Superstring theoryをめぐって(放談室)

* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Vertex Algebras and Algebraic Curves

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.

Quantization of Lie bialgebras, II

This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting examples of such algebras. In particular, we construct a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix, which is a quantum deformation of the affine vertex operator algebra. The simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.

Quantization of Lie bialgebras, V

Fock space representations of affine Lie algebras are studied. Explicit forms of correction terms adding to the currentsF

 i
 
(z) are determined. It is proved that the Sugawara energy-momentum tensor on the Fock spaces is quadratic in free bosons. Furthermore, screening operators are constructed. This implies the existence of generalized hypergeometric integrals satisfying the Knizhnik-Zamolodchikov equation.

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Fock Space Representations of Affine Lie Algebras and Integral Representations in the Wess-Zumino-Witten Models

Investigated is a variant of the Wess-Zumino-Witten model called a twisted WZW model, which is associated to a certain Lie group bundle on a family of elliptic curves. The Lie group bundle is a non-trivial bundle with flat connection and related to the classical elliptic r-matrix. (The usual (non-twisted) WZW model is associated to a trivial group bundle with trivial connection on a family of compact Riemann surfaces and a family of its principal bundles.) The twisted WZW model on a fixed elliptic curve at the critical level describes the XYZ Gaudin model. The elliptic Knizhnik-Zamolodchikov equations associated to the classical elliptic r-matrix appear as flat connections on the sheaves of conformal blocks in the twisted WZW model.

Twisted wess-zumino-witten models on elliptic curves

We construct an integral representation of solutions of the Knizhnik–Zamolodchikov–Bernard equations, using the Wakimoto modules.

Bosonization and Integral Representation of Solutions¶of the Knizhnik–Zamolodchikov–Bernard Equations

We shall construct the quantized q-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the B\"acklund transformations for Painlev\'e equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quantized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized B\"acklund transformations of q-Painlev\'e equations constructed by Hasegawa. We shall also prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.

Quantum groups and quantization of Weyl group symmetries of Painlev\'e systems

We shall construct the quantized q-analogues of the birational Weyl group ac- tions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Backlund transformations for Painleve equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quan- tized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Backlund transformations of q-Painleve equations constructed by Hasegawa. As a byproduct, we shall prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain.

Gen Kuroki

Papers

Fock Space Representations of Affine Lie Algebras and Integral Representations in the Wess-Zumino-Witten Models

Twisted wess-zumino-witten models on elliptic curves

Bosonization and Integral Representation of Solutions¶of the Knizhnik–Zamolodchikov–Bernard Equations

Quantum groups and quantization of Weyl group symmetries of Painlev\'e systems

Quantum groups and quantization of Weyl group symmetries of Painlevé systems