Chiral zero modes of the SU(n) Wess–Zumino–Novikov–Witten model

We define the chiral zero modes' phase space of the G = SU(n) Wess–Zumino–Novikov–Witten (WZNW) model as an (n − 1)(n + 2)-dimensional manifold q equipped with a symplectic form Ωq involving a Wess–Zumino term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) by This classical system exhibits a Poisson–Lie symmetry that evolves upon quantization into a Uq(sln) symmetry for q a primitive even root of 1. For each (non-degenerate, constant) solution of the classical Yang–Baxter equation we write down explicitly a ρ(M) satisfying equation () and invert the form Ωq, thus computing the Poisson bivector of the system. The resulting Poisson brackets (PB) appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously in Furlan et al (2000 Preprint hep-th/0003210). We argue that it is advantageous to equate the determinant D of the zero modes' matrix (ajα) to a pseudoinvariant under permutations q-polynomial in the SU(n) weights, rather than to adopt the familiar convention D = 1. A finite-dimensional 'Fock space' operator realization of the factor algebra q/h where h is an appropriate ideal in q for qh = −1, is briefly discussed.