From 3D topological quantum field theories to 4D models with defects

Understand

(2+1) dimensional topological quantum field theories with defect excitations are by now quite well understood, while many questions are still open for (3+1) dimensional TQFTs.

• Here we propose a strategy to lift states and operators of a (2+1) dimensional TQFT to states and operators of a (3+1) dimensional theory with defects.
• The main technical tool are Heegard splittings, which allow to encode the topology of a three--dimensional manifold with line defects into a two--dimensional Heegard surface.
• We apply this idea to the example of BF theory which describes locally flat connections.