Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries

Tsuchiya, Akihiro; Ueno, K.; Yamada, Yasuhiko

doi:10.1016/b978-0-12-385342-4.50020-2

Cited by 174 publications

(284 citation statements)

References 25 publications

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“…Laszlo [16] showed that the Hitchin connection coincides with the connection constructed by Tsuchiya, Ueno and Yamada [21] through the above identification. On the other hand Kirillov [13], [14] constructed a Hermitian product on the conformal block compatible with the Tsuchiya-Ueno-Yamada connection using the representation theory of affine Lie algebras together with the theory of hermitian modular tensor categories; cf.…”

Section: Introductionmentioning

confidence: 62%

An abelianization of the SU(2) WZW model

Yoshida

2006

Ann. Math.

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Section: Introductionmentioning

confidence: 62%

An abelianization of the SU(2) WZW model

Yoshida

2006

Ann. Math.

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“…One more direction was started by the important paper of Tsuchiya, Ueno, Yamada [38] . In contrary to the above mentioned works (except the approach of KnizhnikZamolodchikov) they constructed a projectively flat connection on the space of punctured Riemann surfaces (more precisely, on the moduli space of stable curves with marked points).…”

Section: Introductionmentioning

confidence: 99%

Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras

Schlichenmaier¹,

Sheinman²

1999

Russ. Math. Surv.

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Abstract. Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus g are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann surfaces (respectively of smooth, projective complex curves) with N marked points are introduced. It is shown that the tangent space of the moduli space at an arbitrary moduli point is isomorphic to a certain subspace of the Krichever-Novikov vector field algebra given by the data of the moduli point. This subspace is complementary to the direct sum of the two subspaces containing the vector fields which vanish at the marked points, respectively which are regular at a fixed reference point. For each representation of the affine algebra 3g − 3 + N equations (∂ k + T [e k ])Φ = 0 are given, where the elements {e k } are a basis of the subspace, and T is the Sugawara representation of the centrally extended vector field algebra. For genus zero one obtains the Knizhnik-Zamolodchikov equations in this way. The coefficients of the equations for genus one are found in terms of Weierstraß-σ function.

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“…A mathematical formulation of this model on general algebraic curves is given in [13], where the correlation functions are defined as flat sections of a certain vector bundle over the moduli space of curves. On the projective line P 1 , the correlation functions are realized more explicitly in [12] as functions which take their values in a certain finite-dimensional vector space, and characterized by the system of equations containing the Knizhnik-Zamolodchikov (KZ) equations [11].…”

Section: Introductionmentioning

confidence: 99%

“…Starting from the formulation developed in [13], we derive a system of differential equations which contains the Knizhmk-Zamolodchikov-Bernard equations [1] [9]. Our system completely determines the AT-point functions and is regarded as a natural elliptic analogue of the system obtained in [12] for the projective line.…”

Section: Introductionmentioning

confidence: 99%

Differential Equations Associated to the $SU(2)$ WZNW Model on Elliptic Curves

Suzuki¹

1996

Publ. Res. Inst. Math. Sci.

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We study the SU(2] WZNW model over a family of elliptic curves. Starting from the formulation developed in [13], we derive a system of differential equations which contains the Knizhmk-Zamolodchikov-Bernard equations [1] [9]. Our system completely determines the AT-point functions and is regarded as a natural elliptic analogue of the system obtained in [12] for the projective line. We also calculate the system for the 1-point functions explicitly. This gives a generalization of the results in [7] for si (2, C)-characters. § 0. IntroductionWe consider the Wess-Zumino-Novikov-Witten (WZNW) model. A mathematical formulation of this model on general algebraic curves is given in [13], where the correlation functions are defined as flat sections of a certain vector bundle over the moduli space of curves. On the projective line P 1 , the correlation functions are realized more explicitly in [12] as functions which take their values in a certain finite-dimensional vector space, and characterized by the system of equations containing the Knizhnik-Zamolodchikov (KZ) equations [11]. One aim in the present paper is to have a parallel description on elliptic curves. Namely, we characterize the N-point functions as vector-valued functions by a system of differential equations containing an elliptic analogue of the KZ equations by Bernard [1]. Furthermore we write down this system explicitly in the 1-pointed case.To explain more precisely, first let us review the formulation in [13] roughly. Let g be a simple Lie algebra over C and g the corresponding affine Lie algebra. We fix a positive integer / (called the level) and consider the integrable highest weight modules of g of level /. Such modules are parameterized by the set of highest weight P £ and we denote by J-f A the left module corresponding to leP^. By M 9tN we denote the moduli space of Communicated by T. Miwa, February 9, 1995. 1991 Mathematical Subject Classification (s):

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Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries

Cited by 174 publications

References 25 publications

An abelianization of the SU(2) WZW model

An abelianization of the SU(2) WZW model

Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras

Differential Equations Associated to the $SU(2)$ WZNW Model on Elliptic Curves

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