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A Lie-theoretic Description of the Solution Space of the tt*-Toda Equations

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Abstract

We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt ∗-Toda equations) which were introduced by Cecotti and Vafa. It is known from Guest and Lin (J. Reine Angew. Math. 689, 1–32 2014) Guest et al. (It. Math. Res. Notices 2015, 11745–11784 2015) and Mochizuki (2013, 2014) that these solutions can be parametrized by monodromy data of a certain flat S L n+ 1 ℝ-connection. Using Boalch’s Lie-theoretic description of Stokes data, and Steinberg’s description of regular conjugacy classes of a linear algebraic group, we express this monodromy data as a convex subset of a Weyl alcove of S U n+ 1.

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Acknowledgements

The authors thank Philip Boalch, Ralf Kurbel, and Eckhard Meinrenken for useful conversations. The first author was partially supported by JSPS grant (A) 25247005. He is grateful to the National Center for Theoretical Sciences for excellent working conditions and financial support. The second author was partially supported by MOST grant 104-2115-M-007-004. She is grateful to the Japan Society for the Promotion of Science for the award of a Short-Term Invitation Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo, 169-8555, Japan

    Martin A. Guest

  2. Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan

    Nan-Kuo Ho

  3. National Center for Theoretical Sciences, Taipei, 106, Taiwan

    Nan-Kuo Ho

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  1. Martin A. Guest
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  2. Nan-Kuo Ho
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Correspondence to Martin A. Guest.

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Guest, M.A., Ho, NK. A Lie-theoretic Description of the Solution Space of the tt*-Toda Equations. Math Phys Anal Geom 20, 24 (2017). https://doi.org/10.1007/s11040-017-9255-z

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