  We construct an unwrapped Floer theory for bundles of Liouville sectors. In
particular, we construct a compatible collection of unwrapped Fukaya categories
of fibers of a Liouville bundle, and prove that the two natural constructions
of continuation maps in this setting behave compatibly. These constructions are
exploited in [OT19] to construct homotopically coherent actions of Lie groups
on wrapped Fukaya categories, thereby proving a conjecture from Teleman's 2014
ICM address.
