Chiral zero modes of the SU(n) WZNW 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 manifoldMq equipped with a symplectic form Ωq involving a Wess-Zumino (WZ) term ρ which depends on the monodromy M and is implicitly defined (on an open dense neighbourhood of the group unit) by dρ(M) = 1 3 tr (MdM) . (∗) This classical system exhibits a Poisson-Lie symmetry that evolves upon quantization into an Uq(sln) symmetry for q a primitive even root of 1 . For each (non-degenerate, constant) solution of the classical Yang-Baxter equation (CYBE) we write down explicitly a ρ(M) satisfying Eq.(∗) 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 [31]. We argue that it is advantageous to equate the determinant D of the zero modes’ matrix (aα) to a pseudoinvariant under permutations 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 Mq/Ih , where Ih is an appropriate ideal in Mq for qh = −1 , is briefly discussed.