arXiv subjects
Publications and source records attributed to Kazuhiro Hikami.
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.
We reveal an intimate connection between the quantum knot invariant for torus knot T(s,t) and the character of the minimal model M(s,t), where s and t are relatively prime integers. We show that Kashaev's invariant, i.e., the N-colored Jones polynomial at the N-th root of unity, coincides with the Eichler integral of the character.
Studied is a generalization of Zagier's q-series identity. We introduce a generating function of L-functions at non-positive integers, which is regarded as a half-differential of the Andrews--Gordon q-series. When q is a root of unity, the generating function coincides with the quantum invariant for the torus knot.
We consider a generalization of the vicious walker model. Using a bijection map between the path configuration of the non-intersecting random walkers and the hook Young diagram, we compute the probability concerning the number of walker's movements. Applying the saddle point method, we reveal that the scaling limit gives the Tracy--Widom distribution, which is same with the limit distribution of the largest eigenvalues of the Gaussian unitary ensemble.
We study descendants of inhomogeneous vertex models with boundary reflections when the spin-spin scattering is assumed to be quasi--classical. This corresponds to consider certain power expansion of the boundary-Yang-Baxter equation (or reflection equation). As final product, integrable $su(2)$-spin chains interacting with a long range with $XXZ$ anisotropy are obtained. The spin-spin couplings are non uniform, and a non uniform tunable external magnetic field is applied; the latter can be obtained when the boundary conditions are assumed to be quasi-classical as well. The exact spectrum is achieved by algebraic Bethe ansatz. Having realized the $su(2)$ operators in terms of fermions, the class of models we found turns out to describe confined fermions with pairing force interactions. The class of models presented in this paper is a one-parameter extension of certain Hamiltonians constructed previously. Extensions to $su(n)$-spin open chains are discussed.
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.
We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing polyhedra arise naturally in the classical limit as the saddle point equation of our invariant.
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.
A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra U'_q(A^{(1)}_M) is introduced. It is a crystal theoretic formulation of the generalized box-ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of U'_q(A^{(1)}_{M-1}). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete KP equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.
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{ö} 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.
We give the infinite-dimensional representation for the elliptic $ K $-operator satisfying the boundary Yang-Baxter equation. By restricting the functional space to finite-dimensional space, we construct the elliptic $ K $-matrix associated to Belavin's completely $ \mathbb{Z} $-symmetric $ R $-matrix.
The integrable system is introduced based on the Poisson $ rs $-matrix structure. This is a generalization of the Gaudin magnet, and in SL(2) case isomorphic to the generalized Neumann model. The separation of variables is discussed for both classical and quantum case.
The Yangian symmetry Y(su($n$)) of the Calogero-Sutherland-Moser spin model is reconsidered. The Yangian generators are constructed from two non-commuting su($n$)-loop algebras. The latters generate an infinite dimensional symmetry algebra which is a deformation of the $W_\infty$-algebra. We show that this deformed $W_\infty$-algebra contains an infinite number of Yangian subalgebras with different deformation parameters.
Nonlinear integrable equations, such as the KdV equation, the Boussinesq equation and the KP equation, have the close relation with many-body problem. The solutions of such equations are the same as the restricted flows of the classical Calogero model, which is one-dimensional particle system with inverse square interactions. The KP hierarchy and the Calogero model share the same structure called ``additional symmetry''. This symmetry plays a crucial role in this relation.