On the action of the group of diffeomorphisms of a surface on sections of the determinant line bundle

Let Σ denote a closed oriented surface. There is a natural action of the group Diff⁺(Σ) on sections of the chiral determinant line over the space of gauge equivalence classes of connections. The question we address is whether this action is unitarizable. We introduce a SDiff-equivariant regularization, and we prove the existence of, and explicitly compute, the limit as the regularization is removed. The SDiff unitary representations that arise, both by regularization and after removing the regularization, appear to be new.