Quivers with additive labelings: classification and algebraic entropy

TL;DR

This paper classifies quivers with additive labelings based on their algebraic entropy and establishes a complete classification of Zamolodchikov periodic and integrable quivers, linking dynamics to algebraic structures.

Contribution

It provides a complete classification of quivers with additive labelings, connecting algebraic entropy to subadditive labelings and classifying all Zamolodchikov periodic quivers.

Findings

Zero algebraic entropy corresponds to weakly subadditive labelings.

Classification includes 40 infinite families and 13 exceptional quivers.

Complete classification of Zamolodchikov periodic and integrable quivers.

Abstract

We show that Zamolodchikov dynamics of a recurrent quiver has zero algebraic entropy only if the quiver has a weakly subadditive labeling, and conjecture the converse. By assigning a pair of generalized Cartan matrices of affine type to each quiver with an additive labeling, we completely classify such quivers, obtaining $40$ infinite families and $13$ exceptional quivers. This completes the program of classifying Zamolodchikov periodic and integrable quivers.