Integrals of motion of classical lattice sine-Gordon system

We compute the local integrals of motions of the classical limit of the lattice sine-Gordon system, using a geometrical interpretation of the local sine-Gordon variables. Using an analogous description of the screened local variables, we show that these integrals are in involution. We present some remarks on relations with the situation at roots of 1 and results on another latticisation (linked to the principal subalgebra of $\widehat{s\ell}_{2}$ rather than the homogeneous one). Finally, we analyse a module of ``screened semilocal variables'', on which the whole $\widehat{s\ell}_{2}$ acts.Comment: (references added

Two character formulas for sl2^\hat{sl_2} spaces of coinvariants

We consider $\hat{sl_2}$ spaces of coinvariants with respect to two kinds of ideals of the enveloping algebra U(sl_2\otimes\C[t]). The first one is generated by $sl_2\otimes t^N$, and the second one is generated by $e\otimes P(t), f\otimes R(t)$ where $P(t), R(t)$ are fixed generic polynomials. (We also treat a generalization of the latter.) Using a method developed in our previous paper, we give new fermionic formulas for their Hilbert polynomials in terms of the level-restricted Kostka polynomials and $q$-multinomial symbols. As a byproduct, we obtain a fermionic formula for the fusion product of $sl_3$-modules with rectangular highest weights, generalizing a known result for symmetric (or anti-symmetric) tensors.Comment: LaTeX, 22 pages; very minor change

Functional realization of some elliptic Hamiltonian structures and bosonization of the corresponding quantum algebras

We introduce a functional realization of the Hamiltonian structure on the moduli space of P-bundles on the elliptic curve E. Here P is parabolic subgroup in SL_n. We also introduce a construction of the corresponding quantum algebras.Comment: 20 pages, Amstex, minor change

Geometrical Description of the Local Integrals of Motion of Maxwell-Bloch Equation

We represent a classical Maxwell-Bloch equation and related to it positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra $n_+$ of affine Lie algebra $\hat {sl}_2$ on a Maxwell-Bloch phase space treated as a homogeneous space of $n_+$. A space of local integrals of motion is described using cohomology methods. We show that hamiltonian flows associated to the Maxwell-Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an abelian subalgebra of the nilpotent subalgebra $n_+$ on a Maxwell- Bloch phase space. Possibilities of quantization and latticization of Maxwell-Bloch equation are discussed.Comment: 16 pages, no figures, plain TeX, no macro

Gaudin model and Deligne's category

We show that the construction of the higher Gaudin Hamiltonians associated to the Lie algebra $\mathfrak{gl}_{n}$ admits an interpolation to any complex $n$. We do this using the Deligne's category $\mathcal{D}_{t}$, which is a formal way to define the category of finite-dimensional representations of the group $GL_{n}$, when $n$ is not necessarily a natural number. We also obtain interpolations to any complex $n$ of the no-monodromy conditions on a space of differential operators of order $n$, which are considered to be a modern form of the Bethe ansatz equations. We prove that the relations in the algebra of higher Gaudin Hamiltonians for complex $n$ are generated by our interpolations of the no-monodromy conditions. Our constructions allow us to define what it means for a pseudo-deifferential operator to have no monodromy. Motivated by the Bethe ansatz conjecture for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}_{n\vert n'}$, we show that a ratio of monodromy-free differential operators is a pseudo-differential operator without monodromy.Comment: 35 page

Gaudin Model, Bethe Ansatz and Critical Level

We propose a new method of diagonalization of hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these hamiltonians by restricting certain invariant functionals on tensor products of Wakimoto modules. In conformal field theory language, the eigenvectors are given by certain bosonic correlation functions. Analogues of Bethe ansatz equations naturally appear as Kac-Kazhdan type equations on the existence of certain singular vectors in Wakimoto modules. We use this construction to expalain a connection between Gaudin's model and correlation functions of WZNW models.Comment: 40 pages, postscript-file (references added and corrected