[Paper Review] Canonical approach to the WZNW model

P. Furlan, Ludmil Hadjiivanov+1 more|arXiv (Cornell University)|Oct 27, 2014

Algebraic structures and combinatorial models124 references3 citations

TL;DR

This paper presents a canonical Hamiltonian formulation of the chiral WZNW model, establishing its Poisson-Lie symmetry and deriving its quantum group structure via covariant quantization. It demonstrates that the chiral fields satisfy an exchange algebra realizing the quantum group symmetry, with the monodromy matrix generating the quantum algebra and the zero modes realizing the restricted quantum group $\overline{U}_q(\mathfrak{sl}(2))$ for $q = e^{-i\pi/h}$, leading to braid group representations and non-abelian statistics.

ABSTRACT

The chiral Wess-Zumino-Novikov-Witten (WZNW) model provides the simplest class of rational conformal field theories which exhibit a non-abelian braid-group statistics and an associated "quantum symmetry". The canonical derivation of the Poisson-Lie symmetry of the classical chiral WZNW theory (originally studied by Faddeev, Alekseev, Shatashvili and Gawedzki, among others) is reviewed along with subsequent work on a covariant quantization of the theory which displays its quantum group symmetry.

Motivation & Objective

• To provide a canonical derivation of the Poisson-Lie symmetry in the classical chiral WZNW model, extending earlier work by Faddeev, Alekseev, Shatashvili, and Gaw{\'e}dzki.
• To establish a covariant quantization framework that realizes the quantum group symmetry of the WZNW model, particularly through the exchange algebra of chiral fields.
• To analyze the zero-mode sector and its Fock representation, showing how it realizes the restricted quantum group $\overline{U}_q(\mathfrak{sl}(2))$ for $q = e^{-i\pi/h}$.
• To demonstrate that the quantum monodromy matrix generates a quantum algebra isomorphic to the Drinfeld double, linking the classical and quantum structures.
• To explore the braid group statistics and non-integer statistical dimensions in the context of the $\widehat{su}(2)_k$ model beyond the unitary limit, using the KZ equation and extended quantum groups.

Proposed method

• Formulates the chiral WZNW model using a first-order canonical formalism with a $(D+1)$-form action, enabling the derivation of symplectic structures and Poisson brackets.
• Introduces chiral fields $g_L(x^+)$ and $g_R(x^-)$ via monodromy factorization $g(x^+,x^-) = g_L(x^+) g_R^{-1}(x^-)$, with twisted periodicity $g_C(x+2\pi) = g_C(x)M$.
• Derives the Poisson brackets for chiral fields and zero modes using the classical Yang-Baxter equation and the Drinfeld map, ensuring consistency with the $R$-matrix structure.
• Constructs the quantum exchange algebra for chiral fields $g(x)$, showing that the $R$-matrix satisfies the quantum Yang-Baxter equation and generates the quantum group symmetry.
• Applies the Fock representation to the zero-mode algebra, realizing the restricted quantum group $\overline{U}_q(\mathfrak{sl}(2))$ and its quasitriangular cover $\overline{\overline{U}}_q$.
• Uses Lusztig's extension $\widetilde{U}_q$ of $\overline{U}_q$ to define the KZ equation and braid group representations, linking quantum symmetry to topological statistics.

Experimental results

Research questions

• RQ1 How can the Poisson-Lie symmetry of the classical chiral WZNW model be derived canonically from a first-order Hamiltonian formulation?
• RQ2 What is the structure of the quantum exchange algebra for chiral fields in the WZNW model, and how does it realize quantum group symmetry?
• RQ3 How do the zero modes of the chiral WZNW model realize the restricted quantum group $\overline{U}_q(\mathfrak{sl}(2))$ for $q = e^{-i\pi/h}$, and what is the role of the monodromy matrix?
• RQ4 How does the quantum monodromy matrix generate a quantum algebra isomorphic to the Drinfeld double, and how does this relate to the quantum determinant?
• RQ5 What is the role of the KZ equation and braid group representations in the extended $\widehat{su}(2)_k$ model beyond the unitary limit?

Key findings

• The chiral WZNW model's classical Poisson brackets are derived using a canonical formalism with a $(D+1)$-form action, leading to a consistent symplectic structure and Poisson-Lie symmetry.
• The chiral fields $g(x)$ satisfy an exchange algebra with a constant $R$-matrix, realizing the quantum group symmetry $U_q(\mathfrak{sl}(2))$ in the quantum theory.
• The monodromy matrix $M$ generates a quantum algebra isomorphic to the Drinfeld double of $U_q(\mathfrak{sl}(2))$, with the quantum determinant $\det(M) = 1$ ensuring consistency.
• For $q = e^{-i\pi/h}$, the zero-mode Fock representation realizes the restricted quantum group $\overline{U}_q(\mathfrak{sl}(2))$, and its quasitriangular cover $\overline{\overline{U}}_q$ enables braid group representations.
• The KZ equation in the extended $\widetilde{U}_q$ framework yields braid group representations, confirming non-abelian statistics and non-integer statistical dimensions in the $\widehat{su}(2)_k$ model.
• The quantum determinant $\det(M) = 1$ is rigorously derived using the quantum antisymmetrizer and $R$-matrix relations, confirming the consistency of the quantum algebra structure.