The holonomy braiding for ${\mathcal{U}}_{\xi }({\mathfrak{sl}}_2)$ in terms of geometric quantum dilogarithms

Abstract

We derive an explicit formula for the holonomy $R$-matrix of quantum ${\mathfrak{sl}}_2$ at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.

37 + 4 pages. Revised version, some material moved to arXiv:2509.02365