Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy

G. Carlet; J. van de Leur; H. Posthuma; S. Shadrin

doi:10.1007/s11005-021-01391-4

Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy

G. Carlet
, J. van de Leur
, H. Posthuma
, S. Shadrin

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.

Original language	English
Article number	63
Pages (from-to)	1-67
Number of pages	67
Journal	Letters in Mathematical Physics
Volume	111
Issue number	3
DOIs	https://doi.org/10.1007/s11005-021-01391-4
Publication status	Published - Jun 2021

Bibliographical note

Funding Information:
G. C., H. P., and S. S. were supported by the Netherlands Organization for Scientific Research. G. C. is supported by the ANER grant “FROBENIUS” of the Region Bourgogne-Franche-Comté. The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002).

Publisher Copyright:
© 2021, The Author(s).

Keywords

Catalan numbers
Frobenius manifolds
Hirota equations
KP hierarchy
Lax equations

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Carlet2021_Article_HigherGeneraCatalanNumbersAndHFinal published version, 823 KBLicence: CC BY

Cite this

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title = "Higher genera Catalan numbers and Hirota equations for extended nonlinear Schr{\"o}dinger hierarchy",

abstract = "We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck{\textquoteright}s dessins d{\textquoteright}enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schr{\"o}dinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.",

keywords = "Catalan numbers, Frobenius manifolds, Hirota equations, KP hierarchy, Lax equations",

author = "G. Carlet and Leur, \{J. van de\} and H. Posthuma and S. Shadrin",

note = "Funding Information: G. C., H. P., and S. S. were supported by the Netherlands Organization for Scientific Research. G. C. is supported by the ANER grant “FROBENIUS” of the Region Bourgogne-Franche-Comt{\'e}. The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002). Publisher Copyright: {\textcopyright} 2021, The Author(s).",

year = "2021",

month = jun,

doi = "10.1007/s11005-021-01391-4",

language = "English",

volume = "111",

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journal = "Letters in Mathematical Physics",

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publisher = "Springer Netherlands",

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AU - Carlet, G.

AU - Leur, J. van de

AU - Posthuma, H.

AU - Shadrin, S.

N1 - Funding Information: G. C., H. P., and S. S. were supported by the Netherlands Organization for Scientific Research. G. C. is supported by the ANER grant “FROBENIUS” of the Region Bourgogne-Franche-Comté. The IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002). Publisher Copyright: © 2021, The Author(s).

PY - 2021/6

Y1 - 2021/6

N2 - We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.

AB - We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.

KW - Catalan numbers

KW - Frobenius manifolds

KW - Hirota equations

KW - KP hierarchy

KW - Lax equations

UR - https://www.scopus.com/pages/publications/85105481482

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DO - 10.1007/s11005-021-01391-4

M3 - Article

SN - 0377-9017

VL - 111

SP - 1

EP - 67

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 3

M1 - 63

ER -

Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy

Abstract

Bibliographical note

Keywords

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