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arXiv:hep-th/9610202 · International Journal of Modern Physics A

A Quantum Gauge Group Approach to the 2D SU(n) WZNW Model

Paolo Furlan, Ludmil K. Hadjiivanov, Ivan T. Todorov

Published Jan 10, 1997• 13 citations
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Abstract

The cannonical quantization of the WZNW model provides a complete set of exchange relation in the enlarged chiral state spaces that include the Gauss components M±, [Formula: see text] of the monodromy matrices M, [Formula: see text]. Regarded as new dynamical variables, the elements of M and [Formula: see text] cannot be identified — they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject we show that the definition of the appropriate physical subspace of the SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n = 2) case we explicitly construct the subspace of Uq(sl(2)) ⊗ Uq(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.

DOI
10.1142/s0217751x97000049

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The cannonical quantization of the WZNW model provides a complete set of exchange relation in the enlarged chiral state spaces that include the Gauss components M±, [Formula: see text] of the monodromy matrices M, [Formula: see text]. Regarded as new dynamical variables, the elements of M and [Formula: see text] cannot be identified — they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject we show that the definition of the appropriate physical subspace of the SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n = 2) case we explicitly construct the subspace of Uq(sl(2)) ⊗ Uq(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.
The canonical quantization of the WZNW model provides a complete set of exchange relations in the enlarged chiral state spaces that include the Gauss components $M_\\pm$, ${\\bar M}_\\pm$ of the monodromy matrices MMM, barM\\bar MbarM. Regarded as new dynamical variables, the elements of MMM and barM\\bar MbarM cannot be identified -- they satisfy different exchange relations. Accordingly, the two dimensional theory expressed in terms of the left and right movers' fields does not automatically respect monodromy invariance. Continuing our recent work on the subject, we show that the definition of the appropriate physical subspace of the SU(n)SU(n)SU(n) WZNW model involves an extended quantum group symmetry in a finite dimensional Fock space. In the simplest (n=2n=2n=2) case we explicitly construct the subspace of Uq(sl(2))otimesUq(sl(2))U_q(sl(2))\\otimes U_q(sl(2))Uq​(sl(2))otimesUq​(sl(2)) invariant vectors. Monodromy invariance is also restored in a weak sense.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.
By permitting canonical transformations that are nonlinear in fermion creation and annihilation operators, we show that the space of canonical transformations of ordinary spin-1/2 operators local to a point in space is SU(2)\ensuremath{\bigotimes}SU(2)\ensuremath{\bigotimes}U(1)\ensuremath{\bigotimes}Z2{\mathit{Z}}_{2}Z2​. We identify those subgroups that form local and global gauge symmetries of the Hubbard-Heisenberg model on and off half filling. Our systematic method of generating all ocal canonical transformations enables us to discover a ``nonlinear'' local U(1) gauge symmetry of the Heisenberg-Hubbard model that remains a local symmetry away from half filling. The paper presents this group together with all other known canonical transformations in a unified framework.