Open-closed maps and spectral local systems

Abstract

Let $X$ be a graded Liouville domain. Fix a pair of infinite loop spaces $\hat{I}¨=(\hat{I}˜\to \hat{I}¦)$ living over $(BO\to BU)$. This determines a spectral Fukaya category $\mathcal{F}(X;\hat{I}¨)$ whenever $TX$ lifts to $\hat{I}¦$, containing closed exact Lagrangians $L$ for which $TL$ lifts compatibly to $\hat{I}˜$; and by Bott periodicity and index theory, a Thom spectrum $R$ with bordism theory $R_{\ast }$. This paper has two main goals: we incorporate rank one spectral local systems $\hat{I}¾:L\to BG{L}_1(R)$ into the spectral category; and we prove that the bordism class $[(L,\hat{I}¾)]$ 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 $\hat{I}\cdot \in \ddot{I}{€}_1^{st}$ on $\hat{I}¾$. 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 $BG{L}_1(R)\subseteq K(R)\to R$. The companion paper \cite{PS4} shows that (for Lagrangians which themselves admit spectral lifts) one can lift quasi-isomorphisms from $\mathbb{Z}$ to $\hat{I}¨$ 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.

Minor changes