The main aim of this project is development of the theory of low-dimensional conformal and massive integrable models. The principal goals are: 1) further development of the technique of computation of correlation functions and form-factors; 2) investigation of quantum symmetries of integrable models.
2.2 Background
Integrable 1+1-dimensional models are of large importance for the modern physics since they preserve many features of 3+1-dimensional models of the quantum field theory (e.g., the mass renormalization, the asymptotic freedom, the dimensional transmutation phenomenon) and, on the other hand, they can be studied in non-perturbative ways. Also, some one-dimensional integrable models, in particular, various spin chains are closely related to two-dimensional exactly solvable lattice models that have many applications in the statistical physics. Conformal models that are specific massless limits of massive integrable models play a distinguished role in this context. Furthermore, 1+1-dimensional integrable models, especially the conformal models, have numerous applications in the condensed matter physics (e.g., in the theory of the quantum Hall effect) and in the string theory. Finally, the development of the integrable models is intimately connected with the development of the quantum groups - the new branch of mathematics.
Unlike the conformal field theory, massive integrable models do not possess a universal symmetry structure. This makes their investigation much more involved. Especially difficult task is calculation of correlation functions that give more detailed information about the system. Although the algebraic Bethe ansatz allows us to do this in principle, the real computations are quite cumbersome. As an alternative, one may start with computing the corresponding form-factors first. However, there is no systematic way for the form-factor calculations yet. This motivates the first goal of the present project:
1) to develop further the technique of computation of correlation functions and form-factors for conformal and off-critical (i.e. massive) integrable models. The second goal is:
2) to extend our knowledge of quantum symmetries of massive and conformal integrable models.
The motivation for this goal is many-fold. In particular, for massive integrable models such a knowledge is very instrumental in finding certain characteristics of a model in question, e.g. the mass spectrum. Models possessing the same or similar symmetry structure can often be treated by the same methods, e.g. the algebraic Bethe ansatz. Moreover, one can construct a new integrable model appropriately deforming the symmetry of a known model (for instance, the XXX, XXZ and XYZ spin chains). In the conformal field theory the knowledge of symmetry is also very important. It allows to classify conformal models, to describe their field content, to find general equations for their correlation functions. Furthermore, deformations and extensions of the conformal symmetries (q-deformations of the Virasoro algebra, the W-algebras) appear as dynamical symmetries in certain off-critical models. Finally, investigations of quantum symmetries of integrable models produce a lot of new mathematical results (in particular, in the domain of quantum groups and W-algebras).
Computation of correlation functions and form-factors.
The following sub-tasks are planned:
1.1 to construct new, simpler, representations of multi-point form-factors for various operators in the quantum sine-Gordon theory employing the off-shell Bethe ansatz technique;
1.2 to calculate numerically various two-point correlation functions in the sine-Gordon and massive Thirring models;
1.3 to focus on the breather sector of the sine-Gordon theory and apply the general method to the computation of the form-factors related to various affine Toda field theories.
1.4 to compute exactly the expectation values of exponential fields in the super-symmetric sine-Gordon and perturbed super-symmetric minimal models using already known reflection amplitudes of the super Liouville model.
1.5 to derive integral representations for correlation functions of the WZNW on a torus by means of the off-shell Bethe ansatz method and generalize this construction to higher genus Riemann surfaces;
1.6 to construct Bethe ansatz solution of Gaudin magnets corresponding to the exceptional Lie algebras;
1.7 to construct vertex type elliptic generalization of the KZ equation, to solve it with the help of the off-shell Bethe ansatz, and to study its connection (a gauge transformation) with the elliptic Gaudin magnet.
1.8 to calculate correlation functions of spin chains with non-periodic boundary conditions at the free fermion points employing the recently found twisting elements for certain quantum super-algebras.
Investigation of symmetries of integrable models.
The following sub-tasks are planned:
2.1 to construct a q-deformation of the super-symmetric Virasoro algebra and to develop its bosonization at higher (>1) levels;
2.2 to study extensions of Wq,p(sl(n)) algebra, obtained from the elliptic algebra, by defining and solving the corresponding cocycle equation.
2.3 to investigate fermionic sum forms and product forms of characters for super-symmetric conformal models.
2.4 to find twisting elements for finite dimensional quantum algebras and super-algebras of low rank and investigate changes of physical characteristics in models related by these twists;
2.5 to study transformations of dual Hopf algebras generated by twisting of original quasi-triangular Hopf algebras.
2.6 to construct integrable deformations of the spherical top associated with outer automorphisms of the Lie algebra e(3,C);
2.7 to develop non-canonical Sturm transformations for the Toda lattices and the Stackel systems;
2.8 to investigate generalizations of the Stackel systems related to the covering of hyper-elliptic curves.
2.9 to study representations of the Weyl algebras arising in 1+1 dimensional lattice models and to establish their connection to the Tomita theory of modular algebras;
2.10 to construct lattice evolution operators for models related to higher rank quantum algebras.
The collaboration between the groups is planned in the following fields:
a) the Berlin and the Yerevan teams -- the form-factor program;
b) the Annecy-le-Vieux and the Yerevan teams -- the dynamical symmetries of off-critical models, i.e. deformed Virasoro and W-algebras;
c) the Annecy-le-Vieux and the St.Petersburg teams -- applications of the quantum groups to integrable models;
d) the Berlin and the St.Petersburg teams -- the characters of the conformal and the super-conformal models.
3.2 Deliverables, Exploitation & Dissemination of Results
Regular reports on the work progress will be sent to INTAS. During the whole project duration the results will be reported at local scientific seminars of the institutions to which the teams belong and at conferences and workshops in which team members will participate. The results will be published as local preprints, electronic preprints and then in international journals.
Report Schedule
Reports must be submitted to the INTAS Secretariat as set out in Article 9 of the General Conditions (Annex II) in the format specified in the INTAS Procedures and Guidelines updated regularly on the Internet site
http://www.intas.be
It is planned to organize the coordination meeting of all the teams in Berlin (March 2000) and then to hold workshops in St.Petersburg (September 2000), Annecy-le-Vieux (May 2001), and Yerevan (October 2001). In addition, members of different teams will have short-term visits to the institutions of their co-workers in order to make joint research more effective. Distribution of information related to the project as well as discussions concerning the concrete tasks will be held between the teams by e-mail.