String Functions for Affine Lie Algebras Integrable Modules

The recursion relations of branching coefficients $k_{\xi}^{(\mu)}$ for a module $L_{\frak{g}\downarrow \frak{h}}^{\mu}$ reduced to a Cartan subalgebra $\frak{h}$ are transformed in order to place the recursion shifts $\gamma \in \Gamma_{\frak{a}\subset \frak{h}}$ into the fundamental Weyl chamber. The new ensembles $F\Psi$ (the "folded fans") of shifts were constructed and the corresponding recursion properties for the weights belonging to the fundamental Weyl chamber were formulated. Being considered simultaneously for the set of string functions (corresponding to the same congruence class $\Xi_{v}$ of modules) the system of recursion relations constitute an equation $\mathbf{M}_{(u)}^{\Xi_{v}} \mathbf{m}_{(u)}^{\mu}= \delta_{(u)}^{\mu}$ where the operator $\mathbf{M}_{(u)}^{\Xi_{v}}$ is an invertible matrix whose elements are defined by the coordinates and multiplicities of the shift weights in the folded fans $F\Psi$ and the components of the vector $\mathbf{m}_{(u)}^{\mu}$ are the string function coefficients for $L^{\mu}$ enlisted up to an arbitrary fixed grade $u$. The examples are presented where the string functions for modules of $\frak{g}=A_{2}^{(1)}$ are explicitly constructed demonstrating that the set of folded fans provides a compact and effective tool to study the integrable highest weight modules.Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Quantum Inverse Scattering Method and (Super)Conformal Field Theory

In this paper we consider the possibility of application of the quantum inverse scattering method for studying the superconformal field theory and it's integrable perturbations. The classical limit of the considered constructions is based on $\hat{osp}(1|2)$ super-KdV hierarchy. The quantum counterpart of the monodromy matrix corresponding to the linear problem associated with the L-operator is introduced. Using the explicit form of the irreducible representations of $\hat{osp}_q(1|2)$, the ``fusion relations'' for the transfer-matrices (i.e. the traces of the monodromy matrices in different representations) are obtained.Comment: LaTeX2e, 15 pages, Theor. Math. Phys., 2005, in pres

Group Theoretical Structure and Inverse Scattering Method for super-KdV Equation

Using the group-theoretical approach to the inverse scattering method the supersymmetric Korteweg-de Vries equation is obtained by application of the Drinfeld-Sokolov reduction to osp(1|2) loop superalgebra. The direct and inverse scattering problems are discussed for the corresponding Lax pair.Comment: LaTeX2e, 19 pages, Zapiski Nauchnih Seminarov POMI (Steklov Institute), vol. 291, 185-205, 2002 (in russian); Engl. transl. : Journal of Math. Sci., Kluwer, in pres

Superconformal Field Theory and SUSY N=1 KdV Hierarchy II: The Q-operator

The algebraic structures related with integrable structure of superconformal field theory (SCFT) are introduced. The SCFT counterparts of Baxter's Q-operator are constructed. The fusion-like relations for the transfer-matrices in different representations and their truncations are obtained.Comment: LaTeX2e, elsart.cls, 17 pages, Nuclear Physics B, 2005, in pres

Integrable Structure of Superconformal Field Theory and Quantum super-KdV Theory

The integrable structure of the two dimensional superconformal field theory is considered. The classical counterpart of our constructions is based on the $\hat{osp}(1|2)$ super-KdV hierarchy. The quantum version of the monodromy matrix associated with the linear problem for the corresponding L-operator is introduced. Using the explicit form of the irreducible representations of $\hat{osp}_q(1|2)$, the so-called "fusion relations" for the transfer matrices considered in different representations of $\hat{osp}_q(1|2)$ are obtained. The possible integrable perturbations of the model (primary operators, commuting with integrals of motion) are classified and the relation with the supersymmetric $\hat{osp}(1|2)$ Toda field theory is discussed.Comment: LaTeX2e, elsart.cls, 11 pages, subm. to Physics Letters