Constant connections, quantum holonomies and the Goldman bracket (Q2642821)

This paper is one of the series of papers of the present authors. Investigations are in the context of the \(2+1\) dimensional quantum gravity with negative cosmological constant, on \(\mathbb R\times T^2\). Here there are studied constant matrix valued connections applied to the piecewise linear paths between integer points in \(\mathbb R^2\). The authors obtain the \(q\)-deformed surface group representation based on the observation that matrices for homotopic paths are related by phase factor which depends on signed area between paths. The relevant action of the modular group and the Goldman bracket are discussed as well.