An infinite family of tight, not semi-fillable contact three-manifolds

We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and the non-triviality of the Kronheimer-Mrowka monopole invariants of symplectic fillings.