Chiral WZNW Research Articles

Quasi-Hamiltonian bookkeeping of WZNW defects

We interpret the chiral WZNW model with general monodromy as an infinite dimensional quasi-Hamiltonian dynamical system. This interpretation permits to explain the totality of complicated cross-terms in the symplectic structures of various WZNW defects solely in terms of the single concept of the quasi-Hamiltonian fusion. Translated from the WZNW language into that of the moduli space of flat connections on Riemann surfaces, our result gives a compact and transparent characterization of the symplectic structure of the moduli space of flat connections on a surface with k handles, n boundaries and m Wilson lines.

Dynamical r matrices and chiral WZNW phase space

The dynamical generalization of the classical Yang-Baxter equation that governs the possible Poisson structures on the space of chiral WZNW fields with generic monodromy is reviewed. It is explained that for particular choices of the chiral WZNW Poisson brackets this equation reduces to the CDYB equation recently studied by Etingof--Varchenko and others. Interesting dynamical r-matrices are obtained for generic monodromy as well as by imposing Dirac constraints on the monodromy.

The chiral WZNW phase space as a quasi-Poisson space

It is explained that the chiral WZNW phase space is a quasi-Poisson space with respect to the ‘canonical’ Lie quasi-bialgebra which is the classical limit of Drinfeld's quasi-Hopf deformation of the universal enveloping algebra. This exemplifies the notion of quasi-Poisson–Lie symmetry introduced recently by Alekseev and Kosmann–Schwarzbach, and also permits us to generalize certain dynamical twists considered previously in this example.

Chiral extensions of the WZNW phase space, Poisson–Lie symmetries and groupoids

The chiral WZNW symplectic form Ω ρ chir is inverted in the general case. Thereby a precise relationship between the arbitrary monodromy dependent 2-form appearing in Ω ρ chir and the exchange r-matrix that governs the Poisson brackets of the group valued chiral fields is established. The exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang–Baxter (YB) equation and Poisson–Lie (PL) groupoids are constructed that encode this equation analogously as PL groups encode the classical YB equation. For an arbitrary simple Lie group G, exchange r-matrices are found that are in one-to-one correspondence with the possible PL structures on G and admit them as PL symmetries.

Classical Wakimoto realizations of chiral WZNW Bloch waves

It is well-known that the chiral WZNW Bloch waves satisfy a quadratic classical exchange algebra which implies the affine Kac-Moody algebra for the corresponding currents. We here obtain a direct derivation of the exchange algebra by inverting the symplectic form on the space of Bloch waves, and give a completely algorithmic construction of its generalized free field realizations that extend the classical Wakimoto realizations of the current algebra.

The chiral WZNW phase space and its Poisson-Lie groupoid

The precise relationship between the arbitrary monodromy dependent 2-form appearing in the chiral WZNW symplectic form and the ‘exchange r-matrix' that governs the corresponding Poisson brackets is established. Generalizing earlier results related to diagonal monodromy, the exchange r-matrices are shown to satisfy a new dynamical generalization of the classical modified Yang-Baxter equation, which is found to admit an interpretation in terms of (new) Poisson-Lie groupoids. Dynamical exchange r-matrices for which right multiplication yields a classical or a Poisson-Lie symmetry on the chiral WZNW phase space are presented explicitly.

N = 2 CHIRAL WZNW MODEL IN SUPERSPACE

We study the Poisson bracket algebra of the (N = 2)-supersymmetric chiral WZNW model in superspace. It involves two classical r matrices, one of which comes from the geometrical constraints implied by N = 2 supersymmetry. The phase space itself consists of superfields satisfying constraints involving this r matrix. An attempt is made to relax these constraints. The symmetries of the model are investigated.

Quadratic brackets from symplectic forms

We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite-dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is the appearance of quadratic Poisson brackets for group-like variables. It is believed that upon quantization they lead to quadratic exchange algebras.

Path integral derivation of characters for Kac-Moody groups

The character of an arbitrary Kac-Moody group is formally expressed as a partition function of an extension of the chiral WZNW model on a torus. Using a stationary phase approximation, I evaluate the path integral explicitly and obtain the Weyl-Kac formula for the character.

Quantum groups and WZNW models

An explanation of the appearance of quantum groups in chiral WZNW models is given. Invariance of the theory under quantum group action is discussed.