Open-closed maps and spectral local systems

Let $X$ be a graded Liouville domain. Fix a pair of infinite loop spaces $\Psi = (\Theta \to \Phi)$ living over $(BO \to BU)$. This determines a spectral Fukaya category $\mathcal{F}(X;\Psi)$ whenever $TX$ lifts to $\Phi$, containing closed exact Lagrangians $L$ for which $TL$ lifts compatibly to $\Theta$; and by Bott periodicity and index theory, a Thom spectrum $R$ with bordism theory $R_*$. This paper has two main goals: we incorporate rank one spectral local systems $\xi: L \to BGL_1(R)$ into the spectral category; and we prove that the bordism class $[(L,\xi)]$ defined by the open-closed map differs from the class $[L]$ by a multiplicative two-torsion element in $R^0(L)^{\times}$ determined by an action of the stable homotopy class of the Hopf map $\eta \in \pi_1^{st}$ on $\xi$. Methods include a twisting construction associating flow categories to spectral local systems, and a model for the open-closed map incorporating Schlichtkrull's construction of the trace map $BGL_1(R) \subseteq K(R) \to R$. The companion paper \cite{PS4} shows that (for Lagrangians which themselves admit spectral lifts) one can lift quasi-isomorphisms from $\mathbb{Z}$ to $\Psi$ at the cost of introducing rank one local systems. Together with the open-closed computation given here, this gives an essentially complete picture of the bordism-theoretic consequences of quasi-isomorphism in the classical exact Fukaya category.