Quantum Monodromy Representations Of Braids In Thickened Riemann Surfaces

The talk will consist in introducing a notion of (integrable) quantum connection in quantum principal bundles over differentiable manifolds with a ribbon Hopf algebra as a quantum structural group. For quantum connections, we define parallel transport by introducing quantum non-abelian integration. In the integrable case, for which we introduce a new notion of the quantum Maurer-Cartan equation, the parallel transport depends only on the homotopy class of a path. This allows us to speak about quantum monodromy. Using the latter, we introduce the notion of a quantum monodromy representation of the braid homotopy group of an oriented surface and Reshetikhin-Turaev-type invariants with local coefficients. Both constructions combine the topology of the surface and the interlacement of strands in the braid diagrams or of edges in ribbon graphs. (joint with Majid Bigdeli)