2,088 research outputs found
Explicit Free Parameterization of the Modified Tetrahedron Equation
The Modified Tetrahedron Equation (MTE) with affine Weyl quantum variables at
N-th root of unity is solved by a rational mapping operator which is obtained
from the solution of a linear problem. We show that the solutions can be
parameterized in terms of eight free parameters and sixteen discrete phase
choices, thus providing a broad starting point for the construction of
3-dimensional integrable lattice models. The Fermat curve points parameterizing
the representation of the mapping operator in terms of cyclic functions are
expressed in terms of the independent parameters. An explicit formula for the
density factor of the MTE is derived. For the example N=2 we write the MTE in
full detail. We also discuss a solution of the MTE in terms of bosonic
continuum functions.Comment: 28 pages, 3 figure
Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
In this paper we consider three-dimensional quantum q-oscillator field theory
without spectral parameters. We construct an essentially big set of eigenstates
of evolution with unity eigenvalue of discrete time evolution operator. All
these eigenstates belong to a subspace of total Hilbert space where an action
of evolution operator can be identified with quantized discrete BKP equations
(synonym Miwa equations). The key ingredients of our construction are specific
eigenstates of a single three-dimensional R-matrix. These eigenstates are
boundary states for hidden three-dimensional structures of U_q(B_n^1) and
U_q(D_n^1)$.Comment: 13 page
Quantum 2+1 evolution model
A quantum evolution model in 2+1 discrete space - time, connected with 3D
fundamental map R, is investigated. Map R is derived as a map providing a zero
curvature of a two dimensional lattice system called "the current system". In a
special case of the local Weyl algebra for dynamical variables the map appears
to be canonical one and it corresponds to known operator-valued R-matrix. The
current system is a kind of the linear problem for 2+1 evolution model. A
generating function for the integrals of motion for the evolution is derived
with a help of the current system. The subject of the paper is rather new, and
so the perspectives of further investigations are widely discussed.Comment: LaTeX, 37page
New series of 3D lattice integrable models
In this paper we present a new series of 3-dimensional integrable lattice
models with colors. The case generalizes the elliptic model of our
previous paper. The weight functions of the models satisfy modified tetrahedron
equations with states and give a commuting family of two-layer
transfer-matrices. The dependence on the spectral parameters corresponds to the
static limit of the modified tetrahedron equations and weights are
parameterized in terms of elliptic functions. The models contain two free
parameters: elliptic modulus and additional parameter . Also we briefly
discuss symmetry properties of weight functions of the models.Comment: 17 pages, IHEP-93-126, Late
New solution of vertex type tetrahedron equations
In this paper we formulate a new N-state spin integrable model on a
three-dimensional lattice with spins interacting round each elementary cube of
the lattice. This model can be also reformulated as a vertex type model. Weight
functions of the model satisfy tetrahedron equations.Comment: 12 pages, LaTeX, IHEP-94-10
The modified tetrahedron equation and its solutions
A large class of 3-dimensional integrable lattice spin models is constructed.
The starting point is an invertible canonical mapping operator in the space of
a triple Weyl algebra. This operator is derived postulating a current branching
principle together with a Baxter Z-invariance. The tetrahedron equation for
this operator follows without further calculations. If the Weyl parameter is
taken to be a root of unity, the mapping operator decomposes into a matrix
conjugation and a C-number functional mapping. The operator of the matrix
conjugation satisfies a modified tetrahedron equation (MTE) in which the
"rapidities" are solutions of a classical integrable Hirota-type equation. The
matrix elements of this operator can be represented in terms of the
Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of
Gauss functions. The paper summarizes several recent publications on the
subject.Comment: 24 pages, 6 figures using epic/eepic package, Contribution to the
proceedings of the 6th International Conference on CFTs and Integrable
Models, Chernogolovka, Spetember 2002, reference adde
- Vectors for Three Dimensional Models
In this paper we apply the method of psi-vectors to three dimensional
statistical models. This method gives the correspondence between the Bazhanov
-- Baxter model and its vertex formulation. Considering psi-vectors for the
Planar model, we obtain its self-duality.Comment: 11 pages, LaTeX, no figure
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