Quantum groups as generalized gauge symmetries in WZNW models. Part I. The classical model
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Quantum Groups as Generalized Gauge Symmetries in WZNW Models. Part I. The Classical Model1, 2 L. Hadjiivanova, * and P. Furlanb, ** a
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, BG-1784 Sofia, Bulgaria bDipartimento di Fisica dell’ Università degli Studi di Trieste, Strada Costiera 11, I-34014 Trieste, Italy *e-mail: [email protected] **e-mail: [email protected] Abstract⎯Wess–Zumino–Novikov–Witten (WZNW) models over compact Lie groups G constitute the best studied class of (two dimensional, 2D ) rational conformal field theories (RCFTs). A WZNW chiral state space is a finite direct sum of integrable representations of the corresponding affine (current) algebra, and the correlation functions of primary fields are monodromy invariant combinations of left times right sector conformal blocks solving the Knizhnik–Zamolodchikov equation. However, even in this very well understood case of 2D RCFT, the “internal” (gauge) symmetry that governs the ensuing fusion rules remains unclear. On the other hand, the canonical approach to the classical chiral WZNW theory developed by Faddeev, Alekseev, Shatashvili, Gawdzki and Falceto reveals its Poisson–Lie symmetry. After a covariant quantization, the latter gives rise to an associated quantum group symmetry which naturally requires an extension of the state space. This paper contains a review of earlier work on the subject with a special emphasis, in the case G = SU (n), on the emerging chiral “WZNW zero modes” which provide an adequate algebraic description of the internal symmetry structure of the model. Combining further left and right zero modes, one obtains a specific dynamical quantum group, the structure of its Fock representation resembling the axiomatic approach to gauge theories in which a “restricted” quantum group plays the role of a generalized gauge symmetry. DOI: 10.1134/S1063779617040049
CONTENTS 1. INTRODUCTION 1.1. Historical Perspective, Motivation and Goals 1.2. Structure of the Paper 2. 2D AND CHIRAL WZNW MODEL 2.1. Chiral Symmetry Requires a Wess–Zumino Term 2.2. Canonical Formalism 2.3. The Chiral WZNW Model 2.4. Symmetries of 2D and Chiral Symplectic Forms 2.5. Diagonalizing the Monodromy. Bloch Waves and Zero Modes 2.6. Classical r-matrices—Operator Approach 2.7. WZ 2-forms ρ(M) and Solutions of the Classical YBE 1 The article is published in the original. 2 Based in part on the D.Sc. thesis of the first author.
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2.8. Compact Notation for Multiple Tensor Products 3. EXTENDED CHIRAL PHASE SPACE AND POISSON BRACKETS 3.1. Ω(a, Mp) in Terms of Basic Right Invariant 1-forms 3.2. Extending the Zero Modes’ Phase Space 3.3. Zero Modes’ Poisson and Dirac Brackets 3.4. Poisson Brackets for the Bloch Waves 3.5. The Poisson Brackets of the Chiral Field g(x) 3.6. Symmetries of the Chiral PB 3.7. The Classical Right Movers’ Sector; “Bar” Variables 3.8. Back to the 2D WZNW Model APPENDIX A. COMP
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