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A Mirror Symmetric Solution to the Quantum Toda Lattice

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Abstract

We give a representation-theoretic proof of a conjecture from Rietsch (Adv Math 217:2401–2442, 2008) providing 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 and totally negative critical point of the ‘superpotential’ in every mirror fiber.

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Authors and Affiliations

  1. Department of Mathematics, King’s College London, Strand, London, UK

    Konstanze Rietsch

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  1. Konstanze Rietsch
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Correspondence to Konstanze Rietsch.

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Communicated by Y. Kawahigashi

The author is supported by EPSRC advanced fellowship EP/S071395/1.

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Rietsch, K. A Mirror Symmetric Solution to the Quantum Toda Lattice. Commun. Math. Phys. 309, 23–49 (2012). https://doi.org/10.1007/s00220-011-1308-8

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