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

We derive an explicit formula for the holonomy $R$-matrix of quantum ${\mathfrak{s}\mathfrak{l}}_2$ 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.