Two-dimensional variational systems on the root lattice $Q(A_N)$

Abstract

We study certain two-dimensional variational systems, namely pluri-Lagrangian systems on the root lattice $Q(A_N)$. Here, we follow the scheme which was already used to define two-dimensional pluri-Lagrangian systems on the lattice ${\mathbb{Z}}^N$ and three-dimensional pluri-Lagrangian systems on the lattice ${\mathbb{Z}}^N$ as well as on $Q(A_N)$. We will show that the two-dimensional pluri-Lagragian systems on $Q(A_N)$ are more general than the ones on ${\mathbb{Z}}^N$, in the sense that they can encode several different pluri-Lagrangian systems on ${\mathbb{Z}}^N$. This also means that the variational formulation of several systems of certain hyperbolic equations, so-called quad-equations, can be obtained from one and the same pluri-Lagrangian system on $Q(A_N)$.

17 pages, 3 figures. arXiv admin note: text overlap with arXiv:1506.00729