Constant connections, quantum holonomies and the Goldman bracket

Abstract

In the context of $2+1$-dimensional quantum gravity with negative cosmological constant and topology $\mathbb{R} \times T^2$, constant matrix-valued connections generate a $q$-deformed representation of the fundamental group, and signed area phases relate the quantum matrices assigned to homotopic loops. Some features of the resulting quantum geometry are explored, and as a consequence a quantum version of the Goldman bracket is obtained.