Global Fukaya category II: singular connections, quantum obstruction theory, and other applications

Abstract

In part I, using the theory of $\infty$-categories, we constructed a natural ``continuous action'' of $\operatorname {Ham} (M, \omega) $ on the Fukaya category of a closed monotone symplectic manifold. Here we show that this action is generally homotopically non-trivial, i.e implicitly the main part of a conjecture of Teleman. We use this to give various applications. For example we find new curvature constraint phenomena for smooth and singular $\mathcal{G}$-connections on principal $\mathcal{G}$-bundles over $S ^{4}$, where $\mathcal{G}$ is $\operatorname {PU} (2)$ or $\operatorname {Ham} (S ^{2} )$. Even for the classical group $\operatorname {PU} (2)$, these phenomena are invisible to Chern-Weil theory, and are inaccessible to known Yang-Mills theory and quantum characteristic classes techniques. So this can be understood as one application of Floer theory and the theory of $\infty$-categories in basic differential geometry. We also develop, based on this $\infty$-categorical Fukaya theory, some basic new integer valued invariants of smooth manifolds, called quantum obstruction. On the way we also construct what we call quantum Maslov classes, which are higher degree variants of the relative Seidel morphism. This also leads to new applications in Hofer geometry of the space of Lagrangian equators in $S^2$.