Quantum Dilogarithms and New Integrable Lattice Models in Three Dimensions

TL;DR

The paper addresses the construction of integrable three-dimensional lattice models with commuting layer-to-layer transfer matrices by leveraging quantum dilogarithm identities. It develops a unified framework where Boltzmann weights are built from dilogarithms satisfying inversion and pentagon relations, yielding both vertex and IRC formulations and a restricted star-triangle relation that underpins integrability through the Zamolodchikov tetrahedron equation. Three explicit dilogarithm examples (Faddeev, Andersen–Kashaev, Woronowicz) are used to generate corresponding 3D vertex and IRC models, with exact partition-function results in the infinite-lattice limit demonstrated for the Faddeev case via angle-parameterized data and the Lobachevsky function. The work connects quantum-group–style integrability to geometric structures and paves the way for further exact analyses and extensions (bkms2/bkms3), highlighting rich interactions between special functions, tetrahedron equations, and 3D lattice models with potential geometric interpretations in the quasi-classical limit.

Abstract

In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.