  Let $(M,\omega_M)$ be a monotone or negatively monotone symplectic manifold,
or a Weinstein manifold. One can construct an "action" of $H^1(M,\mathbb{G}_m)$
on the Fukaya category (wrapped Fukaya category in the exact case) that
reflects the action of $Symp^0(M,\omega_M)$ on the set of Lagrangian branes. A
priori this action is only analytic. The purpose of this work is to show the
algebraicity of this action under some assumptions.
  We use this to prove a tameness result for the sheaf of Lagrangian Floer
homology groups obtained by moving one of the Lagrangians via global symplectic
isotopies. We also show the algebraicity of the locus of $z\in
H^1(M,\mathbb{G}_m)$ that fix a Lagrangian brane in the Fukaya category. The
latter has applications to Lagrangian flux. Finally, we prove a statement in
mirror symmetry: in the Weinstein case, assume that $M$ is mirror to an affine
or projective variety $X$, that there exists an exact Lagrangian torus
$L\subset M$ such that $H^1(M)\to H^1(L)$ is surjective, and that $L$ is sent
to a smooth point of $x\in X$ under the mirror equivalence. Then we construct a
Zariski chart of $X$ containing $x$, that is isomorphic to
$H^1(L,\mathbb{G}_m)$, and such that other points of this chart correspond to
non-exact deformations of $L$ (possibly equipped with unitary local systems).
In particular, this implies rationality of the irreducible component containing
$x$; however, it is stronger.
  Under our assumptions, one can construct an algebraic action of
$H^1(M,\mathbb{G}_m)$, namely the action by non-unitary local systems. By
combining techniques from family Floer homology and non-commutative geometry,
we prove that this action coincides with the geometric action mentioned in the
first paragraph. We use this to deduce the theorems above.
