Integrable Systems and Two-Dimensional Gravity from Conformally Reduced WZNW Model

Imposing constraints with an integer ordering on WZNW model a large series of conformal invariant integrable systems will result. In this letter, a general approach for imposing the first and the second class constraints based on an arbitrary grading scheme of the Lie algebras of the WZNW groups is presented. The first order constraints correspond to integrable systems containing super Toda and conformal affine Toda systems as examples and are related to two-dimensional induced gravity, whilst the second order constraints correspond to supersymmetric-like integrable systems containing super Toda and conformal affine super Toda systems (for super WZNW groups) and are conjectured to be related to twodimensional induced supergravity.     

W-algebras for generalized Toda theories

The generalized Toda theories obtained in a previous paper by the conformal reduction of WZNW theories possess a new class ofW-algebras, namely the algebras of gauge-invariant polynomials of the reduced theories. An algorithm for the construction of base-elements for theW-algebras of all such generalized Toda theories is found, and theW-algebras for the maximalSL(N,R) generalized Toda theories are constructed explicitly, the primary field basis being identified.     

Vertex Operators – From a Toy Model to Lattice Algebras

Within the framework of the discrete Wess–Zumino–Novikov–Witten theory we analyze the structure of vertex operators on a lattice. In particular, the lattice analogues of operator product expansions and braid relations are discussed. As the main physical application, a rigorous construction for the discrete counterpart g _n $ of the group valued field g(x) is provided. We study several automorphisms of the lattice algebras including discretizations of the evolution in the WZNW model. Our analysis is based on the theory of modular Hopf algebras and its formulation in terms of universal elements. Algebras of vertex operators and their structure constants are obtained for the deformed universal enveloping algebras . Throughout the whole paper, the abelian WZNW model is used as a simple example to illustrate the steps of our construction. Received: 16 December 1996 / Accepted: 5 May 1997     

Legendre's relation and the quantum equivalence osp(4|4)1 = osp(2|2)−2 ⊕ su(2)0

Using explicit results for the four-point correlation functions of the Wess–Zumino–Novikov–Witten (WZNW) model we discuss the conformal embedding osp(4|4)₁ = osp(2|2)₋₂ ⊕ su(2)₀. This embedding has emerged in Bernard and LeClair's recent paper cond-mat/003075. Given that the osp(4|4)₁ WZNW model is a free theory with power law correlation functions, whereas the su(2)₀ and osp(2|2)₋₂ models are CFTs with logarithmic correlation functions, one immediately wonders whether or not it is possible to combine these logarithms and obtain simple power laws. Indeed, this very concern has been raised in a revised version of cond-mat/003075. In this paper we demonstrate how one may recover the free field behaviour from a braiding of the solutions of the su(2)₀ and osp(2|2)₋₂ Knizhnik–Zamolodchikov equations. We do this by implementing a procedure analogous to the conformal bootstrap programme Nucl. Phys. B 241 (1984) 333. Our ability to recover such simple behaviour relies on a remarkable identity in the theory of elliptic integrals known as Legendre's relation.     

Constrained current algebras and g/u(1) conformal field theories

Operator quantization of the WZNW theory invariant with respect to an affine Lie algebra with a constrained subalgebra is performed using Dirac's procedure. Upon quantization the initial energy-momentum tensor is replaced by the g/u(1)^d coset construction. The WZNW theory with a constrained current is equivalent to the su(2)/u(1) conformal field theory.     

ClassicalA n -W-geometry

By analyzing theextrinsic geometry of two dimensional surfaces chirally embedded inC P ⁿ (theC P ⁿ W-surface [1]), we give exact treatments in various aspects of the classical W-geometry in the conformal gauge: First, the basis of tangent and normal vectors are defined at regular points of the surface, such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the (conformal)A _n -Toda Lax pair. Since the latter is known to be intrinsically related with the W symmetries, this gives the geometrical meaning of theA _n W-Algebra. Second, W-surfaces are put in one-to-one correspondence with solutions of the conformally-reduced WZNW model, which is such that the Toda fields give the Cartan part in the Gauss decomposition of its solutions. Third, the additional variables of the Toda hierarchy are used as coordinates ofC P ⁿ . This allows us to show that W-transformations may be extended as particular diffeomorphisms of this target-space. Higher-dimensional generalizations of the WZNW equations are derived and related with the Zakharov-Shabat equations of the Toda hierarchy. Fourth, singular points are studied from a global viewpoint, using our earlier observation [1] that W-surfaces may be regarded as instantons. The global indices of the W-geometry, which are written in terms of the Toda fields, are shown to be the instanton numbers for associated mappings of W-surfaces into the Grassmannians. The relation with the singularities of W-surface is derived by combining the Toda equations with the Gauss-Bonnet theorem.     

Heterotic Toda fields

A non-left-right symmetric conformal integrable Toda field theory is constructed. It is found that the conformal algebra for this model is the product of a left chiral W_r+1 algebra and a right chiral W_r+1² algebra. The general classical solution is constructed out of the chiral vectors satisfying the so-called classical exchange algebra. In addition, we derived an explicit Wronskian type solution in relation to the constrained WZNW theory. We also showed that the A_∞ limit of this model is precisely the (B₂, C₁) flow of the standard Toda lattice hierarchy.     

A New Current Algebra and the Reflection Equation

We establish an explicit algebra isomorphism between the quantum reflection algebra for the U_q([^(sl₂)]) R{U_q(\widehat{sl_2}) R}-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).     

Entropy Treatment of Evolution Algebras

In this paper, by introducing an entropy of Markov evolution algebras, we treat the isomorphism of S-evolution algebras. A family of Markov evolution algebras is defined through the Hadamard product of structural matrices of non-negative real S-evolution algebras, and their isomorphism is studied by means of their entropy. Furthermore, the isomorphism of S-evolution algebras is treated using the concept of relative entropy.     

Factorization of Correlation Functions in Coset Conformal Field Theories

We use the conformal Ward identities to study the structure of correlation functions in coset conformal field theories. For a large class of primary fields of arbitrary g/h theory, a factorization anzatz is found. Corresponding correlation functions are explicitly expressed in terms of correlation functions of two independent WZNW theories for g and h.     

Spinor algebras

We consider supersymmetry algebras in space–times with arbitrary signature and minimal number of spinor generators. The interrelation between super Poincaré and super conformal algebras is elucidated. Minimal super conformal algebras are seen to have as bosonic part a classical semisimple algebra naturally associated to the spin group. This algebra, the Spin(s,t)-algebra, depends both on the dimension and on the signature of space–time. We also consider maximal super conformal algebras, which are classified by the orthosymplectic algebras.     

Kac-Moody symmetries of critical ground states

The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the su(2)_k=1, su(3)_k=1, and the su(4)_k=1 Kac-Moody algebra, respectively. Our approach is based on the Frenkel-Kac-Segal vertex operator construction of level-one Kac-Moody algebras.     

甲基铝氧烷纳米管分子结构的理论研究

用密度泛函理论对[(AlOMe)₂]_n、[(AlOMe)₃]_n和[(AlOMe)₄]_n (n=1～10)作为重复单元组成的甲基铝氧烷(MAO)纳米管进行了研究,计算了其所有体系的结合能和总能量．结果表明,[(AlOMe)₃]_n和[(AlOMe)₄]_n具有稳定的纳米管结构,在所有研究系统中n=3时具有最稳定的结构．考察发现,[(Al₅O₅)]_n和[(Al₇O₇)]_n两种结构的二聚体具有无规则、扭曲的结构,不能继续增长形成纳米管结构．     

Unitary construction of extended conformal algebras

Particular examples and the general structure of extended conformal symmetries in coset conformal field theories are discussed. Discrete series of unitary representations, whose existence had been previously conjectured, are constructed for a class of extended conformal algebras introduced by Fateev and Zamolodchikov (FZ). The construction is a generalisation of the coset construction of the discrete series for the superconformal algebra using the coset spaces so(N) ⊕ su(N)/so(N), for fixed N. The N = 3 series is the FZ S₃ algebra and the N = 4 series consists of two commuting copies of the superconformal algebra. A general method for analysing the extended conformal symmetries present in a particular coset theory and of constructing discrete series of representations of extended symmetry algebras is outlined.     

WZNW models and gauged WZNW models based on a family of solvable Lie algebras

A family of solvable self-dual Lie algebras that are not double extensions of Abelian algebras and, therefore, cannot be obtained through a Wigner contraction, is presented. We construct WZNW and gauged WZNW models based on the first two algebras in this family. We also analyze some general phenomena arising in such models.     

Fusion rings and geometry

The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising inN=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the typeSU(N) _k /SU(n–1) are theN=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism.Supported in part by NSF grant PHY 89-04035 supplemented by funds from NASA     

From geometric quantization to conformal field theory

Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant)r-matrices and this geometrical approach.     

The A-D-E classification of minimal andA 1 (1) conformal invariant theories

We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the Virasoro or of theA ₁ ⁽¹⁾ Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.     

Affine Toda Fields as Restriced WZNW Model

The Sine-Gordon equation is derived from the conformally invariant WZNW model by imposing constraints.The action,equation of motion,canonical equal-time Poisson braket and energy-momentum tensor of S.G.E. are obtained,and the absence of conformal invariance and the complete integrability of S.G.E. are explained.The restricted WZNW model is related to the nonlinear sigma model.In addition,the SL(n,R) affine Toda fields and the SL(2,R) conformal affine Toda fields are also derived from the restricted WZNW model.