A graph TQFT for hat Heegaard Floer homology

In this paper we introduce an extension of the hat Heegaard Floer TQFT which allows cobordisms with disconnected ends. Our construction goes by way of sutured Floer homology, and uses some elementary results from contact geometry. We provide some model computations, which allow us to realize the $H_1(Y;\mathbb{Z})/\text{Tors}$ action and the first order term, $\partial_1$, of the differential of $CF^\infty$ as cobordism maps. As an application we prove a conjectured formula for the action of $\pi_1(Y,p)$ on $\hat{HF}(Y,p)$. We provide enough model computations to completely determine the new cobordism maps without the use of any contact geometric constructions.