The Yang-Baxter (YB) deformations of Wess-Zumino-Witten (WZW) model on the Heisenberg Lie group ($H_4$) are examined. We proceed to obtain the nonequivalent solutions of (modified) classical Yang-Baxter equation ((m)CYBE) for the $ h_4$ Lie algebra by using its corresponding automorphism transformation. Then we show that YB deformations of $H_4$ WZW model are split into ten nonequivalent backgrounds including metric and $B$-field such that some of the metrics of these backgrounds can be transformed to the metric of $H_{4}$ WZW model while the antisymmetric $B$-fields are changed. The rest of the deformed metrics have a different isometric group structure than the $H_{4}$ WZW model metric. As an interesting result, it is shown that all new integrable backgrounds of the YB deformed $ H_4$ WZW model are conformally invariant up to two-loop order. In this way, we obtain the general form of the dilaton fields satisfying the vanishing beta-function equations of the corresponding $σ$-models.
We construct the Lax-pair, the classical monodromy matrix and the corresponding solution of the Yang--Baxter equation, for a two-parameter deformation of the Principal chiral model for a simple group. This deformation includes as a one-parameter subset, a class of integrable gauged WZW-type theories interpolating between the WZW model and the non-Abelian T-dual of the principal chiral model. We derive in full detail the Yangian algebra using two independent methods: by computing the algebra of the non-local charges and alternatively through an expansion of the Maillet brackets for the monodrom…
We show how so-called Yang-Baxter (YB) deformations of sigma models, based on an R-matrix solving the classical Yang-Baxter equation (CYBE), give rise to marginal current-current deformations when applied to the Wess-Zumino-Witten (WZW) model. For non-compact groups these marginal deformations are more general than the ones usually considered, since they can involve a non-abelian current subalgebra. We classify such deformations of the AdS(3) x S(3) string.
We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) to classify the YB deformations of WZW models based on the Lie supergroups. We obtain the inequivalent solutions (classical r-matrices) of the (m)GCYBE for the $gl(1|1)$ and $({\cal C}^3 +{\cal A})$ Lie superalgebras in the non-standard basis, in such a way that the corresponding automorphism transformations are employed. Then, the YB deform…
We show that the Wess-Zumino-Novikov-Witten (WZW) model on the Heisenberg Lie group $H_{4}$ has Jacobi-Lie symmetry with four dual Lie groups. We construct Jacobi-Lie T-dual sigma models with one of their Jacobi-Lie bialgebra and show that the original model is equivalent to the $H_{4}$ WZW model. The conformality of the dual sigma model up to one-loop order is also investigated.
A WZW model for the non-semi-simple D-dimensional Heisenberg group, which directly generalizes the E_{2}^{c} group, is constructed. It is found to correspond to an D-dimensional string background of plane-wave type with physical Lorentz signature D-2. Perturbative and non-perturbative considerations lead both to an integer central charge c=D.
We construct two-parameter families of integrable $λ$-deformations of two-dimensional field theories. These interpolate between a CFT (a WZW/gauged WZW model) and the non-Abelian T-dual of a principal chiral model on a group/symmetric coset space. In examples based on the $SU(2)$ WZW model and the $SU(2)/U(1)$ exact coset CFT, we show that these deformations are related to bi-Yang-Baxter generalisations of $η$-deformations via Poisson-Lie T-duality and analytic continuation. We illustrate the quantum behaviour of our models under RG flow. As a byproduct we demonstrate that the bi-Yang-Baxter $…
We show that the WZW model on the Heisenberg Lie group $H_4$ has Poisson-Lie symmetry only when the dual Lie group is ${ A}_2 \oplus 2{ A}_1$. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group $H_4$ and its dual pair, ${ A}_2 \oplus 2{ A}_1$, as the target space in such a way that the original model is the same as the $H_4$ WZW model. Furthermore, we show that the dual model is conformal up to two loops order. Finally, we discuss $D$-branes and the worldsheet boundary conditions defined by a gluing matrix on the $H_4$ WZW model. …
Yang-Baxter deformed sigma models on $\mathrm{SU}(N)/\mathrm{SO}(N)$
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；HSE University（高等经济学院）；Institute for Theoretical and Mathematical Physics MSU（莫斯科大学理论与数学物理研究所）
We point out the existence of nonlinear $σ$-models on group manifolds which are left symmetric and right Poisson-Lie symmetric. We discuss the corresponding rich T-duality story with particular emphasis on two examples: the anisotropic principal chiral model and the $SL(2,C)/SU(2)$ WZW model. The latter has the de Sitter space as its (conformal) non-Abelian dual.
Trigonometric degeneration of the Baxter-Belavin elliptic r matrix is described by the degeneration of the twisted WZW model on elliptic curves. The spaces of conformal blocks and conformal coinvariants of the degenerate model are factorised into those of the orbifold WZW model.
We review the notion of (anomalous) Poisson-Lie symmetry of a dynamical system and we outline the Poisson-Lie symmetric deformation of the standard WZW model from the vantage point of the twisted Heisenberg double.
The four reggeized gluon states for non-vanishing Lorentz conformal spin $n_h$ are considered. To calculate their spectrum the Q-Baxter method is used. As a result we describe normalizable trajectory-like states, which form continuous spectrum, as well as discrete point-like solutions, which turn out to be non-normalizable. The point-like solutions exist due to symmetry of the Casimir operator where conformal weights $(h,\bar h) \to (h,1-\bar h)$.
We diagonalize Q-operators for rational homogeneous sl(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.
Formulating quantum integrability for nonultralocal models (NM) parallel to the familiar approach of inverse scattering method is a long standing problem. After reviewing our result regarding algebraic structures of ultralocal models, we look for the algebra underlying NM. We propose an universal equation represented by braided Yang-Baxter equation and able to derive all basic equations of the known models like WZWN model, nonabelian Toda chain, quantum mapping etc. As further useful application we discover new integrable quantum NM, e.g. mKdV model, anyonic model, Kundu-Eckhaus equation and d…
We prove the conjecture of Sfetsos, Siampos and Thompson that suitable analytic continuations of the Poisson-Lie T-duals of the bi-Yang-Baxter sigma models coincide with the recently introduced generalized lambda models. We then generalize this result by showing that the analytic continuation of a generic sigma model of "universal WZW-type" introduced by Tseytlin in 1993 is nothing but the Poisson-Lie T-dual of a generic Poisson-Lie symmetric sigma model introduced by Klimcik and Severa in 1995.
We construct Q-operators for the open spin-1/2 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang-Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.
We analyze a completely integrable two-dimensional quantum-mechanical model that emerged in the recent studies of the compound gluonic states in multi-color QCD at high energy. The model represents a generalization of the well-known homogenous Heisenberg spin magnet to infinite-dimensional representations of the SL(2,C) group and can be reformulated within the Quantum Inverse Scattering Method. Solving the Yang-Baxter equation, we obtain the R-matrix for the SL(2,C) representations of the principal series and discuss its properties. We explicitly construct the Baxter Q-operator for this model …
We build the trigonometric solutions of the Yang-Baxter equation that can not be obtained from quantum groups in any direct way. The solution is obtained using the construction suggested recently from the rational conformal field theory corresponding to the WZW model on $SO(3)_{4 R}=SU(2)_{4 R} / Z_{2}$. We also discuss the full elliptic solution to the Yang-Baxter equation whose critical limit corresponds to the trigonometric solution found below.
We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop turns out to be very useful for several applications: we prove that the left and the right actions of a group $G$ on its twisted Heisenberg double $(D,κ)$ realize the (anomalous) Poisson-Lie symmetries and we explain in a very transparent way the concept of the Poisson-Lie subsymmetry and that of Poisson-Lie symplectic reduction. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasi-a…
We show explicitly how to construct the quantum Lax pair for systems with open boundary conditions. We demonstrate the method by applying it to the Heisenberg XXZ model with general integrable boundary terms.
We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. The Lax pair with spectral parameter is constructed by means of a quantum $R$-matrix satisfying the associative Yang-Baxter equation. Equations of motion for ${\rm gl}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when ${\rm rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models…
A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classical r-matrix satisfying the modified Yang-Baxter equation. The corresponding quantum group is studied and its R-matrix is explicitly calculated.
In this paper we study the abelian cosets of the H(4) WZW model. They coincide or are related to several interesting three-dimensional backgrounds such as the Melvin model, the conical point-particle space-times and the null orbifold. We perform a detailed CFT analysis of all the models and compute the coset characters as well as some typical three-point couplings of coset primaries.
In this paper it is shown that a quantum observable algebra, the Heisenberg-Weyl algebra, is just given as the Hopf algebraic dual to the classical observable algebra over classical phase space and the Plank constant is included in this scheme of quantization as a compatible parameter living in the quantum double theory.In this sense,the quantum Yang-Baxter equation naturally appears as a necessary condition to be satisfied by a canonical elements,the universal R-matrix,intertweening the quantum and classical observable algebras. As a byproduct,a new ``quantum group'' is obtained as the quantu…
We present two dimensional sigma model by using the Nambu structure on a manifold in general, and on a Lie group as a special case. Then, we consider model constructed from Nambu structure of order three and obtain conditions under which this model is equivalent to a WZW model. Furthermore, we present an example for this case on the four dimensional Heisenberg Lie group. Finally, as another example we show that the model constructed with Nambu structure of order three on the central extension of the 2D Poincare Lie group is integrable.
We find solutions of the Yang-Baxter equation acting on tensor product of arbitrary representations of the superalgebra sl(2|1). Based on these solutions we construct the local Hamiltonians for integrable homogeneous periodic chains and open chains.
Employing a simple calculation method obtained by M.-H. Kato, we calculate the three loop renormalization group in the $su(2)$ coset conformal field theory with a slightly relevant perturbation and the $su(2)$ Wess-Zumino-Witten model with a particular invariant marginal perturbation. Zamolodchikov's $c$-theorem, exact data of the perturbation operator and a known exact form of the operator product coefficient enable us to calculate the beta function, the gamma function and the $c$-function to three loop order. This result gives the logarithmic finite size correction to the ground state energy…
In this paper we study the diagonal cosets of the non-compact H4 WZW model. Generalising earlier work by Antoniadis and Obers, we provide an exact world-sheet description for several families of non-maximally symmetric gravitational plane waves with background NS fluxes. We show that the sigma-models that correspond to an asymmetric action of the gauge group smoothly interpolate between singular and non-singular plane waves. We also analyse the representations of the coset chiral algebra and derive the spectrum of all the models.
We present a complete description of the spectrum of compound states of reggeized gluons in QCD in multi-colour limit. The analysis is based on the identification of these states as ground states of noncompact Heisenberg SL(2,C) spin magnet. A unique feature of the magnet, leading to many unusual properties of its spectrum, is that the quantum space is infinite-dimensional and conventional methods, like the Algebraic Bethe Ansatz, are not applicable. Our solution relies on the method of the Baxter Q-operator. Solving the Baxter equations, we obtained the explicit expressions for the eigenvalue…
We compute the braided groups and braided matrices $B(R)$ for the solution $R$ of the Yang-Baxter equation associated to the quantum Heisenberg group. We also show that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra $U_{q}(h)$, and use this result to derive an action of $U_{q}(h)$ on the braided groups. We then demonstrate the various covariance properties using the braided Heisenberg group as an explicit example. In addition, the braided Heisenberg group is found to be self-dual. Finally, we discuss a physical application to a sys…
Exact conformal field theories (CFTs) are obtained by using the approach of Poisson-Lie (PL) T-duality in the presence of spectators. We explicitly construct some non-Abelian T-dual $σ$-models (here as the PL T-duality on a semi-Abelian double) on $2+2$-dimensional target manifolds $M \approx O \times \bf G$ and ${\tilde M} \approx O \times {\bf {\tilde G}}$, where $\bf G$ and ${\bf {\tilde G}}$ as two-dimensional real non-Abelian and Abelian Lie groups act freely on $M$ and $\tilde M$, respectively, while $O$ is the orbit of $\bf G$ in $M$. The findings of our study show that the original mod…
The Skyrme model, though it admits correctly a wide range of static properties of the nucleon, does not seem to reproduce properly the scattering behavior of nucleons at high energies. In this paper we present a $T\bar T$-like deformation of it, inspired by a 1+1 dimensional model, in which boosted nucleons behave like shock waves. The scattering of the latter saturates the Froissart bound. We start by showing that 1+1 dimensional $T\bar T$ deformations of the free abelian pion action are in fact generalizations of the old Heisenberg model for nucleon-nucleon scattering, yielding the same satu…
We study a boundary version of the gauged WZW model with a Poisson-Lie group G as the target. The Poisson-Lie structure of G is used to define the Wess-Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson-Lie group G^* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov-Tian-Shansky
The gauge action of the Lie group $G$ on the chiral WZNW phase space ${\cal M}_{\check G}$ of quasiperiodic fields with $\check G$-valued monodromy, where $\check G\subset G$ is an open submanifold, is known to be a Poisson-Lie (PL) action with respect to any coboundary PL structure on $G$, if the Poisson bracket on ${\cal M}_{\check G}$ is defined by a suitable monodromy dependent exchange $r$-matrix. We describe the momentum map for these symmetries when $G$ is either a factorisable PL group or a compact simple Lie group with its standard PL structure. The main result is an explicit one-to-o…
