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Toric Networks, Geometric R-Matrices and Generalized Discrete Toda Lattices

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Abstract

We use the combinatorics of toric networks and the double affine geometric R-matrix to define a three-parameter family of generalizations of the discrete Toda lattice. We construct the integrals of motion and a spectral map for this system. The family of commuting time evolutions arising from the action of the R-matrix is explicitly linearized on the Jacobian of the spectral curve. The solution to the initial value problem is constructed using Riemann theta functions.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Science, Chiba University, Chiba, 263-8522, Japan

    Rei Inoue

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Thomas Lam

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55414, USA

    Pavlo Pylyavskyy

Authors
  1. Rei Inoue
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  2. Thomas Lam
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  3. Pavlo Pylyavskyy
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Corresponding author

Correspondence to Rei Inoue.

Additional information

Communicated by A. Borodin

R. Inoue was partially supported by JSPS KAKENHI Grant Number 26400037.

T. Lam was partially supported by NSF Grants DMS-1160726, DMS-1464693, and a Simons Fellowship.

P. Pylyavskyy was partially supported by NSF Grants DMS-1148634, DMS-1351590, and Sloan Fellowship.

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Inoue, R., Lam, T. & Pylyavskyy, P. Toric Networks, Geometric R-Matrices and Generalized Discrete Toda Lattices. Commun. Math. Phys. 347, 799–855 (2016). https://doi.org/10.1007/s00220-016-2739-z

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  • DOI: https://doi.org/10.1007/s00220-016-2739-z

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