Ramanujan’s Tau Function

2017 ◽  
pp. 311-328 ◽  
Author(s):  
Michael D. Hirschhorn
Keyword(s):  
Tau Function

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 Intertwining operator and integrable hierarchies from topological strings

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.

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

2003 ◽  
Vol 36 (12) ◽  
pp. 3107-3136 ◽  
Author(s):  
Vladimir A Kazakov ◽  
Andrei Marshakov
Keyword(s):  
Matrix Model ◽  
Tau Function ◽  
Complex Curve

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

1999 ◽  
Vol 14 (07) ◽  
pp. 1001-1013 ◽  
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.

scholarly journals Construction of KdV flow I. Tau-Function via Weyl Function

2018 ◽  
Vol 14 (3) ◽  
pp. 297-335 ◽  
Author(s):  
SHINICHI KOTANI ◽  
Keyword(s):  
Weyl Function ◽  
Tau Function

On the Hirota Representation of Soliton Equations with One Tau-Function

2001 ◽  
Vol 70 (3) ◽  
pp. 605-608 ◽  
Author(s):  
Franklin Lambert ◽  
Ignace Loris ◽  
Johan Springael ◽  
Ralph Willox
Keyword(s):  
Tau Function ◽  
Soliton Equations

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.

NONVANISHING OF THE RAMANUJAN TAU FUNCTION IN SHORT INTERVALS

2005 ◽  
Vol 01 (01) ◽  
pp. 45-51 ◽  
Author(s):  
EMRE ALKAN ◽  
ALEXANDRU ZAHARESCU
Keyword(s):  
Delta Function ◽  
Gap Function ◽  
Tau Function ◽  
Short Intervals ◽  
Ramanujan Tau Function

We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals.

Bilinear approach to supersymmetric AKNS system; multiple dressing of fermionic amplitudes

2020 ◽  
Vol 35 (17) ◽  
pp. 2050143 ◽  
Author(s):  
Corina N. Babalic ◽  
A. S. Carstea
Keyword(s):  
Bilinear Form ◽  
Tau Function ◽  
Bilinear Formalism ◽  
Akns System ◽  
Bilinear Approach

In this paper, we investigate the super-bilinear formalism and interaction of supersoliton solutions for the supersymmetric AKNS system [H. Aratyn and A. Das, Mod. Phys. Lett. A 13, 1185 (1998)]. It is shown that the super-bilinear form involves an auxiliary fermionic tau function and the supersolitons have less freedom than expected (concerning the dependence on fermionic parameters). Also, the interaction between super-solitons is much more complicated than in the case of real supersymmetric equations (KdV, mKdV, Sawada–Kotera, Sine–Gordon, etc.), involving four types of dressing of the fermionic parameters. These solutions are general compared to the ones found in Ref. 1 which contain no fermionic parameter.

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