  We consider the algebraic effective couplings for open superstring massless
modes in the framework of the $A_\infty$ theory in the small Hilbert space.
Focusing on quartic algebraic couplings, we reduce the effective action of the
$A_\infty$ theory to the Berkovits one where we have already shown that such
couplings are fully computed from contributions at the boundary of moduli
space, when the massless fields under consideration are appropriately charged
under an ${\cal N}\!=\!2$ $R$-symmetry. Here we offer a proof of localization
which is in the small Hilbert space. We also discuss the flat directions of the
obtained quartic potentials and give evidence for the existence of exactly
marginal deformations in the $D3/D(-1)$ system in the framework of string field
theory.
