A mirror symmetric solution to the quantum Toda lattice

2007/05/22 by Konstanze Rietsch, Rietsch, Konstanze
Mathematics · Physics and Astronomy · #14M17 #14N35 #57T15 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.RT #msc:14M17 #msc:14N35 #msc:57T15

paper · pdf · doi:10.48550/arxiv.0705.3202

25 pages, various improvements including to references, to appear in CMP

openalex publication_date 2007/05/22 · arxiv created 2011/03/26 · arxiv updated 2011/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use representation theory to construct integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SL(n)/B in a uniform way to arbitrary type and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive critical point of the 'superpotential' in every mirror fiber.

Citations

• Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
• Wild Harmonic Bundles and Wild Pure Twistor D-modules
• Lie Group Representations on Polynomial Rings

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