  We study an $A_\infty$ category associated to Legendrian links in
$\mathbb{R}^3$ whose objects are $n$-dimensional representations of the
Chekanov-Eliashberg differential graded algebra of the link. This
representation category generalizes the positive augmentation category and we
conjecture that it is equivalent to a category of sheaves of microlocal rank
$n$ constructed by Shende, Treumann, and Zaslow. We establish the cohomological
version of this conjecture for a family of Legendrian $(2,m)$ torus links.
