Integral and theta formulae for solutions of \(sl_N\) Knizhnik-Zamolodchikov equation at level zero. (English) Zbl 0959.32028

Summary: The solution of the \(sl_N\) Knizhnik-Zamolodchikov (KZ) equations at level 0 are studied. We present the integral formula which is obtained as a quasi-classical limit of the integral formula of the form factors of the \(SU(N)\) invariant Thirring model due to F. Smirnov [Form factors in completely integrable models of quantum field theory. World Scientific, Singapore (1992; Zbl 0788.46077)]. A proof is given that those integrals satisfy the \(sl_N\) KZ equation of level 0. The relation of the integral formulae with the chiral Szegő kernel is clarified. As a consequence the integral formula with the special choice of cycles is rewritten in terms of the Riemann theta functions associated with the \(Z_N\) curve. This formula gives a generalization of Smirnov’s formula in the case of \(sl_2\).


32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
81T10 Model quantum field theories
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
17B81 Applications of Lie (super)algebras to physics, etc.
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras


Zbl 0788.46077
Full Text: DOI arXiv


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