Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 059, 27 pages      arXiv:2403.16845      https://doi.org/10.3842/SIGMA.2025.059

Discrete Lagrangian Multiforms for ABS Equations II: Tetrahedron and Octahedron Equations

Jacob J. Richardson a and Mats Vermeeren b
a) School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK
b) Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

Received March 26, 2024, in final form July 02, 2025; Published online July 18, 2025

Abstract
We present four types of discrete Lagrangian 2-form associated to the integrable quad equations of the ABS list. These include the triangle Lagrangian that has traditionally been used in the Lagrangian multiform description of ABS equations, the trident Lagrangian that was central to Part I of this paper, and two Lagrangians that have not been studied in the multiform setting. Two of the Lagrangian 2-forms have the quad equations, or a system equivalent to the quad equations, as their Euler-Lagrange equations, and one produces the tetrahedron equations. This is in contrast to the triangle Lagrangian 2-form, which produces equations that are weaker than the quad equations (they are equivalent to two octahedron equations). We use relations between the Lagrangian 2-forms to prove that the system of quad equations is equivalent to the combined system of tetrahedron and octahedron equations. Furthermore, for each of the Lagrangian 2-forms, we study the double zero property of the exterior derivative. In particular, this gives a possible variational interpretation to the octahedron equations.

Key words: discrete integrability; Lagrangian multiforms; variational principles.

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