Operator realization of the SU(2) WZNW model

Abstract

Decoupling the chiral dynamics in the canonical approach to the WZNW model requires an extended phase space that includes left and right monodromy variables. Earlier work on the subject, which traced back the quantum qroup symmetry of the model to the Lie-Poisson symmetry of the chiral symplectic form, left some open questions: - How to reconcile the monodromy invariance of the local 2D group valued field (i.e., equality of the left and right monodromies) with the fact that the latter obey different exchange relations? - What is the status of the quantum group symmetry in the 2D theory in which the chiral fields commute? - Is there a consistent operator formalism in the chiral and in the extended 2D theory in the continuum limit? We propose a constructive affirmative answer to these questions for G=SU(2) by presenting the chiral quantum fields as sums of chiral vertex operators and q-Bose creation and annihilation operators.Comment: 18 pages, LATE

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

“…, generated by multiple of h powers of a i α (Section 2.3). Unlike earlier work [14,4], here we do not set to zero the maximal ideal I h , thus admitting indecomposable representations of U q in the Fock space representation of A q displayed in Section 2.3.…”

“…Our starting point is the algebra of the zero modes a = (a i α ) of a chiral group valued field [1,20,23,14] and its Fock space representation. The quantum matrix a intertwines chiral vertex operators (with diagonal monodromy) and quantum group covariant chiral fields.…”

Quantum matrix algebra for theSU(n) WZNW model

“…Chiral fields admit an expansion into chiral vertex operators (CVO) [63] which diagonalize the monodromy and are expressed in terms of the currents' degrees of freedom with "zero mode" coefficients that are independent of the world sheet coordinate [2,15,35,36,38]. Such a type of quantum theory has been studied in the framework of lattice current algebras (see [28,40,30,3,16] and references therein).…”

“…Their operator interpretation exhibits some puzzling features: the presence of noninteger ("quantum") statistical dimensions (that appear as positive real solutions of the fusion rules [64]) contrasted with the local ("Bose") commutation relations (CR) of the corresponding 2D fields. The gradual understanding of both the factorization property and the hidden braid group statistics (signaled by the quantum dimensions) only begins with the development of the canonical approach to the model (for a sample of references, see [6,27,28,40,30,8,34,35,36,38]) and the associated splitting of the basic group valued field g : S 1 × R → G into chiral parts. The resulting zero mode extended phase space displays a new type of quantum group gauge symmetry: on one hand, it is expressed in terms of the quantum universal enveloping algebra U q (G) , a deformation of the finite dimensional Lie algebra G of G -much like a gauge symmetry of the first kind; on the other, it requires the introduction of an extended, indefinite metric state space, a typical feature of a (local) gauge theory of the second kind.…”

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

“…Among these structures are the quadratic exchange algebras that encode the Poisson brackets (PBs) of the chiral group valued fields, g C (x C ) for C = L, R, which yield the general solution of the WZNW field equation as g(x L , x R ) = g L (x L )g −1 R (x R ). These exchange algebras were investigated intensively at the beginning of the decade ( [4] - [15]) motivated by the idea to understand the quantum group properties of the WZNW model [16] directly by means of canonical quantization [17,18,19]. In accordance with the general philosophy of quantum groups [20], the Poisson-Lie (PL) symmetries of the chiral fields should be the most relevant in this respect.…”