Zamolodchikov integrability via rings of invariants

Pavlo Pylyavskyy

Published 2015-06-17Version 1

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.