Integral and Theta Formulae for Solutions of $s{l}_N$ Knizhnik–Zamolodchikov Equation at Level Zero

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 $s{l}_N$ 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 $s{l}_2$.