1,397 research outputs found

### 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 $\hat{sl_2}$ 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

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