Ansatz of Hans Bethe for a two-dimensional Bose gas

The method of q-oscillator lattices, proposed recently in [hep-th/0509181], provides the tool for a construction of various integrable models of quantum mechanics in 2+1 dimensional space-time. In contrast to any one dimensional quantum chain, its two dimensional generalizations -- quantum lattices -- admit different geometrical structures. In this paper we consider the q-oscillator model on a special lattice. The model may be interpreted as a two-dimensional Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe Ansatz: the p-particles' wave function is the sum of plane waves. Consistency conditions is the set of 2p equations for p one-particle wave vectors. These "Bethe Ansatz" equations are the main result of this paper.Comment: LaTex2e, 12 page

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

Tetrahedron equations and nilpotent subalgebras of U_q(sl_n)

A relation between q-oscillator R-matrix of the tetrahedron equation and decompositions of Poinkare-Birkhoff-Witt type bases for nilpotent subalgebras of U_q(sl_n) is observed.Comment: 4 page

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

Super-tetrahedra and super-algebras

In this paper we give a detailed classification scheme for three-dimensional quantum zero curvature representation and tetrahedron equations. This scheme includes both even and odd parity components, the resulting algebras of observables are either Bose q-oscillators or Fermi oscillators. Three-dimensional $R$-matrices intertwining variously oriented tensor products of Bose and Fermi oscillators and satisfying tetrahedron and super-tetrahedron equations are derived. The 3d->2d compactification reproduces U_q(gl(n|m)) super-algebras and their representation theory.Comment: LaTeX, 27 page