Fredholm determinants and the {$\tau$} function for the Kadomtsev-Petviashvili hierarchy

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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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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The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

Advances in Mathematical Physics ◽

10.1155/2018/1642139 ◽

2018 ◽

Vol 2018 ◽

pp. 1-18

Author(s):

Pierre Gaillard

Keyword(s):

Rational Solutions ◽

Fredholm Determinants ◽

Infinite Hierarchy

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.

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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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