On contact invariants in bordered Floer homology

In this paper, we define contact invariants in bordered sutured Floer homology. Given a contact 3-manifold with convex boundary, we apply a result of Zarev (arxiv:1010.3496) to derive contact invariants in the bordered sutured modules $\widehat{BSA}$ and $\widehat{BSD}$, as well as in bimodules $\widehat{BSAA},$ $\widehat{BSDD},$ $\widehat{BSDA}$ in the case of two boundary components. In the connected boundary case, our invariants appear to agree with bordered contact invariants defined by Alishahi-Földvári-Hendricks-Licata-Petkova-Vértesi (arxiv:2011.08672) whenever the latter are defined, although ours can be defined in broader contexts. We prove that these invariants satisfy a pairing theorem, which is a bordered extension of the Honda-Kazez-Matić gluing map (arxiv:0705.2828) for sutured Floer homology. We also show that there is a correspondence between certain ${\mathcal{A}}_{\mathrm{\infty }}$ operations in bordered type-$A$ modules and bypass attachment maps in sutured Floer homology. As an application, we characterize the Stipsicz-Vértesi map from $SFH$ to $\widehat{HFK}$ as an ${\mathcal{A}}_{\mathrm{\infty }}$ action on $\widehat{CFA}$. We also apply the immersed curve interpretation of Hanselman-Rasmussen-Watson (arxiv:1604.03466) to prove results involving contact surgery.