 #20G20 #37J35 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG) #hep-th #math-ph #math.AG #math.MP #math.SG #msc:14H60 #msc:14J32 #msc:14J60 #msc:20G20 #msc:37J35

paper · pdf · doi:10.48550/arxiv.math/0009081

21 pages, LaTeX with packages amsfonts, amssym

openalex publication_date 2000/09/08 · arxiv created 2001/01/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By normalizing 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 hyperkahler 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).

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