Open-closed maps and spectral local systems

Let $X$ be a graded Liouville domain. Fix a pair of infinite loop spaces $Ψ= (Θ\to Φ)$ living over $(BO \to BU)$. This determines a spectral Fukaya category $\mathcal{F}(X;Ψ)$ whenever $TX$ lifts to $Φ$, containing closed exact Lagrangians $L$ for which $TL$ lifts compatibly to $Θ$; 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 $ξ: L \to BGL_1(R)$ into the spectral category; and we prove that the bordism class $[(L,ξ)]$ 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 $η\in π_1^{st}$ on $ξ$. 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 $Ψ$ 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.

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