Ideas from conformal field theory are applied to symplectic four-manifolds through the use of modular functors to ‘linearise’ Lefschetz fibrations. In Chern–Simons theory, this leads to the study of parabolic vector bundles of conformal blocks. Motivated by the Hard Lefschetz theorem, the author shows that the bundles of SU(2) conformal blocks associated to Kähler surfaces are Brill–Noether special, although the associated flat connexions may be irreducible if the surface is simply connected and not spin.