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 Uq(B (1) n ) and Uq(D (1) n ). Introduction Quantum q-oscillator system [1,2] in three-dimensional space-time is the result of canonical quantization of a Hamiltonian form [16] of discrete three-wave equations [5, 6, 12]. In the most general form [12, 19] the discrete three-wave equations involve some extra parameters (spectral parameters in quantum world) and correspond to a generic AKP-type hierarchy of integrable systems. There are two special choices of spectral parameters corresponding to the discrete-differential geometry [3, 15] of discrete conjugate nets (syn. quadrilateral nets) [4, 6, 10] – either circular nets (syn. orthogonal nets) in Euclidean space or ortho-chronous hyperbolic nets in Minkowski space. There are a lot of equations associated with discrete nets, here we mean equations for angular data (rotation coefficients) [1, 10]. Algebraically, circular and hyperbolic nets are distinguished by a signature of determinant of rotation matrix. For the latter case of hyperbolic nets the equations of motion admit two constrains reducing a number of degrees of freedom of Cauchy problem twice. One constraint corresponds to discrete BKP equations (syn. Miwa equations) [14]. Discrete BKP equations appear in discrete differential geometry in many ways [11], constraint for hyperbolic net just clearly shows the reduction. The other constraint is a real form on equations of motion (curiously, we discuss in fact six-wave equations, they become three-wave upon this reality condition). 1991 Mathematics Subject Classification. 20G42, 81Txx 82B20, 82B23 .