205 research outputs found

### On the Space of KdV Fields

The space of functions A over the phase space of KdV-hierarchy is studied as
a module over the ring D generated by commuting derivations. A D-free
resolution of A is constructed by Babelon, Bernard and Smirnov by taking the
classical limit of the construction in quantum integrable models assuming a
certain conjecture. We propose another D-free resolution of A by extending the
construction in the classical finite dimensional integrable system associated
with a certain family of hyperelliptic curves to infinite dimension assuming a
similar conjecture. The relation of two constructions is given.Comment: 13 page

### Cohomologies of Affine Jacobi Varieties and Integrable Systems

We study the affine ring of the affine Jacobi variety of a hyper-elliptic
curve. The matrix construction of the affine hyper-elliptic Jacobi varieties
due to Mumford is used to calculate the character of the affine ring. By
decomposing the character we make several conjectures on the cohomology groups
of the affine hyper-elliptic Jacobi varieties. In the integrable system
described by the family of these affine hyper-elliptic Jacobi varieties, the
affine ring is closely related to the algebra of functions on the phase space,
classical observables. We show that the affine ring is generated by the highest
cohomology group over the action of the invariant vector fields on the Jacobi
variety.Comment: 33 pages, no figure

### Baxter equations and Deformation of Abelian Differentials

In this paper the proofs are given of important properties of deformed
Abelian differentials introduced earlier in connection with quantum integrable
systems. The starting point of the construction is Baxter equation. In
particular, we prove Riemann bilinear relation. Duality plays important role in
our consideration. Classical limit is considered in details.Comment: 28 pages, 1 figur

### Deriving bases for Abelian functions

We present a new method to explicitly define Abelian functions associated
with algebraic curves, for the purpose of finding bases for the relevant vector
spaces of such functions. We demonstrate the procedure with the functions
associated with a trigonal curve of genus four. The main motivation for the
construction of such bases is that it allows systematic methods for the
derivation of the addition formulae and differential equations satisfied by the
functions. We present a new 3-term 2-variable addition formulae and a complete
set of differential equations to generalise the classic Weierstrass identities
for the case of the trigonal curve of genus four.Comment: 35page

### The Monodromy Matrices of the XXZ Model in the Infinite Volume Limit

We consider the XXZ model in the infinite volume limit with spin half quantum
space and higher spin auxiliary space. Using perturbation theory arguments, we
relate the half infinite transfer matrices of this class of models to certain
$U_q(\hat{sl_2})$ intertwiners introduced by Nakayashiki. We construct the
monodromy matrices, and show that the one with spin one auxiliary space gives
rise to the L operator.Comment: 19 page

### A generalization of the q-Saalschutz sum and the Burge transform

A generalization of the q-(Pfaff)-Saalschutz summation formula is proved.
This implies a generalization of the Burge transform, resulting in an
additional dimension of the ``Burge tree''. Limiting cases of our summation
formula imply the (higher-level) Bailey lemma, provide a new decomposition of
the q-multinomial coefficients, and can be used to prove the Lepowsky and Primc
formula for the A_1^{(1)} string functions.Comment: 18 pages, AMSLaTe

### Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations

Solutions of the qKZ equation associated with the quantum affine algebra Uq(^sl2) and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of q-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko

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