This project is maintained by ysykimura
2023年5月30日(火) 15:00~16:30
i-Siteなんば S1での対面およびZoomのハイブリッド形式
Modularity of Nahm sums and periodicity phenomena in cluster algebras.
Nahm sums are certain q-hypergeometric series that appear in various areas of mathematics such as knot theory, partition theory, and conformal field theory. Nahm conjectured a connection between the modularity of Nahm sums and algebraic K-theory. As a significant development regarding conjectured relationship, Calegari, Garoufalidis, and Zagier showed that the element of the Bloch group associated with a modular Nahm sum is a torsion element. Their proof is based on the relationship between the asymptotic expansion of Nahm sums and the Chern class map in algebraic K-theory. After reviewing their works, I will explain the relation to the theory of cluster algebras. In this setting, we consider cluster modular group elements instead of Bloch group elements. Torsion elements correspond to the automorphisms on cluster algebras of finite order, and typical examples appear as the periodicity of the Y-systems conjectured by Zamolodchikov and proved by Fomin and Zelevinsky. In general, we expect that there is a modular Nahm sum associated with each periodic Y-system. I will give a classification of periodic Y-systems of rank 2, and see that they actually correspond to modular Nahm sums.