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

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 $({\partial }_k+T[e_k])\Phi =0$ are given, where the elements $\{e_k\}$ are a basis of the subspace, and $T$ is the affine 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-$\sigma$ function.

42 pages, Amstex

References in corpus (6)

• The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I
• Degenerations of Generalized Krichever Novikov algebras on Tori
• The KZB equations on Riemann surfaces
• Differential Operator Algebras on compact Riemann Surfaces
• Knizhnik-Zamolodchikov-Bernard equations on Riemann surfaces
• Sugawara Construction and Casimir Operators for Krichever-Novikov Algebras

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• R-Matrix Structure of Hitchin System in Tyurin Parameterization
• Towards Vertex Algebras of Krichever-Novikov Type, Part I
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• Higher Genus Affine Lie Algebras of Krichever -- Novikov Type
• Certain families of Polynomials arising in the study of hyperelliptic Lie algebras
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• Global Deformations of the Witt Algebra of Krichever-Novikov Type
• Elements of a Global Operator Approach to Wess-Zumino-Novikov-Witten Models
• Lie superalgebras of Krichever-Novikov type and their central extensions
• DJKM algebras I: Their Universal Central Extension