Star-triangle equations and some properties of algebraic curves connected with the integrable chiral Potts model
type: Item
instance of: scholarly article
title: Star-triangle equations and some properties of algebraic curves connected with the integrable chiral Potts model
zbMATH Open document ID: 0703.34051
DOI: 10.1007/BF01158372
published in: Mathematical Notes
publication date: 1989-01-01T00:00:00Z
review text: The authors obtain the following results concerning the integrable model of Potts. Firstly, they calculate the explicite form of the factor R(p,q,r) appearing in the startriangle equation. The second result is the explicit expression for the elements of matrix B-periods of the curve \[ x^ Ny^ N+x^ N+y^ N+1/k^ 2=0,\text{ for } N=3. \] The third result is the equation of Piccard-Fuchs for the above curve. In general, if an algebraic curve of genus g depending upon some parameter, every integral along an arbitrary closed contour of holomorphic differential satisfies, as functions of this parameter, some ordinary linear differential equation of order 2g (equation of Piccard and Fuchs). The authors prove that for the above curve with \(N=3\), the equation can be strongly simplified and reduced to hypergeometric equations of special form, such as \[ a(1-a)\ddot z_ 1+(1-2a)\dot z_ 1-(2/9)z_ 1=0, \] where \(\dot z_ 1=dz_ 1/da\).
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 4152893
zbMATH Keywords: Piccard-Fuchs equation
zbMATH Keywords: hypergeometric equations
Wikidata QID: Q57936908
author: V. B. Matveev
author: Aleksandr Olegovich Smirnov
MaRDI profile type: Publication
cites work: Free energy of the solvable chiral Potts model.
cites work: A conjecture for the superintegrable chiral Potts model
cites work: On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations
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full work available at URL: https://doi.org/10.1007/bf01158372
OpenAlex ID: W1984339694
description: scientific article; zbMATH DE number 4152893