TY - JOUR

T1 - Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy

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 - http://www.scopus.com/inward/record.url?scp=85105481482&partnerID=8YFLogxK

U2 - 10.1007/s11005-021-01391-4

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 -