Baxter Equation for Quantum Discrete Boussinesq Equation

Studied is the Baxter equation for the quantum discrete Boussinesq equation. We explicitly construct the Baxter $\mathcal{Q}$ operator from a generating function of the local integrals of motion of the affine Toda lattice field theory, and show that it solves the third order operator-valued difference equation.Comment: 25 page

Difference equation of the colored Jones polynomial for torus knot

We prove that the N-colored Jones polynomial for the torus knot T_{s,t} satisfies the second order difference equation, which reduces to the first order difference equation for a case of T_{2,2m+1}. We show that the A-polynomial of the torus knot can be derived from this difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for T_{2,2m+1}.Comment: 7 page

Supersymmetric Polychronakos Spin Chain: Motif, Distribution Function, and Character

Degeneracy patterns and hyper-multiplet structure in the spectrum of the su($m|n$) supersymmetric Polychronakos spin chain are studied by use of the "motif''. Using the recursion relation of the supersymmetric Rogers-Szeg{\"o} polynomials which are closely related to the partition function of the $N$ spin chain, we give the representation for motif in terms of the supersymmetric skew Young diagrams. We also study the distribution function for quasi-particles. The character formulae for $N\to \infty$ are briefly discussed.Comment: 24 pages + 1 figure, to appear in Nucl. Phys.

Dimensional Reduction by Conformal Bootstrap

The dimensional reductions in the branched polymer and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. The small size minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to D'=D-2 dimensional Yang-Lee edge singularity and to pure D'=D-2 dimensional Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$, and for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.Comment: 23 page, 13 figure

Hyperbolicity of Partition Function and Quantum Gravity

We study a geometry of the partition function which is defined in terms of a solution of the five-term relation. It is shown that the 3-dimensional hyperbolic structure or Euclidean AdS_3 naturally arises in the classical limit of this invariant. We discuss that the oriented ideal tetrahedron can be assigned to the partition function of string.Comment: 16 pages, 4 figure