scholarly journals Tau-Functions and Monodromy Symplectomorphisms

2021 ◽  
Author(s):  
M. Bertola ◽  
D. Korotkin
Keyword(s):  
Generating Function ◽  
Symplectic Form ◽  
Hamiltonian Formulation ◽  
Matrix Structure ◽  
Tau Function ◽  
Tau Functions ◽  
Log Canonical ◽  
Monodromy Map

AbstractWe derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock–Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo–Miwa–Ueno tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

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.

scholarly journals A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

2018 ◽  
Vol 07 (04) ◽  
pp. 1840001
Author(s):  
A. N. W. Hone ◽  
F. Zullo
Keyword(s):  
Taylor Series ◽  
Taylor Expansion ◽  
Tau Function ◽  
Tau Functions ◽  
Sigma Function ◽  
Special Cases ◽  
Hirota Bilinear Equation ◽  
Order Four ◽  
Hirota Bilinear ◽  
Bilinear Equation

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

scholarly journals Tau Function and Moduli of Meromorphic Quadratic Differential

2022 ◽  
Author(s):  
Dmitry Korotkin ◽  
◽  
Peter Zograf ◽  
Keyword(s):  
Moduli Spaces ◽  
Quadratic Differential ◽  
Algebraic Curves ◽  
Picard Group ◽  
Quadratic Differentials ◽  
Tau Function ◽  
Tau Functions

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

scholarly journals BKP tau-functions as square roots of KP tau-functions

2021 ◽  
Author(s):  
Johan W van de Leur
Keyword(s):  
Recent Result ◽  
Square Root ◽  
Tau Function ◽  
Tau Functions ◽  
Square Roots

Abstract It is well-known that a BKP tau-function is the square root of a certain KP tau-function, provided one puts the even KP times equal to zero. In this paper we compute for all polynomial BKP tau-function its corresponding KP ”square”. We also give, in the polynomial case, a representation theoretical proof of a recent result by Alexandov, viz. that a KdV tau-function becomes a BKP tau-function when one divides all KdV times by 2.

scholarly journals NEW SOLVABLE MATRIX INTEGRALS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 276-293 ◽  
Author(s):  
A. YU. ORLOV
Keyword(s):  
Complex Matrices ◽  
Tau Function ◽  
Tau Functions ◽  
Matrix Argument ◽  
Matrix Integrals ◽  
The Matrix

We generalize the Harish-Chandra-Itzykson-Zuber and certain other integrals (the Gross-Witten integral, the integrals over complex matrices and the integrals over rectangle matrices) using a notion of the tau function of the matrix argument. In this case one can reduce multi-matrix integrals to integrals over eigenvalues, which in turn are certain tau functions. We also consider a generalization of the Kontsevich integral.

scholarly journals ON CONNECTION BETWEEN TOPOLOGICAL LANDAU-GINZBURG GRAVITY AND INTEGRABLE SYSTEMS

1995 ◽  
Vol 10 (29) ◽  
pp. 4161-4178 ◽  
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.

Applications in random matrix theory of a PIII’ τ-function sequence from Okamoto’s Hamiltonian formulation

2021 ◽  
pp. 2250014
Author(s):  
Dan Dai ◽  
Peter J. Forrester ◽  
Shuai-Xia Xu
Keyword(s):  
Generating Function ◽  
Bessel Functions ◽  
Matrix Theory ◽  
Logarithmic Derivative ◽  
Hamiltonian Formulation ◽  
Linear Statistic ◽  
Toeplitz Determinants ◽  
Hard Edge ◽  
Toeplitz Determinant ◽  
Function Sequence

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

scholarly journals Asymptotic symmetries of Yang-Mills fields in Hamiltonian formulation

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Roberto Tanzi ◽  
Domenico Giulini
Keyword(s):  
Symplectic Form ◽  
Hamiltonian Formulation ◽  
Hamiltonian Formalism ◽  
Poincaré Group ◽  
Asymptotic Symmetry ◽  
Abelian Gauge ◽  
Poincare Group ◽  
Yang Mills ◽  
Clear Cut ◽  
Asymptotic Symmetries

Abstract We investigate the asymptotic symmetry group of the free SU(N )-Yang-Mills theory using the Hamiltonian formalism. We closely follow the strategy of Henneaux and Troessaert who successfully applied the Hamiltonian formalism to the case of gravity and electrodynamics, thereby deriving the respective asymptotic symmetry groups of these theories from clear-cut first principles. These principles include the minimal assumptions that are necessary to ensure the existence of Hamiltonian structures (phase space, symplectic form, differentiable Hamiltonian) and, in case of Poincaré invariant theories, a canonical action of the Poincaré group. In the first part of the paper we show how these requirements can be met in the non-abelian SU(N )-Yang-Mills case by imposing suitable fall-off and parity conditions on the fields. We observe that these conditions admit neither non-trivial asymptotic symmetries nor non-zero global charges. In the second part of the paper we discuss possible gradual relaxations of these conditions by following the same strategy that Henneaux and Troessaert had employed to remedy a similar situation in the electromagnetic case. Contrary to our expectation and the findings of Henneaux and Troessaert for the abelian case, there seems to be no relaxation that meets the requirements of a Hamiltonian formalism and allows for non-trivial asymptotic symmetries and charges. Non-trivial asymptotic symmetries and charges are only possible if either the Poincaré group fails to act canonically or if the formal expression for the symplectic form diverges, i.e. the form does not exist. This seems to hint at a kind of colour-confinement built into the classical Hamiltonian formulation of non-abelian gauge theories.

The 2-Component BKP Grassmannian and Simple Singularities of Type D

2019 ◽  
Author(s):  
Jipeng Cheng ◽  
Todor Milanov
Keyword(s):  
The Other ◽  
Simple Singularity ◽  
Tau Function ◽  
Tau Functions ◽  
Gaussian Type ◽  
Type D ◽  
Other Hand ◽  
Bkp Hierarchy ◽  
Simple Singularities

Abstract It was proved in 2010 that the principal Kac–Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac–Wakimoto hierarchy. We find explicitly the point in the Grassmannian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.

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