Strong L-spaces and left-orderability

We introduce the notion of a strong L-space, a closed, oriented threemanifold admitting a Heegaard diagram whose associated Heegaard Floer complex has rank equal to the order of the first homology of the manifold. Examples of strong Lspaces include the branched double covers of alternating links in the three-sphere. We prove that the fundamental group of a strong L-space is not left-orderable.