Zamolodchikov integrability via rings of invariants

TL;DR

The paper introduces Zamolodchikov integrability, an affine analog of periodicity in certain recursions, and proves it for specific Dynkin diagram products using cluster algebra structures.

Contribution

It proposes a new integrability conjecture for affine Dynkin diagram products and proves it in a specific case using cluster algebra techniques.

Findings

Conjecture holds for $A_m imes A_{2n-1}^{(1)}$

Uses cluster structures in rings of invariants

Extends Zamolodchikov periodicity to affine case

Abstract

Zamolodchikov periodicity is periodicity of certein recursions associated with box products $X \square Y$ of two finite type Dynkin diagrams. We suggest an affine analog of Zamolodchikov periodicity, which we call Zamolodchikov integrability. We conjecture that it holds for products $X \square Y$, where $X$ is a finite type Dynkin diagram and $Y$ is an extended Dynkin diagram. We prove this conjecture for the case of $A_m \square A_{2n-1}^{(1)}$. The proof employs cluster structures in certain classical rings of invariants, previously studied by S. Fomin and the author.