Categorified Braid Group Action 2: Relating Type-B and Type-A Braid Group Actions

Continuing from last week’s talk, we will create a similar story for the type-$B_n$ Artin braid group $\mathcal{A}(B_n)$. On the topological side, this group is isomorphic to the mapping class group of the disk with $n+1$ marked points $\mathbb{D}_{n+1}$, where we fix the first marked point. This topological interpretation then induces a natural action on curves in $\mathbb{D}_{n+1}$.

On the categorical side, we construct a suitable zigzag algebra $\mathscr{B}_n$, where we show that there is a categorical $\mathcal{A}(B_n)$-action on $\text{Kom}^b(\mathscr{B}_n\text{-prgrmod})$. Following Khovanov-Seidel’s idea, we then attempt to relate the topological and categorical actions by providing the type-$B_n$ version of the recipe that assigns complexes from curves. Rather than showing that this assignment actually intertwines the braid action directly, we will take a detour through relating the type-$B_n$ and type-$A_{2n-1}$ braid groups, both topologically and categorically.

This research is in collaboration with KieSeng Nge.