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

TL;DR

This paper explores the classical phase space of chiral zero modes in the SU(n) WZNW model, revealing a Poisson-Lie symmetry that quantizes into an U_q(sl_n) quantum group, with explicit symplectic and Poisson structures.

Contribution

It explicitly constructs the symplectic form and Poisson brackets for the zero modes, linking classical structures to quantum exchange relations and proposing a new perspective on the zero modes' determinant.

Findings

Derived explicit symplectic form for zero modes

Connected classical Poisson brackets to quantum exchange relations

Proposed determinant D as a q-polynomial pseudoinvariant

Abstract

We define the chiral zero modes' phase space of the G=SU(n) Wess-Zumino-Novikov-Witten model as an (n-1)(n+2)-dimensional manifold M_q equipped with a symplectic form involving a special 2-form - the Wess-Zumino (WZ) term - which depends on the monodromy M. This classical system exhibits a Poisson-Lie symmetry that evolves upon quantization into an U_q(sl_n) symmetry for q a primitive even root of 1. For each constant solution of the classical Yang-Baxter equation we write down explicitly a corresponding WZ term and invert the symplectic form thus computing the Poisson bivector of the system. The resulting Poisson brackets appear as the classical counterpart of the exchange relations of the quantum matrix algebra studied previously. We argue that it is advantageous to equate the determinant D of the zero modes' matrix to a pseudoinvariant under permutations q-polynomial in the SU(n) weights, rather than to adopt the familiar convention D=1.