Geometric Solutions of the Strict KP Hierarchy

In this paper we show first of all that for solutions of the strict KP hierarchy it is sufficient to work in a standard setting. Further we discuss a minimal realization of the hierarchy and present the scaling invariance of the Lax equations of the hierarchy.

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Soliton Equations and their Algebro-Geometric Solutions

10.1017/cbo9780511543203 ◽

2008 ◽

Cited By ~ 34

Author(s):

Fritz Gesztesy ◽

Helge Holden ◽

Johanna Michor ◽

Gerald Teschl

Keyword(s):

Soliton Equations ◽

Geometric Solutions

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Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

Journal of Nonlinear Mathematical Physics ◽

10.1080/14029251.2014.975520 ◽

2014 ◽

Vol 21 (4) ◽

pp. 473-508

Author(s):

Yu Hou ◽

Engui Fan

Keyword(s):

Two Component ◽

Geometric Solutions

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Genus expansion of matrix models and ћ expansion of KP hierarchy

Journal of High Energy Physics ◽

10.1007/jhep12(2020)038 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

A. Andreev ◽

A. Popolitov ◽

A. Sleptsov ◽

A. Zhabin

Keyword(s):

Matrix Models ◽

Matrix Model ◽

Special Interest ◽

Hermitian Matrix ◽

Geometric Meaning ◽

Kp Hierarchy ◽

Hurwitz Numbers ◽

Kp Equations ◽

Genus Expansion ◽

Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

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Padé approximants on Riemann surfaces and KP tau functions

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00585-2 ◽

2021 ◽

Vol 11 (4) ◽

Author(s):

Marco Bertola

Keyword(s):

Riemann Surface ◽

Riemann Surfaces ◽

Hilbert Problem ◽

Line Bundles ◽

Fixed Degree ◽

Tau Function ◽

Tau Functions ◽

Deformation Parameters ◽

Higher Genus ◽

Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

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Intertwining operator and integrable hierarchies from topological strings

Journal of High Energy Physics ◽

10.1007/jhep05(2021)216 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Jean-Emile Bourgine

Keyword(s):

Topological Strings ◽

Vertex Operator ◽

Integrable Hierarchies ◽

Kp Hierarchy ◽

Tau Function ◽

Time Deformation ◽

Toroidal Algebra ◽

Crystal Model ◽

Melting Crystal ◽

Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

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