Rational reductions of the 2D-Toda hierarchy and mirror symmetry

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External organisations

  • Korteweg-de Vries Institute for Mathematics
  • Université Catholique de Louvain
  • Université de Bourgogne Franche-Comté


We introduce and study a two-parameter family of symmetry reductions of the twodimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1+1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's q-deformed Gelfand-Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki-Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the Ablowitz-Ladik hierarchy to Gromov-Witten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi-Yau threefolds obtained from minimal resolutions of the local orbifold line.


Original languageEnglish
Pages (from-to)835-880
Number of pages46
JournalJournal of the European Mathematical Society
Issue number3
Early online date15 Feb 2017
Publication statusPublished - 2017


  • 2DToda, Ablowitz-Ladik, Gromov-Witten, Integrable hierarchies, Mirror symmetry, Rational reductions

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