# Integral and Theta Formulae for Solutions of $sl_{N}$ Knizhnik–Zamolodchikov Equation at Level Zero

### Atsushi Nakayashiki

Kyushu University, Fukuoka, Japan

## Abstract

The solutions of the sin 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. A proof is given that those integrals satisfy $sl_{N}$ KZ equation of level 0. The relation of the integral formulae with the chiral Szegö 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 $sl_{2}$.

## Cite this article

Atsushi Nakayashiki, Integral and Theta Formulae for Solutions of $sl_{N}$ Knizhnik–Zamolodchikov Equation at Level Zero. Publ. Res. Inst. Math. Sci. 34 (1998), no. 5, pp. 439–486

DOI 10.2977/PRIMS/1195144514