Genus expansion of matrix models and ћ expansion of KP hierarchy

Andreev, A. A.; Popolitov, A.; Слепцов, Алексей Васильевич; Zhabin, A.N.

doi:10.1007/jhep12(2020)038

Cited by 8 publications

(18 citation statements)

References 78 publications

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“…All mentioned examples of τ -functions and many others have a geometric expansion over compact Riemann surfaces (genus expansion). Genus expansion for τ -functions coincides with expansion in parameter h for the h-KP hierarchy [14]. The introduction of the h parameter slightly modifies the hierarchy and allows one, among other things, to obtain solutions of the classical KP hierarchy for h = 1 and dispersionless KP for h → 0 [15,16].…”

Section: Introductionmentioning

confidence: 76%

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Combinatorics of KP hierarchy structural constants

et al. 2021

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We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on $$\tau $$ τ -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study.

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Section: Introductionmentioning

confidence: 76%

“…From the other hand the Fay identity for h-KP hierarchy has the form (14). Now, using replacement z i → 1 y i and the fact that ∂ 1 = ∂ x = ∂ we obtain exactly (30).…”

Section: Connection With Kp Hierarchymentioning

confidence: 91%

Combinatorics of KP hierarchy structural constants

et al. 2021

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“…It is known from [12] that in addition to the standard set of Hamiltonians generating the flows of ∂/∂t 1 d , there is a an additional set of commuting Hamiltonians. The corresponding flows extend the tau-function Z, and this extension is identified with D, where ∂/∂t 2 d are the flows of this additional set of Hamiltonians.…”

Section: Digression 3 (On Special Integrability)mentioning

confidence: 99%

“…It is proved in[36, Theorem 5.2] that Z is a tau-function of the KP hierarchy. (More precisely, one should speak of -KP hierarchy in the sense of[52,56] for = 2 , see[2]. )…”

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confidence: 99%

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

et al. 2021

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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.

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“…. It is proved in [30,Theorem 5.2] that Z is a tau-function of the KP-hierarchy (more precisely, one should speak of -KP hierarchy in the sense of [48,44]for = ǫ 2 , see [2]). In particular, it is proved in [30, Theorem 5.2] that Z takes the following form:…”

Section: 4mentioning

confidence: 99%

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

Carlet,

van de Leur,

Posthuma

et al. 2020

Preprint

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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

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Cited by 8 publications

References 78 publications

Combinatorics of KP hierarchy structural constants

Combinatorics of KP hierarchy structural constants

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

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

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