The fields of Topological Quantum Field Theory (TQFT) and Quantum Topology came from the realisation that QFT is sensitive to the topology of the underlying space(time). One of the crowning achievements of the field is Witten's 2+1D Chern−Simons TQFT and the formidable challenge of its mathematical formalisation is ongoing to this day. The development of the mathematical counterpart relies on the geometric quantisation of the moduli space/stack of bundles over a Riemann surface. The topological nature of this construction, which is clear from the physics perspective but well hidden behind the obvious dependence on the complex structure in the mathematical construction, is witnessed by the Hitchin connection, which identifies the Hilbert spaces obtained for different complex structures. I will explain my previous work on identifying the Hitchin connection with the KZ connection from conformal field theory, and discuss my related current work on topologising the string-theoretic BBB brane quantisation of the hyperkähler moduli space of Higgs bundles on a Riemann surface.