1998 Fiscal Year Final Research Report Summary
Diflenence equation versions of integrable systems and geometric structures in the background
| Project/Area Number |
09640004
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tohoku University |
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Principal Investigator |
KUROKI Gen Mathematical Institute, Tohoku University Research Associate, 大学院・理学研究科, 助手 (10234593)
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| Project Period (FY) |
1997 – 1998
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| Keywords | integrable system / Yang-Baxter equation / quantum conformal field theory / quantum group / affine Lie algebra |
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| Research Abstract |
First the investigator constructed a twisted Wess-Zumino-Witten (WZW) model on elliptic curves and found an algebro-geometric interpretation of the elliptic Gaudin model.
The twisted WZW model on elliptic curves is a conformal field theory which possesses certain non-trivial 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 Knizhnik-Zamolodchikov equations, are equal to the elliptic classical gamma-matrices of Belavin and Drinfeld.
The elliptic Gaudin model is the quantum integrable system introduced as a quasi-classical limit of a certain spin chain model. The commuting Hamiltonians of the model are also described by the elliptic classical gamma-matrices.
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 second-order elliptic Gaudin Hamiltonians can be derived from the Ward-Takahashi identity of the energy-momentum tensor defined by the Sugawara construction.
Second he constructed integrable representations of solutions of Knizhnik-Zamolodchikov-Bernard (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 gamma-operators 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.
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Research Products
(4 results)
