Infinitely many universally tight torsion free contact structures with vanishing Ozsv\'ath-Szab\'o contact invariants

Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo invariant vanishes. We also discuss the relation between these invariants and an invariant on T^3 and construct other examples of new phenomena in Heegaard--Floer theory. Along the way, we prove two conjectures of K Honda, W Kazez and G Matic about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.