Bordered Floer theory, links in $S^1\times S^2$, and (non-)formality

An early paper of Ozsváth–Szabó establishes the existence of a spectral sequence from the reduced Khovanov homology of the mirror of a link to the Heegaard Floer homology of its branched double cover. We will describe an analogue of this spectral sequence from Rozansky’s categorified stable SU(2) Witten–Reshetikhin–Turaev invariant of links in $S^1\times S^2$ to the Hochschild homology of an $A_\infty$-bimodule defined using bordered Heegaard Floer homology. Along the way, we will see that the algebras over which these bimodules are defined are nontrivial $A_\infty$-deformations of Khovanov’s arc algebras, in stark contrast to Abouzaid–Smith's symplectic arc algebras which (over Q) are formal.