Categorical action of the extended braid group of affine type $A$

Abstract : Using a quiver algebra of a double cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type $A$ on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.