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

SIGMA 21 (2025), 058, 30 pages      arXiv:2501.13012      https://doi.org/10.3842/SIGMA.2025.058

Discrete Lagrangian Multiforms for ABS Equations I: Quad 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
Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums.

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

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