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

Paolo Furlan, Ludmil K. Hadjiivanov, Ivan T. Todorov, Trieste (Italy) International Centre for Theoretical Physics

Published Jan 1, 1996• 13 citations
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Abstract

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 $M$, $\\bar M$. Regarded as new dynamical variables, the elements of $M$ and $\\bar M$ 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 $U_q(sl(2))\\otimes U_q(sl(2))$ invariant vectors. Monodromy invariance is also restored in a weak sense.

Related papers

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

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

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.

arXiv:hep-th/9610202•International Journal of Modern Physics A•Jan 10, 1997•13 citations

Canonical approach to 2D WZNW model, non-Abelian bosonization, and anomalies

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We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge-invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multicomponent Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete ${D}_{3}$ gauge group, are presented.

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BETHE ANSATZ AND QUANTUM GROUPS

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The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when $q$ is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact $SU(2)_q$ invariance on the lattice. The algebra of the Yang-Baxter (YB) and $SU(2)_q$ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the $SU(2)_q$ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of $SU(2)_q$. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS($p$) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of $SU(2)_q$, at $q=e^{iπ/p}$, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J

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Local canonical transformations of fermions

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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.
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.
We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space H. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1/g) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g) electric field terms describe all their interactions. In the weak coupling (g → 0) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group SO(4, 2) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.
We examine the Kogut-Susskind formulation of lattice gauge theories under the light of fermionic and bosonic degrees of freedom that provide a description useful to the development of quantum simulators of gauge-invariant models. We consider both discrete and continuous gauge groups and adopt a realistic multicomponent Fock space for the definition of matter degrees of freedom. In particular, we express the Hamiltonian of the gauge theory and the Gauss law in terms of Fock operators. The gauge fields are described in two different bases based on either group elements or group representations. This formulation allows for a natural scheme to achieve a consistent truncation of the Hilbert space for continuous groups, and provides helpful tools to study the connections of gauge theories with topological quantum double and string-net models for discrete groups. Several examples, including the case of the discrete D3{D}_{3}D3​ gauge group, are presented.
The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide quantum group representations even when qqq is a root of unity. We start by solving the six-vertex model with fixed boundary conditions (FBC) that guarantee exact SU(2)qSU(2)_qSU(2)q​ invariance on the lattice. The algebra of the Yang-Baxter (YB) and SU(2)qSU(2)_qSU(2)q​ generators turns to close. The infinite spectral parameter limit of the YB generators yields {\bf cleanly} the SU(2)qSU(2)_qSU(2)q​ generators. The Bethe Ansatz states constructed for FBC are shown to be {\bf highest weights} of SU(2)qSU(2)_qSU(2)q​. The higher level Bethe Ansatz equations (BAE, describing the physical excitations) are explicitly derived for FBC. We then solve the RSOS(ppp) models on the light--cone lattice with fixed boundary conditions by disentangling the type II representations of SU(2)qSU(2)_qSU(2)q​, at q=eiπ/pq=e^{iπ/p}q=eiπ/p, from the full SOS spectrum obtained through Algebraic Bethe Ansatz. The RSOS states are those with quantum spin $J
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.