Classical r-matrix and exchange algebra in WZNW and Toda theories

In this paper we study a class of theories of free particles on the complex plane satisfying a non-Abelian statistics. This kind of particles are generalizations of the anyons and are sometimes called plectons. The peculiarity of these theories is that they are associated to free conformal field theories defined on Riemann surfaces with a discrete and non-Abelian group of authomorphisms Dm. More explicitly, the plectons appear here as “induced vertex operators” that simulate, on the complex plane, the nontrivial topology of the Riemann surface. In order to express the local exchange algebra of the particles, one is led to introduce an R matrix satisfying a multiparameter generalization of the usual Yang-Baxter equations. It is interesting that analogous generalizations have already been investigated in connection with integrable models, in which the spectral parameter takes its values on a Riemann surface that is in many respects similar to the Riemann surfaces we are studying here. The explicit form of the R matrices mentioned above can be also used to define a multiparameter version of the quantum complex hyperplane.

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BOSONIC SUPERCONFORMAL AFFINE TODA THEORY: EXCHANGE ALGEBRA AND DRESSING SYMMETRY

International Journal of Modern Physics A ◽

10.1142/s0217751x93001533 ◽

1993 ◽

Vol 08 (21) ◽

pp. 3773-3789 ◽

Cited By ~ 4

Author(s):

LIU CHAO ◽

BO-YU HOU

Keyword(s):

Semiclassical Limit ◽

Group Symmetry ◽

Hamiltonian Reduction ◽

Classical Solutions ◽

Conformal Invariant ◽

Reconstruction Formula ◽

R Matrix ◽

Exchange Algebra ◽

Integrable Field Theory ◽

Integrable Field

We propose and investigate a new conformal invariant integrable field theory called bosonic superconformal affine Toda theory. This theory can be viewed either as the affine generalization of the so-called bosonic superconformal Toda theory studied by the authors sometime earlier, or as the generalization to the case of half-integer conformal weights of the conformal affine Toda theory, and can also be obtained from the Hamiltonian reduction of WZNW theory (with an affine WZNW group). The fundamental Poisson stracture is established in terms of the classical r matrix. Then the exchange algebra for the chiral vectors is obtained as well as the reconstruction formula for the classical solutions. The dressing transformations of the fundamental fields are found explicitly, and the Poisson-Lie structure of the dressing group is also constructed with the aid of classical exchange algebras, which turns out to be the semiclassical limit of the quantum affine group. The conformal breaking orbit of the model is also studied, which is called bosonic super loop Toda theory in the context. In addition, the quantum exchange relation and quantum group symmetry are discussed briefly.

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Symmetric space λ-model exchange algebra from 4d holomorphic Chern-Simons theory

Journal of High Energy Physics ◽

10.1007/jhep12(2021)004 ◽

2021 ◽

Vol 2021 (12) ◽

Author(s):

David M. Schmidtt

Keyword(s):

Symmetric Space ◽

Sigma Model ◽

Hamiltonian Formalism ◽

Simons Theory ◽

Symmetry Principle ◽

R Matrix ◽

Exchange Algebra ◽

Coset Spaces ◽

Chern Simons ◽

Chern Simons Theory

Abstract We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.

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FROM INTEGRABILITY TO CONFORMAL SYMMETRY: BOSONIC SUPERCONFORMAL TODA THEORIES

International Journal of Modern Physics A ◽

10.1142/s0217751x93000448 ◽

1993 ◽

Vol 08 (06) ◽

pp. 1105-1123 ◽

Cited By ~ 6

Author(s):

BO-YU HOU ◽

LIU CHAO

Keyword(s):

Lie Algebra ◽

Equations Of Motion ◽

Conformal Symmetry ◽

Vertex Operators ◽

Reduction Procedure ◽

R Matrix ◽

Exchange Algebra ◽

Order Constraints ◽

Lax Equations ◽

Special Case

In this paper we study the conformal integrable models obtained from conformal reductions of WZNW theory associated with second order constraints. These models are called bosonic superconformal Toda models due to their conformal spectra and their resemblance to the usual Toda theories. From the reduction procedure we get the equations of motion and the linearlized Lax equations in a generic Z gradation of the underlying Lie algebra. Then, in the special case of principal gradation, we derive the classical r matrix, fundamental Poisson relation, exchange algebra of chiral operators and find out the classical vertex operators. The result shows that our model is very similar to the ordinary Toda theories in that one can obtain various conformal properties of the model from its integrability.

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