Y-systems and generalized associahedra

TL;DR

This paper proves the periodicity conjecture for Y-systems associated with any finite root system, provides explicit formulas for the related rational functions, and introduces simplicial complexes generalizing associahedra.

Contribution

It establishes the periodicity of Y-systems for all finite root systems and constructs new simplicial complexes generalizing known polytopes.

Findings

Proved periodicity conjecture for all finite root systems.

Derived explicit Laurent polynomial formulas for Y-system functions.

Constructed simplicial complexes generalizing associahedra and cyclohedra.

Abstract

We prove, for an arbitrary finite root system, the periodicity conjecture of Al.B.Zamolodchikov concerning Y-systems, a particular class of functional relations arising in the theory of thermodynamic Bethe ansatz. Algebraically, Y-systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the underlying root system. In the course of proving periodicity, we obtain explicit formulas for all these rational functions, which turn out to always be Laurent polynomials. In a closely related development, we introduce and study a family of simplicial complexes that can be associated to arbitrary root systems. In type A, our construction produces Stasheff's associahedron, whereas in type B, it gives the Bott-Taubes polytope, or cyclohedron. We enumerate the faces of these complexes, prove that their geometric realization is always a sphere, and describe them in concrete combinatorial terms for the classical types ABCD.