A qq-anaolg of the sixth Painlev\'e equation

A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of $q$-hypergeometric functions are also discussed.Comment: 8 pages, LaTeX file (Two misprints corrected

Isomonodromic deformation with an irregular singularity and the Theta function

We will study a monodromy preserving deformation with an irregular singular point and determine the $\tau$ function of the monodromy preserving deformation by the elliptic $\theta$ function moving the argument $z$ and the period $\Omega$.Comment: 30 page

Toda Lattice Hierarchy and Zamolodchikov's Conjecture

In this letter, we show that certain Fredholm determinant $D(\lambda;t)$, introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit of soliton solution for the Toda lattice hierarchy with 2-periodic reduction condition.Comment: 6 pages, LaTeX file, no figure

Factorization of R-matrix and Baxter's Q-operator

The general rational solution of the Yang-Baxter equation with the symmetry algebra sl(2) can be represented as the product of the simpler building blocks denoted as R-operators. The R-operators are constructed explicitly and have simple structure. Using the R-operators we construct the two-parametric Baxter's Q-operator for the generic inhomogeneous periodic XXX spin chain. In the case of homogeneous XXX spin chain it is possible to reduce the general Q-operator to the much simpler one-parametric operator.Comment: 17 page

On the Construction of Trigonometric Solutions of the Yang-Baxter Equation

We describe the construction of trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two irreducible representations of a quantum algebra U_q(\G). Our method is a generalization of the tensor product graph method to the case of two different representations. It yields the decomposition of the R-matrix into projection operators. Many new examples of trigonometric R-matrices (solutions to the spectral parameter dependent Yang-Baxter equation) are constructed using this approach.Comment: latex file, 29 pages, Universitaet Bielefeld and University of Queensland preprint, BI-TP-94/13, UQMATH-94-02 (minor correction: in eq. (4.63) the number 32 should be replaced by 36 and in eq. (4.64) -16 becomes -18 and -10 becomes -8.

Free Boson Representation of qq-Vertex Operators and their Correlation Functions

A bosonization scheme of the $q$-vertex operators of \uqa for arbitrary level is obtained. They act as intertwiners among the highest weight modules constructed in a bosonic Fock space. An integral formula is proposed for $N$-point functions and explicit calculation for two-point function is presented.Comment: 22 pages, latex file, UT-618 (revised version

The BCS theory of q-deformed nucleon pairs - qBCS

We construct a coherent state of q-deformed zero coupled nucleon pairs distributed in several single-particle orbits. Using a variational approach, the set of equations of qBCS theory, to be solved self consistently for occupation probabilities, gap parameter Delta, and the chemical potential lambda, is obtained. Results for valence nucleons in nuclear degenerate sdg major shell show that the strongly coupled zero angular momentum nucleon pairs can be substituted by weakly coupled q-deformed zero angular momentum nucleon pairs. A study of Sn isotopes reveals a well defined universe of (G, q) values, for which qBCS converges. While the qBCS and BCS show similar results for Gap parameter Delta in Sn isotopes, the ground state energies are lower in qBCS. The pairing correlations in N nucleon system, increase with increasing q (for q real).Comment: 8 pages, REVTEX, 3 eps figure

An Integrable Model of Quantum Gravity

We present a new quantization scheme for $2D$ gravity coupled to an $SU(2)$ principal chiral field and a dilaton; this model represents a slightly simplified version of stationary axisymmetric quantum gravity. The analysis makes use of the separation of variables found in our previous work [1] and is based on a two-time hamiltonian approach. The quantum constraints are shown to reduce to a pair of compatible first order equations, with the dilaton playing the role of a ``clock field''. Exact solutions of the Wheeler-DeWitt equation are constructed via the integral formula for solutions of the Knizhnik-Zamolodchiokov equations.Comment: 12 page

Free Boson Realization of Uq(slN^)U_q(\widehat{sl_N})

We construct a realization of the quantum affine algebra $U_q(\widehat{sl_N})$ of an arbitrary level $k$ in terms of free boson fields. In the $q\!\rightarrow\! 1$ limit this realization becomes the Wakimoto realization of $\widehat{sl_N}$. The screening currents and the vertex operators(primary fields) are also constructed; the former commutes with $U_q(\widehat{sl_N})$ modulo total difference, and the latter creates the $U_q(\widehat{sl_N})$ highest weight state from the vacuum state of the boson Fock space.Comment: 24 pages, LaTeX, RIMS-924, YITP/K-101