Quivers with subadditive labelings: classification and integrability

TL;DR

This paper classifies quivers with subadditive labelings, links them to integrability via linear recurrences in $T$-system dynamics, and explores their properties including solitonic behavior, extending previous work on Zamolodchikov periodicity.

Contribution

It provides a complete classification of quivers with subadditive labelings and establishes their integrability properties, connecting to recurrence relations and solitonic phenomena.

Findings

Classified all quivers with subadditive labelings.

Linked integrability to linear recurrence relations in $T$-systems.

Identified solitonic behavior in affine slices of tropical $T$-systems.

Abstract

Strictly subadditive, subadditive and weakly subadditive labelings of quivers were introduced by the second author, generalizing Vinberg's definition for undirected graphs. In our previous work we have shown that quivers with strictly subadditive labelings are exactly the quivers exhibiting Zamolodchikov periodicity. In this paper, we classify all quivers with subadditive labelings. We conjecture them to exhibit a certain form of integrability, namely, as the $T$-system dynamics proceeds, the values at each vertex satisfy a linear recurrence. Conversely, we show that every quiver integrable in this sense is necessarily one of the $19$ items in our classification. For the quivers of type $\hat A \otimes A$ we express the coefficients of the recurrences in terms of the partition functions for domino tilings of a cylinder, called \emph{Goncharov-Kenyon Hamiltonians}. We also consider tropical $T$-systems of type $\hat A \otimes A$ and explain how affine slices exhibit solitonic behavior, i.e. soliton resolution and speed conservation. Throughout, we conjecture how the results in the paper are expected to generalize from $\hat A \otimes A$ to all other quivers in our classification.