A braid group action on an $A_{\mathrm{\infty }}$-category for zigzag algebras

We construct differential graded enhancements of the zigzag algebras which were used by Khovanov, Seidel and Thomas to produce categorical braid group actions. These enhancements are related to $p$-differential graded structures by a version of Koszul duality. We prove that the minimal model $A_{\mathrm{\infty }}$-structure on the zigzag algebras is {\em not} formal. We construct a braid group action in this setting and suggest a symplectic interpretation.