Let $\Sigma$ denote a closed oriented surface. There is a natural action of the group ${\rm Diff}^{+}(\Sigma )$ 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 $S\Diff$-equivariant regularization, and we prove the existence of, and explicitly compute, the limit as the regularization is removed. The $S\Diff$ unitary representations that arise, both by regularization and after removing the regularization, appear to be new.