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YANG-BAXTER AND KAC-MOODY ALGEBRAS IN FIELD THEORY WITH SOLITONS

Hector J. de Vega

Published Jan 1, 1985• 0 citations
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Abstract

The integrability of some important physical systems with a finite number of degrees of freedom relies on purely algebraic properties. One can solve the Coulomb potential and the harmonic oscillator both classically and quantum mechanically, using the appropriate dynamical algebra only [1]. Integrable systems with an infinite number of degrees of freedom are those with an infinite number of conserved quantities. They exhibit loop algebras, Yang-Baxter algebras and Kac-Moody algebras as symmetries. The investigation of theories has shown in recent years that Yang-Baxter algebras are the basic concepts of integrability [2]. The commutativity of a family of transfer matrices both in field theory and statistical mechanics follows directly from the Yang-Baxter algebra of monodromy matrices. Since the transfer matrix is a generating functional of local and non-local charges (including the hamiltonian) one gets an infinite set of commuting and conserved quantities. So, we can say that the theory is integrable. We are using the word integrable in the large sense explained above. Moreover, theories can be often explicitly solved to a large extent [2, 3]. In this brief report I will concentrate on a specific class of field theories. A complete account can be found in refs. 4-7. This class

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THE YANG–BAXTER SYMMETRY IN FIELD THEORY

Hector J. de Vega

This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of $q-$deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of $S{\hat U}(2)_q$). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix $τ(ł)$ are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit $τ(ł)$ turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].

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Functional relations and the Yang-Baxter algebra

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Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are inti-mately connected with Baxter’s concept of commuting transfer matri-ces. This concept has culminated in the celebrated Yang-Baxter equa-tion which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spec-trum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.

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The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.

International Journal of Modern Physics A•Jun 1, 1989•262 citations
The integrability of some important physical systems with a finite number of degrees of freedom relies on purely algebraic properties. One can solve the Coulomb potential and the harmonic oscillator both classically and quantum mechanically, using the appropriate dynamical algebra only [1]. Integrable systems with an infinite number of degrees of freedom are those with an infinite number of conserved quantities. They exhibit loop algebras, Yang-Baxter algebras and Kac-Moody algebras as symmetries. The investigation of theories has shown in recent years that Yang-Baxter algebras are the basic concepts of integrability [2]. The commutativity of a family of transfer matrices both in field theory and statistical mechanics follows directly from the Yang-Baxter algebra of monodromy matrices. Since the transfer matrix is a generating functional of local and non-local charges (including the hamiltonian) one gets an infinite set of commuting and conserved quantities. So, we can say that the theory is integrable. We are using the word integrable in the large sense explained above. Moreover, theories can be often explicitly solved to a large extent [2, 3]. In this brief report I will concentrate on a specific class of field theories. A complete account can be found in refs. 4-7. This class
This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of q−q-q−deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of SU^(2)qS{\hat U}(2)_qSU^(2)q​). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix τ(ł)τ(ł)τ(ł) are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit τ(ł)τ(ł)τ(ł) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].
Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the celebrated Yang-Baxter equation which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spectrum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``XXXXXXXXX spin models --- quasi-XXXXXXXXX spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables.
Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are inti-mately connected with Baxter’s concept of commuting transfer matri-ces. This concept has culminated in the celebrated Yang-Baxter equa-tion which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spec-trum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.
<!-- *** Custom HTML *** --> The Yang–Baxter algebras (YBA) are introduced and formulated in a general way stressing graphical methods. Their various physical applications are then exposed: lattice statistical models, integrable field theories and factorizable S-matrices. The Bethe Ansatz (BA) and its generalizations provide the explicit solutions of all these models using the appropiate YBA. The six-vertex model solution is exposed. YB algebras and their associated physical models are classified in terms of simple Lie algebras. It is exposed how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is exposed as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YB algebras. To conclude braid and quantum groups are derived from the YBA in the limit of infinite spectral parameter.
The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.