Diflenence equation versions of integrable systems and geometric structures in the background
Project/Area Number  09640004 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Tohoku University 
Principal Investigator 
KUROKI Gen(1998) Mathematical Institute, Tohoku University Research Associate, 大学院・理学研究科, 助手 (10234593)
長谷川 浩司 東北大, 理学(系)研究科, 講師 (30208483)

CoInvestigator(Kenkyūbuntansha) 
黒木 玄 東北大学, 大学院・理学研究科, 助手 (10234593)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,500,000 (Direct Cost : ¥3,500,000)
Fiscal Year 1998 : ¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 1997 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  integrable system / YangBaxter equation / quantum conformal field theory / quantum group / affine Lie algebra / 可積系 / 共形場理論 / ヤン・バクスター方程式 / アフィン・リー環 / 楕円R行列 / ルイセナス系 / 差分可積分系 / WZW模型 
Research Abstract 
First the investigator constructed a twisted WessZuminoWitten (WZW) model on elliptic curves and found an algebrogeometric interpretation of the elliptic Gaudin model. The twisted WZW model on elliptic curves is a conformal field theory which possesses certain nontrivial flat Lie algebra bundles on elliptic curves as gauge symmetry. Coefficients of the linear differential equations satisfied by conformal blocks of the model, the elliptic KnizhnikZamolodchikov equations, are equal to the elliptic classical gammamatrices of Belavin and Drinfeld. The elliptic Gaudin model is the quantum integrable system introduced as a quasiclassical limit of a certain spin chain model. The commuting Hamiltonians of the model are also described by the elliptic classical gammamatrices. In fact the elliptic Gaudin model can be identified with the twisted WZW model on elliptic curves at the critical level and hence the generating function of secondorder elliptic Gaudin Hamiltonians can be derived from the WardTakahashi identity of the energymomentum tensor defined by the Sugawara construction. Second he constructed integrable representations of solutions of KnizhnikZamolodchikovBernard (KZB) equations from the Wakimoto modules over an affine Lie algebra. The KZB equation is a linear differential equation of connection type with coefficients described by the dynamical elliptic classical gammaoperators and can be identified with the equation satisfied by the conformal blocks of the WZW model defined on a family of pairs of a pointed elliptic curve and a flat Lie algebra bundle. Applying the theory of the Wakimoto modules to the latter interpretation of the equation, we can obtain integrable representations of solutions of it. The integral formulas can be regarded as elliptic function versions of hypergeometric functions of several variables.

Report
(3results)
Research Output
(8results)