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With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representation of the partition function of the six-vertex model with a non-diagonal reflecting end under domain wall boundary condition., Comment: Latex file, 24 pages, 4 figures

With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representations of the scalar products of Bethe states of the model., Comment: Latex file, 28 pages, based on the talk given by W. -L. Yang at Statphys 24, Cairns, Australia, 19-23 July, 2010

A Higgs-Yang Mills monopole scattering spherical symmetrically along light cones is given. The left incoming anti-self-dual \alpha plane fields are holomorphic, but the right outgoing SD \beta plane fields are antiholomorphic, meanwhile the diffeomorphism symmetry is preserved with mutual inverse affine rapidity parameters \mu and \mu^{-1}. The Dirac wave function scattering in this background also factorized respectively into the (anti)holomorphic amplitudes. The holomorphic anomaly is realized by the center term of a quasi Hopf algebra corresponding to an integrable conform affine massive field. We find explicit Nahm transformation matrix(Fourier-Mukai transformation) between the Higgs YM BPS (flat) bundles (D modules) and the affinized blow up ADHMN twistors (perverse sheafs). Thus establish the algebra for the Hecke-'t Hooft operators in the Hecke correspondence of the geometric Langlands Program., Comment: Identical to the version 2 and only the acknowledgement is replaced

In this paper, we give the general forms of the minimal $L$ matrix (the elements of the $L$-matrix are $c$ numbers) associated with the Boltzmann weights of the $A_{n-1}^1$ interaction-round-a-face (IRF) model and the minimal representation of the $A_{n-1}$ series elliptic quantum group given by Felder and Varchenko. The explicit dependence of elements of $L$-matrices on spectral parameter $z$ are given. They are of five different forms (A(1-4) and B). The algebra for the coefficients (which do not depend on $z$) are given. The algebra of form A is proved to be trivial, while that of form B obey Yang-Baxter equation (YBE). We also give the PBW base and the centers for the algebra of form B., Comment: 23 pages

By extending the dressing symmetric action of IIB string in $AdS_5\times S^5$ to the $D_3$ brane, we find a gauged WZW action of Higgs Yang-Mills field including the 2-cocycle of axially anomaly. The left and right twistor structure of left and right $\alpha $-planes glue into an ambitwistor. The symmetry group of Nahm equations is central extended to an affine group, thus we explain why the spectral curve is given by affine Toda., Comment: v2: strongly revised version (especially in important section 5), notation unified, typos corrected, referrences added

In this paper we interpret the hidden symmetry of the moduli space of IIB superstring on $AdS_{5}\times S^{5}$ in terms of the chiral embedding in $AdS_{5}$, which turns to be the $\mathbb{CP}^{3}$ conformal affine Toda model. We review how the position $\mu $ of poles in the Riemann-Hilbert formulation of dressing transformation and how the value of loop parameters $\mu $ in the vertex operator of affine algebra determines the moduli space of the soliton solutions, which describes the moduli space of the Green-Schwarz superstring. We show also how this affine SU(4) symmetry affinize the conformal symmetry in the twistor space, and how a soliton string corresponds to a Robinson congruence with twist and dilation spin coefficients $\mu $ of twistor., Comment: Final version, Misprints corrected, Note added

In this paper, we motivate how the Hodge dual related with S-duality gives the hidden symmetry in the moduli space of IIB string. Utilizing the static $% \kappa $-symmetric Killing gauge, if we take the Hodge dual of the vierbeins keeping the connection invariant, the duality of Maure-Cartan equations and the equations of motion becomes manifest. Thus by twistly transforming the vierbein, we can express the BPR currents as the Lax connections by a unique spectral parameter. Then we construct the generators of the infinitesimal dressing symmetry, the related symmetric algebra becomes the affine $% gl(2,2|4)^{(1)}$, which can be used to find the classical $r$ matrix., Comment: 16 pages, no figures,typo crrected and references added

In this paper, we construct the common eigenstates of "translation" operators $\{U_{s}\}$ and establish the generalized $Kq$ representation on integral noncommutative torus $T^{2N}$. We then study the finite rotation group $G$ in noncommutative space as a mapping in the $Kq$ representation and prove a Blocking Theorem. We finally obtain the complete set of projection operators on the integral noncommutative orbifold $T^{2N}/G$ in terms of the generalized $Kq$ representation. Since projectors are soliton solutions on noncommutative space in the limit $\alpha ^{\prime}B_{ij}\to \infty (\Theta_{ij}/\alpha ^{\prime}\to 0)$, we thus obtain all soliton solutions on that orbifold $T^{2N}/G$., Comment: 31 pages

In this paper, we construct a closed form of projectors on the integral noncommutative orbifold $T^2/Z_6$ in terms of elliptic functions by $GHS$ construction. After that, we give a general solution of projectors on $% T^{2}/Z_{6}$ and $T^{2}/Z_{3}$ with minimal trace and continuous reduced matrix $M(k,q_{0})$.The projectors constructed by us possess symmetry and manifest covariant forms under $Z_{6}$ rotation. Since projectors correspond to the soliton solutions of field theory on the noncommutative orbifold, we thus present a series of corresponding manifest covariant soliton solutions., Comment: 33 pages, no figures

In this paper, we explicitly construct a series of projectors on integral noncommutative orbifold $T^2/Z_4$ by extended $GHS$ constrution. They include integration of two arbitary functions with $Z_4$ symmetry. Our expressions possess manifest $Z_{4}$ symmetry. It is proved that the expression include all projectors with minimal trace and in their standard expansions, the eigen value functions of coefficient operators are continuous with respect to the arguments $k$ and $q$. Based on the integral expression, we alternately show the derivative expression in terms of the similar kernal to the integral one.Since projectors correspond to soliton solutions of the field theory on the noncommutative orbifold, we thus present a series of corresponding solitons., Comment: 18 pages, no figure; referrences added

We study the tensor product of the {\it higher spin representations} (see the definition in Sect. 2.2) of the elliptic quantum group $E_{\tau,\eta}(sl_n)$. The transfer matrices associated with the $E_{\tau,\eta}(sl_n)$-module are exactly diagonalized by the nested Bethe ansatz method. Some special cases of the construction give the exact solution for the $Z_n$ Belavin model and for the elliptic $A_{n-1}$ Ruijsenaars-Schneider model., Comment: 23 pages, latex file, to appear in Nucl. Phys. B

After review the quantum Hall effect on the fuzzy two-sphere $S^2$ and Zhang and Hu's 4-sphere $S^4$, the incompressible quantum Hall fluid on $S^2$, $S^4$ and torus are discussed respectively. Next, the corresponding Laughlin wavefunctions on $S^2$ are also given out. The ADHM construction on $S^4$ is discussed. We also point out that on torus, the incompressible quantum Hall fluid is related to the integrable Gaudin model and the solution can be given out by the Yang Bethe ansatz., Comment: 13 pages, no figures, plain latex. Talk given by Bo-Yu Hou at the International conference of the String theory, Beijing, August 17-19, 2002

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$, we construct the basis of Hilbert space ${\ca$H}_n$ in terms of $\theta$ functions of the positions $z_i$ of $n$ solitons. The wrapping around the torus generates the algebra ${\cal A}_n$, which is the $Z_n \times Z_n$ Heisenberg group on $\theta$ functions. We find the generators $g$ of an local elliptic $su(n)$, w$transform covariantly by the global gauge transformation of ${\cal A}$By acting on ${\cal H}_n$ we establish the isomorphism of ${\cal A}_n$$g$. We embed this $g$ into the $L$-matrix of the elliptic Gaudin and$models to give the dynamics. The moment map of this twisted cotangent $su_n({\cal T})$ bundle is matched to the $D$-equation with Fayet-Illiopoulos source term, so the dynamics of the N.C. solitons becomes that of the brane. The geometric configuration $(k, u)$ of th$spectral curve ${\rm det}|L(u) - k| = 0$ describes the brane configuration, with the dynamical variables $z_i$ of N.C. solitons as$moduli $T^{\otimes n} / S_n$. Furthermore, in the N.C. Chern-Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map eqaution $the N.C. $su_n({\cal T})$ cotangent bundle with marked points. The eigenfunction of the Gaudin differential $L$-operators as the Laughli$wavefunction is solved by Bethe ansatz., Comment: 25 pages, plain latex, no figures

We develop the description of non-commutative geometry of the 4-dimensional quantum Hall fluid's theory proposed recently by Zhang and Hu. The non-commutative structure of fuzzy $S^{4}$ appears naturally in this theory. The fuzzy monopole harmonics, which are the essential elements in this non-commutative geometry, are explicitly constructed and their obeying the matrix algebra is obtained. This matrix algebra is associative. We also propose a fusion scheme of the fuzzy monopole harmonics of the coupling system from those of the subsystems, and determine the fusion rule in such fusion scheme. By products, we provide some essential ingredients of the theory of SO(5) angular momentum. In particular, the explicit expression of the coupling coefficients, in the theory of SO(5) angular momentum, are given. It is discussed that some possible applications of our results to the 4-dimensional quantum Hall system and the matrix brane construction in M-theory., Comment: latex 22 pages, no figures. some references added. some results are clarified

The relationship between Elliptic Ruijsenaars-Schneider (RS) and Calogero-Moser (CM) models with Sklyanin algebra is presented. Lax pair representations of the Elliptic RS and CM are reviewed. For n=2 case, the eigenvalue and eigenfunction for Lame equation are found by using the result of the Bethe ansatz method., Comment: 20 pages, no figures. This article is posted for archival purpose. Proceeding of International Conference on "Gauge Theory and Integrable Models" held in Kyoto University, Japan (Feb.1999)

We study the algebra ${\cal A}_n$ and the basis of the Hilbert space ${\cal H}_n$ in terms of the $\theta$ functions of the positions of $n$ solitons. Then we embed the Heisenberg group as the quantum operator factors in the representation of the transfer matrice of various integrable models. Finally we generalize our result to the generic $\theta$ case., Comment: Talk given by Bo-Yu Hou at the Joint APCTP-Nankai Symposium. Tianjin (PRC), Oct. 2001. To appear in the proceedings, to be published by Int. J. Mod. Phys. B. 7 pages, latex, no figures

For the noncommutative torus ${\cal T}$, in case of the N.C. parameter $\theta = \frac{Z}{n}$ and the area of ${\cal T}$ is an integer, we construct the basis of Hilbert space ${\cal H}_n$ in terms of $\theta$ functions of the positions $z_i$ of $n$ solitons. The loop wrapping around the torus generates the algebra ${\cal A}_n$. We show that ${\cal A}_n$ is isomorphic to the $Z_n \times Z_n$ Heisenberg group on $\theta$ functions. We find the explicit form for the local operators, which is the generators $g$ of an elliptic $su(n)$, and transforms covariantly by the global gauge transformation of the Wilson loop in ${\cal A}_n$. By acting on ${\cal H}_n$ we establish the isomorphism of ${\cal A}_n$ and $g$. Then it is easy to give the projection operators corresponding to the solitons and the ABS construction for generating solitons. We embed this $g$ into the $L$-matrix of the elliptic Gaudin and C.M. models to give the dynamics. For $\theta$ generic case, we introduce the crossing parameter $\eta$ related with $\theta$ and the modulus of ${\cal T}$. The dynamics of solitons is determined by the transfer matrix $T$ of the elliptic quantum group ${\cal A}_{\tau, \eta}$, equivalently by the elliptic Ruijsenaars operators $M$. The eigenfunctions of $T$ found by Bethe ansatz appears to be twisted by $\eta$., Comment: 26 pages, plain latex, no figure. Rewritten version. Some references added

In this paper, we set up the bi-module of the algebra ${\cal A}$ on fuzzy sphere. Based on the differential operators in moving frame, we generalize the ABS construction into fuzzy sphere case. The applications of ABS construction are investigated in various physical systems., Comment: Latex file without figure, 13 pages

We construct a realization of the central extension of super-Yangian double $DY_{\hbar}(\hat{sl}(M+1|N+1))$ at level-one in terms of free boson fields with a continuous parameter., Comment: 9 pages, latex, reference revised

The q-deformed supersymmetric t-J model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the quantum affine superalgebra $U_q[\hat{sl(2|1)}]$. We give the bosonization of the boundary states. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy., Comment: LaTex file 18 pages

The Lax pair of the Ruijsenaars-Schneider model with interaction potential of trigonometric type based on Dn Lie algebra is presented. We give a general form for the Lax pair and prove partial results for small n. Liouville integrability of the corresponding system follows a series of involutive Hamiltonians generated by the characteristic polynomial of the Lax matrix. The rational case appears as a natural degeneration and the nonrelativistic limit exactly leads to the well-known Calogero-Moser system associated with Dn Lie algebra., Comment: LaTeX2e, 14 pages; more remarks are added in the last section

We study the elliptic C_n and BC_n Ruijsenaars-Schneider models which is elliptic generalization of system given in hep-th/0006004. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parameterized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the systems corresponding to Calogero-Moser and Toda types., Comment: LaTeX2e, 25 pages, 1 table, some references added and rearranged together with misprints corrected

We study the $C_{n}$ and $BC_{n}$ Ruijsenaars-Schneider(RS) models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero-Moser systems., Comment: 20 pages, LaTeX2e, no figures

We study the C_2 Ruijsenaars-Schneider(RS) model with interaction potential of trigonometric type. The Lax pairs for the model with and without spectral parameter are constructed. Also given are the involutive Hamiltonians for the system. Taking nonrelativistic limit, we obtain the Lax pair of C_2 Calogero-Moser model., Comment: LaTeX2e, 10 pages, some misprints corrected and sections rearranged

We propose a rational quantum deformed nonlocal currents in the homogenous space $SU(2)_k/U(1)$, and in terms of it and a free boson field a representation for the Drinfeld currents of Yangian double at a general level $k=c$ is obtained. In the classical limit $\hbar \to 0$, the quantum nonlocal currents become $SU(2)_k$ parafermion, and the realization of Yangian double becomes the parafermion realization of $SU(2)_k$ current algebra., Comment: Latex, 9 pages

In this paper, we give the general crossing relation for boundary reflection matrix $R(\beta)$, which is the extension of the work given by Ghoshal and Zamolodchikov .We also use the first non-trivial extended crossing relation to determine the scaler factor of $R(\beta)$ which is the rational diagonal solution to the boundary Yang-Baxter equation in the case of l=2 and n=3., Comment: Latex 11 pages

Boundary operators and boundary ground states in sine-Gordon model with a fixed boundary condition are studied using bosonization and q-deformed oscillators.We also obtain the form-factors of this model., Comment: Latex 25pages

We derive the exchange relations of the vertex operators of $U_q(A_2^{(2)})$ and show that these vertex operators give the bosonization of the Izergin-Korepin model. We give an integral expression of the correlation functions of the Izergin-Korepin model and derive the difference equations which they satisfy., Comment: RevTex 15 pages

In the framework of the graded quantum inverse scattering method (QISM), we obtain the eigenvalues and eigenvectors of the supersymmetric $t-J$ model with reflecting boundary conditions in FFB background. The corresponding Bethe ansatz equations are obtained., Comment: Latex file, 23 Pages

We demonstrate that in a certain gauge the elliptic Ruijsenaars-Shneider model with N=2 admits a nondynamical r-matrix structure and the corresponding classical r-matrix is the same as that of its non-relativistic counterpart (Calogero-Moser model) in the same gauge.The relation between our (classical)Lax operator and the Lax operator given by Ruijsenaars is also obtained., Comment: 7 pages, Latex file 17k

From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruijsenaars'). The corresponding r-matrix is shown to be the classical $Z_n$-symmetric elliptic r-matrix, which is the same as that obtained in the study of the nonrelativistic version---the $A_{n-1}$ Calogero-Moser model., Comment: Misprints are corrected; LaTex file 15 pages

Elliptic current algebras E_{q,p}(\hat{g}) for arbitrary simply laced finite dimensional Lie algebra g are defined and their co-algebraic structures are studied. It is shown that under the Drinfeld like comultiplications, the algebra E_{q,p}(\hat{g}) is not co-closed for any g. However putting the algebras E_{q,p}(\hat{g}) with different deformation parameters together, we can establish a structure of infinite Hopf family of algebras. The level 1 bosonic realization for the algebra E_{q,p}(\hat{g}) is also established., Comment: LaTeX, 11 pages. This is the new and final version

We use factorized $L$ operator to construct an integrable model with open boundary conditions. By taking trigonometic limit($\tau \to \sqrt{-1}\infty$) and scaling limit($\omega \to 0$), we get a Hamiltonian of a classical integrable system. It shows that this integrable system is similar to those found by Calogero et al., Comment: Latex file, 17 pages

In this paper, we construct a new Lax operator for the elliptic $A_{n-1}$ Calogero-Moser model with general $n(2\leq n$) from the classical dynamical twisting,in which the corresponding r-matrix is purely numeric (nondynamical one). The nondynamical r-matrix structure of this Lax operator is obtained, which is elliptic $Z_n$-symmetric r-matrix., Comment: 15 pages, Latex file 38k

In this paper, we construct a new Lax operator for the elliptic Calogero-Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied . The relation between our Lax operator and the Lax operator given by Krichever is also obtained., Comment: 7 pages, Latex file 17k

With slight modifications in the zero modes contributions, the positive and negative screening currents for the quantum deformed W-algebra W_{q,p}(g) can be put together to form a single algebra which can be regarded as an elliptic deformation of the universal enveloping algebra of \hat{g}, where g is any classical simply-laced Lie algebra., Comment: LaTeX file, 9 pages. Errors in Serre relation corrected. Two references to Awata,H. et al added

We extend the work of Foda et al and propose an elliptic quantum algebra $A_{q,p}(\hat {sl_n})$. Similar to the case of $A_{q,p}(\hat {sl_2})$, our presentation of the algebra is based on the relation $RLL=LLR^*$, where $R$ and $R^*$ are $Z_n$ symmetric R-matrices with the elliptic moduli chosen differently and a factor is also involved. With the help of the results obtained by Asai et al, we realize type I and type II vertex operators in terms of bosonic free fields for $Z_n$ symmetric Belavin model. We also give a bosonization for the elliptic quantum algebra $A_{q,p}(\hat {sl_n})$ at level one., Comment: 17 pages, Latex file 43k

New deformed affine algebras A_{\hbar,\eta}(\hat{g}) are defined for any simply-laced classical Lie algebra g, which are generalizations of the algebra A_{\hbar,\eta}(\hat{sl_2}) recently proposed by Khoroshkin, Lebedev and Pakuliak (KLP). Unlike the work of KLP, we associate to the new algebras the structure of an infinite Hopf family of algebras in contrast to the one containing only finite number of algebras introduced by KLP. Bosonic representation for A_{\hbar,\eta}(\hat{g}) at level 1 is obtained, and it is shown that, by repeated application of Drinfeld-like comultiplications, a realization of A_{\hbar,\eta}(\hat{g}) at any positive integer level can be obtained. For the special case of g=sl_{r+1}, (r+1)-dimensional evaluation representation is given. The corresponding intertwining operators are defined and the intertwining relations are also derived explicitly., Comment: This version has been completely revised. Many corrections are made. In the title we add the word "infinite" to show the difference from Khoroshkin et al. LaTeX, 23 pages

Based on the bosonization of vertex operators for $A_{n-1}^{(1)}$ face model by Asai,Jimbo, Miwa and Pugai, using vertex-face correspondence we obtain vertex operators for Zn symmetric Belavin model,which are constructed by deformed boson oscilllators. The correlation functions are also obtained., Comment: 13 pages, Latex file

We derive a free boson representation of the Yangian double $DY_\hbar(sl_N)$ with arbitrary level k using the observation that there is a correspondence between the q-affine algebra and Yangian double associated with the same Cartan matrix. Vertex operator and screening currents are not obtained in the same way., Comment: LaTeX file, amssymb and amsart used. 13 pages, no figures. Several misprints corrected, some in important formulas

In this paper,we derive a $\hbar$-deformation of the $W_{N}$ algebra and its quantum Miura tranformation. The vertex operators for this $\hbar$-deformed $W_{N}$ algebra and its commutation relations are also obtained., Comment: 16 pages

As the Yangian double with center,which is deformed from affine algebra by the additive loop parameter $\hbar$ ,we get the commuting relation and the bosonization of quantum $\hbar$-deformed Virasoro algebra. The corresponding Miura transformation, associated screening operators and the BRST charge have been studied. Moreover, we also constructe the bosonization for type I and type II intertwiner vertex operators. Finally, we show that the commuting relation of these vertex operators in the case of $p'=r p=r-1$ and $\hbar =\pi$ actually gives the exact scattering matrix of the Restricted sine-Gordon model., Comment: 8 pages

By using the intertwiner and face-vertex correpondence relation, we obtain the Bethe ansatz equation of eight vertex model with open boundary condtitions in the framework of algebraic Bethe ansatz method. The open boundary condition under consideration is the general solution of the reflection equation for eight vertex model with only one restriction on the free parameters of the right side reflecting boundary matrix. The reflecting boundary matrices used in this paper thus may have off-diagonal elements. Our construction can also be used for the Bethe ansatz of SOS model with reflection boundaries., Comment: Latex document, several figures, 85KB

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the Wu-Kadanoff duality between the cube and vertex type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which is corresponding to the three-dimensional Baxter-Bazhanov model. The vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. And we write down the symmetry relations of the weight functions under the actions of the symmetry group $G$ of the cube. The six angles with a constrained condition, appeared in the tetrahedron equation, can be regarded as the six spectrums connected with the six spaces in which the vertex type tetrahedron equation is defined., Comment: 29 pages, latex, 8 pasted figures (Page:22-29)

{}From the cyclic quantum dilogarithm the shift operator is constructed with $q$ is a root of unit and the representation is given for the current algebra introduced by Faddeev $et ~al$. It is shown that the theta-function is factorizable also in this case by using the star-square equation of the Baxter-Bazhanov model., Comment: 9 pages, latex, no figures

In this letter we show that the restricted star-triangle relation introduced by Bazhanov and Baxter can be obtained either from the star-triangle relation of chiral Potts model or from the star-square relation which is proposed by Kashaev $et ~al$ and give a response of the guess which is suggested by Bazhanov and Baxter in Ref. \cite{b2}., Comment: 6 pages, latex file, AS-ITP-94-39

In this paper, we introduce an $N\times N$ matrix $\epsilon^{a\bar{b}}$ in the quantum groups $SU_{q}(N)$ to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on $SU_{q}(N)$, such as the $q$-deformed structure constant ${\bf C}_{IJ}^{~K}$ and the $q$-deformed transposition operator $\Lambda$. From the $q$-gauge covariant condition we define the generalized $q$-deformed Killing form and the $m$-th $q$-deformed Chern class $P_{m}$ for the quantum groups $SU_{q}(N)$. Some useful relations of the generalized $q$-deformed Killing form are presented. In terms of the $q$-deformed homotopy operator we are able to compute the $q$-deformed Chern-Simons $Q_{2m-1}$ by the condition $dQ_{2m-1}=P_{m}$, Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gauge covariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived.