FREE BOSONS AND DISPERSIONLESS LIMIT OF HIROTA TAU-FUNCTION

ON CONNECTION BETWEEN TOPOLOGICAL LANDAU-GINZBURG GRAVITY AND INTEGRABLE SYSTEMS

International Journal of Modern Physics A ◽

10.1142/s0217751x95001923 ◽

1995 ◽

Vol 10 (29) ◽

pp. 4161-4178 ◽

Cited By ~ 11

Author(s):

A. LOSEV ◽

I. POLYUBIN

Keyword(s):

Generating Function ◽

Complex Plane ◽

Integrable Systems ◽

Target Space ◽

Kp Hierarchy ◽

Tau Function ◽

Topological Theory ◽

Topological Gravity ◽

Dispersionless Limit

We study flows on the space of topological Landau-Ginzburg theories coupled to topological gravity. We argue that flows corresponding to gravitational descendants change the target space from a complex plane to a punctured complex plane and lead to the motion of punctures. It is shown that the evolution of the topological theory due to these flows is given by the dispersionless limit of KP hierarchy. We argue that the generating function of correlators in such theories is equal to the logarithm of the tau function of the generalized Kontsevich model.

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FAY-LIKE IDENTITIES OF THE TODA LATTICE HIERARCHY AND ITS DISPERSIONLESS LIMIT

Reviews in Mathematical Physics ◽

10.1142/s0129055x06002838 ◽

2006 ◽

Vol 18 (10) ◽

pp. 1055-1073 ◽

Cited By ~ 7

Author(s):

LEE-PENG TEO

Keyword(s):

Toda Lattice ◽

Tau Function ◽

Toda Hierarchy ◽

Dispersionless Limit ◽

Toda Lattice Hierarchy ◽

Hirota Equations

In this paper, we derive the Fay-like identities of tau function for the Toda lattice hierarchy from the bilinear identity. We prove that the Fay-like identities are equivalent to the hierarchy. We also show that the dispersionless limit of the Fay-like identities are the dispersionless Hirota equations of the dispersionless Toda hierarchy.

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Hamiltonian Aspects of Three-Layer Stratified Fluids

Journal of Nonlinear Science ◽

10.1007/s00332-021-09726-0 ◽

2021 ◽

Vol 31 (4) ◽

Author(s):

R. Camassa ◽

G. Falqui ◽

G. Ortenzi ◽

M. Pedroni ◽

T. T. Vu Ho

Keyword(s):

Hamiltonian Structure ◽

Linear Equations ◽

Hamiltonian Formalism ◽

Reduction Process ◽

Ideal Fluids ◽

Upper Lid ◽

Stratification Structure ◽

Special Solutions ◽

Dispersionless Limit ◽

Unbounded Fluid

AbstractThe theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

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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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Complex curve of the two-matrix model and its tau-function

Journal of Physics A General Physics ◽

10.1088/0305-4470/36/12/315 ◽

2003 ◽

Vol 36 (12) ◽

pp. 3107-3136 ◽

Cited By ~ 64

Author(s):

Vladimir A Kazakov ◽

Andrei Marshakov

Keyword(s):

Matrix Model ◽

Tau Function ◽

Complex Curve

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INTEGRABLE HIERARCHIES AND CONTACT TERMS IN u-PLANE INTEGRALS OF TOPOLOGICALLY TWISTED SUPERSYMMETRIC GAUGE THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x9900049x ◽

1999 ◽

Vol 14 (07) ◽

pp. 1001-1013 ◽

Cited By ~ 5

Author(s):

KANEHISA TAKASAKI

Keyword(s):

Integrable Hierarchies ◽

Gauge Theories ◽

Point Of View ◽

Coupling Constants ◽

Integrable Hierarchy ◽

Supersymmetric Gauge Theories ◽

Tau Function ◽

Single Time ◽

Whitham Equations ◽

Supersymmetric Gauge

The u-plane integrals of topologically twisted N=2 supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mariño and Moore's remark that the blowup formula of the u-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multitime version without spoiling the modular invariance of the blowup formula. The multitime tau function is comprised of a Gaussian factor eQ(t1,t2,…) and a theta function. The time variables tn play the role of physical coupling constants of two-observables In(B) carried by the exceptional divisor B. The coefficients qmn of the Gaussian part are identified to be the contact terms of these two-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.

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