allowing us to prove that certain classes $x$ are periodic. The Fukaya category of $B$ is not known to the same extent, but partial computations are enough to determine that certain classes $x$ are not periodic, since that only requires finding a specific Lagrangian submanifold which can serve as a counterexample.

To conclude this discussion, we should mention that the basic idea is by no means new in homological algebra. From that elevated vantage point, what we are doing (in a rather ad hoc way) is to study algebraic one-parameter subgroups (the elliptic curves $\bar{{\mathscr{S}}}$ which appeared above) inside the derived Picard group [@yekutieli04; @keller04] of the Fukaya category. The idea is to think of elements of Hochschild cohomology as vector fields on an abstract “moduli space of objects”, and that we are asking which vector fields integrate to “periodic flows”. On an informal level, it is clear that this provides an algebraic counterpart to the geometric ideas underlying flux.

[**Structure of the paper.**]{} Section \[sec:theory\] sets up the algebraic theory of families of objects, much of it straightforward. The key to uniqueness results for families is the discussion surrounding Lemma \[th:its-projective\], which is then further developed for our intended 