Dimers for Relativistic Toda Models with Reflective Boundaries

TL;DR

This work constructs dimer graphs for relativistic Toda chains (RTCs) associated with all classical untwisted Lie algebras and selected twisted variants, framing RTCs as cluster integrable systems realized by dimer models on the torus. By embedding Lax matrices and Sklyanin-type reflection matrices into Kasteleyn representations, the authors show that the RTC spectral curves coincide with Seiberg–Witten curves of 5d $\mathcal{N}=1$ pure gauge theories with dual groups $G^\vee$, establishing a unifying dimer/clustering picture across A, B, C, D types and their twists. The paper provides explicit dimer constructions for reflective boundaries (types C, B-1, B-2, D) and derives the corresponding boundary Hamiltonians, monodromies, and spectral curves, including numerous folded/mixture cases and dualities (e.g., ${C}_N^{(1)}$ and $D_{N+1}^{(2)}$). These results bolster the view of semi-simple RTCs as cluster integrable systems and offer concrete templates for future quantization and duality investigations, with connections to 5d gauge theories and brane-configurations.

Abstract

We construct dimer graphs for relativistic Toda chains associated with classical untwisted Lie algebras of A, B, C$_0$, C$_π$, D types and twisted A, D types. We show that the Seiberg-Witten curve of 5d $\mathcal{N}=1$ pure supersymmetric gauge theory of gauge group $G$ is a spectral curve of the relativistic Toda chain of the dual group $G^\vee$.

Figures (16)

• Figure 1: Long root on the end of Dynkin diagram of Lie algebra $\mathfrak{g}$
• Figure 2: Short roots on the end of Dynkin diagram of Lie algebra $\mathfrak{g}$
• Figure 3: Type D boundary of the Dynkin diagram
• Figure 4: The Newton polygon for Toda chain on $N$ sites.
• Figure 5: The bipartite graph $Y^{N,0}$ model associated with type A RTC with Lie algebra ${A}_N^{(1)}$. The horizontal dotted lines (red) on the top and the bottom are identified, as well as the vertical dotted lines (blue) on the left and right. In turn, the bipartite graph is drawn on a torus $T^2$. Left: face variables. Right: gauged edge variables.