On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

Recently, the first-named author gave a classification of 3D consistent 6-tuples of quad-equations with the tetrahedron property; several novel asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can be extended to discrete integrable systems on Z^m. We establish Lagrangian structures and flip-invariance of the action functional for the class of discrete integrable systems involving equations for which some of the biquadratics are non-degenerate and some are degenerate. This class covers, among others, some of the above mentioned novel systems.

💡 Research Summary

The paper investigates the variational (Lagrangian) structure of a class of discrete integrable systems built from asymmetric quad‑equations that are 3‑dimensional (3D) consistent and possess the tetrahedron property. The authors focus on six‑tuples of quad‑equations placed on the faces of an elementary cube, where some of the associated biquadratics are non‑degenerate while others are degenerate. This situation corresponds to equations of type H₄ (four degenerate and two non‑degenerate biquadratics) but excludes the fully degenerate type H₆.

After recalling the basic notions—multi‑affine quad‑polynomials, Möbius equivalence, and the definition of biquadratics—the paper classifies all possible 3D‑consistent six‑tuples that contain at least one H₄ equation and no H₆ equations. Up to independent Möbius transformations of the eight field variables, nine distinct families are obtained. They are organized into three combinatorial patterns of biquadratic placement (Figures 4a–c): (a) all six face equations are of the rhombic Hₖ form with tetrahedron equations of type Qₖ; (b) two pairs of face equations are trap‑ezoidal Hₖ, one pair is Qₖ, and the tetrahedron equations are Hₖ; (c) two pairs are trap‑ezoidal Hₖ, the remaining pair is rhombic Hₖ, and both tetrahedron equations are Hₖ. Each family can be embedded into the full lattice ℤ³ by reflecting the elementary cube, as shown in the earlier work of Boll (2011).

The central technical contribution is the construction of Lagrangian functions for these systems. The authors first derive three‑leg forms for each quad‑equation, which express the equation as a sum of three “leg” functions ψ, φ (and their barred counterparts) depending on a single lattice parameter. Lemma 4.1 shows that, after an appropriate rescaling, each biquadratic depends only on the edge parameter, and Lemma 4.2 provides the explicit three‑leg representations for both the Qₖ and the rhombic Hₖ equations. By introducing uniformizing changes of variables that turn the discriminants into simple quadratic forms, they define edge Lagrangians L(x_i,x_j;α).

Theorem 4.4 proves a “closure” property: for any elementary quadrilateral the sum of the three Lagrangians around the face vanishes (up to a constant), which is the discrete analogue of the Euler–Lagrange equation. Consequently, an action functional S obtained by summing L over all faces of a 3‑dimensional lattice is well defined.

Section 5 establishes the flip‑invariance of this action functional. A flip is a local transformation that replaces a pair of adjacent faces by the complementary pair in a three‑dimensional cube (the elementary 3D flip). Using the three‑leg forms and the closure relation, the authors show that the total contribution of the flipped region to S remains unchanged. This invariance confirms that the variational principle is compatible with the multidimensional consistency of the underlying equations.

In summary, the paper extends the known Lagrangian theory for the symmetric ABS list (type Q) to a broader class that includes asymmetric H₄ equations. It provides a complete classification of admissible six‑tuples, constructs explicit Lagrangians via three‑leg forms, and proves the essential flip‑invariance of the associated action. These results deepen the understanding of discrete integrable systems, open the way for further studies of non‑symmetric lattice models, and suggest possible connections to physical applications such as lattice field theories and discrete geometry.