Lagrangian multiforms and multidimensional consistency

We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for such systems in terms of Lagrangian multiforms. We discuss the connection of this formalism with the notion of multidimensional consistency, and the role of the lattice from the point of view of the relevant variational principle.

💡 Research Summary

The paper investigates the variational structure of two‑dimensional integrable lattice equations, focusing on those classified in the Adler‑Bobenko‑Suris (ABS) list. After a brief historical overview, the authors emphasize the concept of multidimensional consistency: a single quadrilateral equation can be imposed consistently on all two‑dimensional faces of a three‑dimensional cube, leading to a unique value for the field at the opposite corner. This property underlies the integrability of the listed equations (H1‑H3, Q1‑Q4, A1‑A2).

For each equation the authors present a three‑point Lagrangian L(u, u_i, u_j; α_i, α_j). These Lagrangians generate, via a discrete Euler‑Lagrange principle, a seven‑point equation that consists of two copies of the original four‑point relation. The explicit forms involve logarithmic or dilogarithmic functions, reflecting the affine‑linear nature of the underlying equations.

The central result is the “closure relation”. When the three‑point Lagrangians are embedded in a three‑dimensional lattice and acted upon by the forward difference operators Δ_i, the combination Δ₁L₂₃ + Δ₂L₃₁ + Δ₃L₁₂ vanishes on solutions of the lattice system. This identity shows that the Lagrangian defines a closed 2‑form on the multidimensional lattice, providing a variational interpretation of multidimensional consistency. The authors verify the relation explicitly for H1 and, in an appendix, for H3 using identities of the dilogarithm function; the remaining ABS equations are claimed to satisfy the same property.

The paper also discusses the continuous limit, illustrating how the closure property translates into a variational principle for the generating partial differential equation associated with the lattice KdV system. In conclusion, the authors argue that Lagrangian multiforms constitute a natural framework for describing integrable discrete systems, unifying their geometric, algebraic, and variational aspects, and opening avenues for further developments such as quantisation and extensions to higher‑dimensional integrable hierarchies.