  We construct an action of the free group $F_n$ on the homotopy category of
projective modules over a finite dimensional zigzag algebra. The main theorem
in the paper is that this action is faithful. We describe the relationship
between homotopy classes of paths in the punctured disc and complexes of
projective zigzag modules and explore the connection between gradings on the
zigzag algebra and monoids in $F_n$. We use this connection to give homological
constructions of the standard and Bessis dual word length metrics on the free
group.
