6533b850fe1ef96bd12a843a
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Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy
G. CarletJ. Van De LeurH. PosthumaS. ShadrinSub Fundamental MathematicsDep WiskundeFundamental Mathematicssubject
High Energy Physics - TheoryPure mathematicsRank (linear algebra)FOS: Physical sciences[MATH] Mathematics [math]01 natural sciencesCatalan numberMathematics::Algebraic Geometry[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]KP hierarchy0103 physical sciences[NLIN] Nonlinear Sciences [physics][NLIN]Nonlinear Sciences [physics][MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]0101 mathematics[MATH]Mathematics [math]Mathematics::Symplectic GeometryMathematical PhysicsMathematicsHirota equationsPartition function (quantum field theory)ConjectureNonlinear Sciences - Exactly Solvable and Integrable SystemsHierarchy (mathematics)010102 general mathematics[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Statistical and Nonlinear PhysicsMathematical Physics (math-ph)16. Peace & justiceLax equationsManifoldModuli spaceMonotone polygonNonlinear Sciences::Exactly Solvable and Integrable SystemsHigh Energy Physics - Theory (hep-th)010307 mathematical physics[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]Exactly Solvable and Integrable Systems (nlin.SI)Catalan numbersFrobenius manifoldsdescription
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 Schr\"odinger 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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-06-01 | Letters in Mathematical Physics |