Cluster algebras, $SL(N, \mathbb{C})$ flat connections and $3$-manifolds

$SL(N, \mathbb{C})$ Chern–Simons theory have been getting some increasing interest in math and physics in the last years. I'll describe some recently developed techniques inspired by string theoretical constructions, that makes use of the connection between higher Teichmuller theory and cluster algebras to produce topological invariants of hyperbolic $3$-manifolds. As time allows I'll talk about connections with other subjects in this context. They include volume conjecture, defects in quantum field theory and spectral networks.