A Hierarchical Array of Integrable Models

TL;DR

This paper constructs a hierarchical array of new integrable models using advanced gamma functions, revealing both known and novel models, and introduces new mathematical results related to q-gamma and q-deformed functions.

Contribution

It develops a hierarchy of integrable models based on higher Barnes gamma functions, linking known models and discovering new ones through q-deformation and Euler products.

Findings

Established a hierarchy of integrable models with complex gamma functions.

Discovered new mathematical properties of q-gamma and q-deformed Bloch-Wigner functions.

Connected known models within a unified hierarchical framework.

Abstract

Motivated by Harish-Chandra theory, we construct, starting from a simple CDD\--pole $S$\--matrix, a hierarchy of new $S$\--matrices involving ever ``higher'' (in the sense of Barnes) gamma functions.These new $S$\--matrices correspond to scattering of excitations in ever more complex integrable models.From each of these models, new ones are obtained either by ``$q$\--deformation'', or by considering the Selberg-type Euler products of which they represent the ``infinite place''. A hierarchic array of integrable models is thus obtained. A remarkable diagonal link in this array is established.Though many entries in this array correspond to familiar integrable models, the array also leads to new models. In setting up this array we were led to new results on the $q$\--gamma function and on the $q$\--deformed Bloch\--Wigner function.