Pentagon Equation in Deformation Quantization of Character Varieties

Deformation quantizations of the moduli space of framed flat bundles live naturally in the Drinfeld category. Such quantizations were constructed using the so-called quantum fusion by Li-Bland and Ševera. We give a complete characterization of the dependence of this quantization on combinatorial choices, and prove that these quantizations are all isomorphic, forming vertices of an explicitly constructed complete commuting graph of algebra isomorphism. To this end, we extend the Kontsevich integral methods of Le and Murakami from even associators to all associators. Finally, we also prove that bi-invariant Haar measures provide traces on these quantizations.