Fe b 20 09 CLASSICAL INTEGRABLE FIELD THEORIES IN DISCRETE 2 + 1 DIMENSIONAL SPACE - TIME

We study “circular net” (discrete orthogonal net) equations for angular data generalized by external spectral parameters. A criterion defining physical regimes of these Hamiltonian equations is the reality of Lagrangian density. There are four distinct regimes for fields and spectral parameters classified by four types of spherical or hyperbolic triangles. Non-zero external spectral parameters provide the existence of field-theoretical ground states and soliton excitations. Spectral parameters of a spherical triangle correspond to a statistical mechanics; spectral parameters of hyperbolic triangles correspond to three different field theories with massless anisotropic dispersion relations. Introduction The “circular-” or “conic net” (or discrete orthogonal net) equations for angular data [4,5,8] take a selected place among all the classical integrable systems [3] on cubic lattice with AKP-type hierarchy. Algebraically, these equations arise as a Hamiltonian form of discrete three-wave system [6, 20]. The “conic net” equations are classical q → 1 limit of quantum “q-oscillator” model [2] – the top of a pyramid of three-dimensional quantum models – what guarantees in classics the existence of Lagrangian density, energy/action and variational principle [1]. The existence of quantum counterpart is an evident advantage of discrete space-time models with respect to their continuous space-time predecessors [22, 23]. A straightforward geometrical condition for conic net equations is the reality of angular dynamical variables [1] of a circular net in Euclidean target space or of an ortho-chronous hyperbolic net in Minkowski one. However, this discrete differential geometry conditions can be essentially extended by a “physical” condition of reality of action near equilibrium point. A general complex solution of any AKP-type system in finite volume or a solution of Cauchy problem with generic initial data involves the algebraic geometry [15]. Such general solution of discrete “generalized conic net” equations in finite volume is known for a long time [9–14]. It involves a flat algebraic spectral curve Γg of genus g ≤ (N −1)2 for a size N×3 cubic lattice (three-periodic boundary conditions), Θ-functions on Jacobian of Γq, and spectral parameters – three meromorphic functions on Γg. A reduction of Γg to a sphere gives a g-soliton solution 1991 Mathematics Subject Classification. 35Q51, 35Q58, 37K10, 37J35, 70H06, 81Txx .