Categorified Braid Group Action 1: Khovanov-Seidel's Curves and ZigZag Algebra

In the 1920s, Emil Artin firstly introduced the notion of braid explicitly. There are fews ways to describe an Artin braid group. Here, we want to discuss two stories of Artin braid group, or more generally is called the quiver type $A_n$ braid group $\mathcal{A}_n$; one can realise it as a mapping class group of a $n+1-$punctured disc $\mathbb{D}_{n+1}$ and one can also consider its categorical action on the bounded homotopy category Kom$^b(g_{r}p_{r} \mathscr{A}_n - \text{mod})$ of graded projective modules over $\mathscr{A}_n$ zigzag algebra. In 2001, Khovanov and Seidel managed to connect both the topological side and categorical side of $\mathcal{A}_n$ story using their own recipe converting a curve in $\mathbb{D}_{n+1}$ into a chain complex in Kom$^b(g_{r}p_{r} \mathscr{A}_n - \text{mod})$ with the intertwining action of the braid group. In this way, they use the geometric intersection numbers on curves to show that the only braid which acts trivially on curves is the identity braid. Under their recipe, these geometric intersection numbers on two given curves corresponds to the Poincar\’e polynomial of the two graded chain complexes obtained from the given curves, thereby proving the faithfulness of “categorified” braid group action on the category Kom$^b(g_{r}p_{r} \mathscr{A}_n - \text{mod})$. The project that Edmund Heng and I currently work on is to tell the same story for the quiver type $B_n$ braid group $\mathcal{B}_n$ and link the story of $\mathcal{B}_n$ and $\mathcal{A}_n$; these will be told in the sequel to the talk next week.