On integrability of discrete variational systems. Octahedron relations

We elucidate consistency of the so-called corner equations which are elementary building blocks of Euler-Lagrange equations for two-dimensional pluri-Lagrangian problems. We show that their consistency can be derived from the existence of two independent octahedron relations. We give explicit formulas for octahedron relations in terms of corner equations.

💡 Research Summary

The paper investigates the algebraic underpinnings of consistency for “corner equations,” which are the elementary building blocks of discrete Euler‑Lagrange equations in two‑dimensional pluri‑Lagrangian systems. Starting from a discrete 2‑form L defined on oriented elementary squares of the integer lattice ℤ^m, the authors consider the action functional obtained by summing L over any oriented quad‑surface Σ. Critical points of this action lead to discrete Euler‑Lagrange equations; when Σ is the surface of a unit 3‑dimensional cube, these equations reduce to six so‑called corner equations, each involving five of the eight vertex variables of the cube.

The focus is on corner equations that arise from the ABS list of integrable quad‑equations (the classification of multi‑affine, three‑dimensional consistent equations). For each quad‑equation Q(x, x_i, x_j, x_{ij}; α_i, α_j)=0 there exist four three‑leg forms; the corner equations are obtained by taking differences of two such three‑leg forms attached to adjacent faces. By eliminating the auxiliary variable X, the authors rewrite the six corner equations as a system of six polynomial equations E_1,…,E_{12} (equation (5) in the paper). Each E_ℓ is quadratic in its “own” variable and linear in the remaining five.

A key notion introduced is that of a fractional ideal in the polynomial ring R = ℂ