Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger 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'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. Contents 1. The Dubrovin-Frobenius manifold 1.1. The Dubrovin-Frobenius manifold for Catalan numbers 1.2. The canonical coordinates 1.3. The normalised canonical frame 2. The deformed flat connection and the principal hierarchy 2.1. The deformed flat connection 2.2. The superpotential and the deformed flat coordinates 2.3. The calibration 2.4. The principal hierarchy 2.5. The R matrix 3. Givental quantization formalism and potentials 3.1. Symplectic loop space and quantization 3.2. Symplectic transformations and potentials 3.3. Higher genera Catalan numbers 3.4. The descendent potential and the KP hierarchy 4. The period vectors 4.1. Definition and main properties 4.2. Monodromy 4.3. At a special point 4.4. Asymptotics of the period vectors for λ ∼ u i 1 2 G. CARLET, J. VAN DE LEUR, H. POSTHUMA, AND S. SHADRIN 4.5. Asymptotics of the period vectors for λ ∼ ∞ 20 5. The vertex operators 22 5.1. Vertex loop space elements 22 5.2. Asymptotics for λ ∼ u i 22 5.3. The functions W a,b (t, λ) 23 5.4. The functions c a (t, λ) 23 5.5. Splitting 23 5.6. Conjugation by R 24 5.7. Asymptotics at λ ∼ ∞ 25 5.8. Conjugation by S 25 6. The Hirota quadratic equations for the ancestor potential 26 6.1. Definition of Hirota quadratic equations for the ancestor potential 26 6.2. Proof of the ancestor HQE 28 7. The Hirota quadratic equations for the descendent potential 29 7.1. The Hirota equation 29 7.2. Hirota equations for the descendent potential 30 7.3. Explicit form of the Hirota equations 33 8. The Lax formulation of the Catalan hierarchy 34 8.1. Lax representation with difference operators 34 8.2. Lax representation via change of variable 38 8.3. Lax representation from the Hirota equations 41 References 47