arXiv subjects

Atsushi Nakayashiki

Publications and source records attributed to Atsushi Nakayashiki.

33 records · Page 2Linked to original sources

arXiv2004

The Space of Local Fields as a Module over the Ring of Local Integrals of Motion

The chiral space of local fields in Sine-Gordon or the SU(2)-invariant Thirring model is studied as a module over the commutative algebra D of local integrals of motion. Using the recent construction of form factors by means of quantum affine algebra at root of unity due to Feigin et al. we construct a D-free resolution of the space of local fields. In general the cohomologies of the de Rham type complex associated with the space of local fields are determined and shown to be the irreducible representations of the symplectic group Sp(2\infty). Babelon-Bernard-Smirnov's description of the space of local fields automatically incorporated in this framework.

math.QA↗

arXiv2003

Residues of q-Hypergeometric Integrals and Characters of Affine Lie Algebras

We study certain subspaces of solutions to the sl_2 rational qKZ equation at level zero. Each subspace is specified by the vanishing of the residue at a certain divisor which stems from models in two dimensional integrable field theories. We determine the character of the subspace which is parametrized by the number of variables and the sl_2 weight of the solutions. The sum of all characters with a fixed weight gives rise to the branching functions of the irreducible representations of sl_2 in the level one integrable highest weight representations of \hat{sl_2}. It is written in the fermionic form.

math.QA↗

arXiv2003

The Chiral Space of Local Operators in SU(2)-Invariant Thirring Model

The space of local operators in the SU(2) invariant Thirring model (SU(2) ITM) is studied by the form factor bootstrap method. By constructing sets of form factors explicitly we define a susbspace of operators which has the same character as the level one integrable highest weight representation of \hat{sl_2}. This makes a correspondence between this subspace and the chiral space of local operators in the underlying conformal field theory, the su(2) Wess-Zumino-Witten model at level one.

math.QA↗

arXiv2001

On Form Factors of SU(2) Invariant Thirring Model

Integral formulae for form factors of a large family of charged local operators in SU(2) invariant Thirring model are given extending Smirnov's construction of form factors of chargeless local operators in the sine-Gordon model. New abelian symmetry acting on this family of local operators is found. It creates Lukyanov's operators which are not in the above family of local operators in general.

math-ph↗

arXiv2000

On the Cohomology of Theta Divisors of Hyperelliptic Jacobians

We prove that k-th singular cohomology group of the complement of the theta divisor in a hyperelliptic Jacobian is isomorphic to the k-th fundamental representation of the symplectic group Sp(2g,C). This is one of the conjectures in our previous paper.

math.AG↗

arXiv1999

Trace Construction of a Basis for the Solution Space of sl_N qKZ Equation

The trace of intertwining operators over the level one irreducible highest weight modules of the quantum affine algebra of type A_{N-1} is studied. It is proved that the trace function gives a basis of the solution space of the qKZ equation at a generic level. The highest-highest matrix element of the composition of intertwining operators is explicitly calculated. The integral formula for the trace is presented.

math.QA↗

arXiv1998

Integral and Theta Formulae for Solutions of Sl_N Knizhnik-Zamolodchikov Equations at Level Zero

The solutions of the sl_N Knizhnik-Zamolodchikov(KZ) equations at level 0 are studied. We present the integral formula which is obtained as a quasi-classical limit of the integral formula for form factors of the SU(N) invariant Thirring model due to F. Smirnov. A proof is given that those integrals satisfy sl_N KZ equation of level 0. The relation of the integral formulae with the chiral Szego kernel is clarified. As a consequence the integral formula with the special choice of cycles is rewritten in terms of the Riemann theta functions associated with the Z_N curve. This formula gives a generalization of Smirnov's formula for sl_2.

math.QA↗

arXiv1996

On the Thomae formula for $Z_N$ curves

We shall give an elementary and rigorous proof of the Thomae formula for ${\bf Z}_N$ curves which was discovered by Bershadsky and Radul. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szegö kernels and proves the vanishing of zero values of derivatives of theta functions with ${\bf Z}_N$ invariant $1/2N$ characteristics.

alg-geom↗

arXiv1995

Kostka Polynomials and Energy Functions in Solvable Lattice Models

The relation between the charge of Lascoux-Schuzenberger and the energy function in solvable lattice models is clarified. As an application, A.N.Kirillov's conjecture on the expression of the branching coefficient of ${\widehat {sl_n}}/{sl_n}$ as a limit of Kostka polynomials is proved.

q-alg↗

arXiv1995

Crystalline Spinon Basis for RSOS Models

The crystalline spinon basis for the RSOS models associated with $\widehat{sl_2}$ is studied. This basis gives fermionic type character formulas for the branching coefficients of the coset $(\widehat{sl_2})_l \times (\widehat{sl_2})_N/(\widehat{sl_2})_{l+N}$. In addition the path description of the parafermion characters is found as a limit of the spinon description of the string functions.

hep-th↗

arXiv1995

Crystalizing the Spinon Basis

The quasi-particle structure of the higher spin XXZ model is studied. We obtained a new description of crystals associated with the level $k$ integrable highest weight $U_q(\widehat{sl_2})$ modules in terms of the creation operators at $q=0$ (the crystaline spinon basis). The fermionic character formulas and the Yangian structure of those integrable modules naturally follow from this description. We have also derived the conjectural formulas for the multi quasi-particle states at $q=0$.

hep-th↗

arXiv1994

Fusion of the $q$-Vertex Operators and its Application to Solvable Vertex Models

We diagonalize the transfer matrix of the inhomogeneous vertex models of the 6-vertex type in the anti-ferroelectric regime intoducing new types of q-vertex operators. The special cases of those models were used to diagonalize the s-d exchange model\cite{W,A,FW1}. New vertex operators are constructed from the level one vertex operators by the fusion procedure and have the description by bosons. In order to clarify the particle structure we estabish new isomorphisms of crystals. The results are very simple and figure out representation theoretically the ground state degenerations.

hep-th↗

arXiv1993

Vertex operators in solvable lattice models

We formulate the basic properties of q-vertex operators in the context of the Andrews-Baxter-Forrester (ABF) series, as an example of face-interaction models, derive the q-difference equations satisfied by their correlation functions, and establish their connection with representation theory. We also discuss the q-difference equations of the Kashiwara-Miwa (KM) series, as an example of edge-interaction models. Next, the Ising model--the simplest special case of both ABF and KM series--is studied in more detail using the Jordan-Wigner fermions. In particular, all matrix elements of vertex operators are calculated.

hep-th↗

arXiv1992

Correlation Functions of the XXZ model for $Δ<-1$

A new approach to the correlation functions is presented for the XXZ model in the anti-ferroelectric regime. The method is based on the recent realization of the quantum affine symmetry using vertex operators. With the aid of a boson representation for the latter, an integral formula is found for correlation functions of arbitrary local operators. As a special case it reproduces the spontaneous staggered polarization obtained earlier by Baxter.

hep-th↗

arXiv1992

Diagonalization of the XXZ Hamiltonian by Vertex Operators

We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of affine U_q( sl(2) ). Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit --- the $su(2)$-invariant Thirring model.

hep-th↗