The extended D-Toda hierarchy

Cheng, Jipeng; Milanov, Todor

doi:10.1007/s00029-021-00646-1

Cited by 10 publications

(5 citation statements)

References 20 publications

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“…For type A n this integrable hierarchy is the zero-dispersion limit of Carlet's extended bigraded Toda hierarchy [10], and for type D n it is the long-wave limit of the Cheng-Milanov extended D-type hierarchy [11]. For simply-laced R, we expect that the principal hierarchies of Theorem 1.4 should coincide with the dispersionless limit of the Hirota integrable hierarchies constructed by Milanov-Shen-Tseng in [37].…”

Section: Introductionmentioning

confidence: 87%

Mirror symmetry for extended affine Weyl groups

Brini,

van Gemst

2021

Preprint

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We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given by a one-dimensional Landau-Ginzburg superpotential constructed from a suitable degeneration of the family of spectral curves of the affine relativistic Toda chain for the corresponding affine Poisson-Lie group. As applications of our mirror theorem we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko-Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bihamiltonian hierarchies (in both Lax-Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy.

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Section: Introductionmentioning

confidence: 87%

Mirror symmetry for extended affine Weyl groups

Brini,

van Gemst

2021

Preprint

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“…Based on (5.6), it is possible that the particular L found here is a bispectral operator, as defined in [10]. That is, there could exist a suitable differential operator in z with eigenfunctions ψ and ψ and corresponding eigenvalues that are functions of s. Another future direction is to explore additional symmetries of the type D-Toda hierarchy mentioned in [7].…”

Section: Data Availability Statementmentioning

confidence: 99%

“…For a more comprehensive review of the Toda hierarchy and its extension, see [19]. As discussed by Cheng and Milanov, the EBTH is the extension of a Kac-Wakimoto hierarchy in the case A; other recent work on the topic, more specifically on the corresponding extension in the case D, can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

Cox¹,

Sisson²

2022

J. Phys. A: Math. Theor.

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Additional symmetries of the extended bigraded Toda hierarchy (EBTH) were introduced by Bakalov and Wheeless. We describe properties of a Lax operator fixed under these additional symmetries and determine the unique such Lax operator in the special case of the extended Toda hierarchy (ETH). We further present differential equations for the wave functions and their general solutions that hint at a possible connection to the bispectral problem. Based on these solutions, we highlight a correspondence between the wave functions. Finally, we find the form of a tau function for our Lax operator and compute a second solution to the ETH by applying a Darboux transformation presented by Li and Song.

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“…This discovery was enhanced to a mathematical statement called the Toda conjecture [3][4][5]. The Toda conjecture was proved by several methods [6][7][8][9][10] and generalized to CP 1 with orbifold points [11][12][13][14]. The relevant integrable hierarchies are the 1D Toda hierarchy, the bigraded Toda hierarchy [15,16] and a kind of the Kac-Wakimoto hierarchies [17].…”

Section: Introductionmentioning

confidence: 99%

Dressing operators in equivariant Gromov–Witten theory of CP1

Takasaki

2021

J. Phys. A: Math. Theor.

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Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs the equivariant Gromov–Witten theory of C P 1 . A technical clue of their method is a pair of dressing operators on the Fock space of 2D charged free fermion fields. We reformulate these operators as difference operators in the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to the question of why the equivariant Toda hierarchy emerges in the equivariant Gromov–Witten theory of C P 1 . Moreover, the non-equivariant limit of these operators turns out to capture the integrable structure of the non-equivariant Gromov–Witten theory correctly.

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The extended D-Toda hierarchy

Cited by 10 publications

References 20 publications

Mirror symmetry for extended affine Weyl groups

Mirror symmetry for extended affine Weyl groups

The Lax operator fixed under the additional symmetries of the extended Toda hierarchy

Dressing operators in equivariant Gromov–Witten theory of CP1

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