Symplectic four-manifolds and conformal blocks

Abstract

We apply ideas from conformal field theory to study symplectic four-manifolds, by using 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, we show the bundles of SU(2) conformal blocks associated to Kaehler surfaces are Brill-Noether special, although the associated flat connexions may be irreducible if the surface is simply-connected and not spin.