Braid Group Action on $D^b(\mathfrak{M}_η)$

Abstract: We construct an action of the braid group on the bounded derived category of coherent sheaves on hypertoric varieties arising from hyperplane arrangements. Using wall-crossing equivalences associated to paths in the complexified complement of the hyperplane arrangement, we show that these equivalences yield a functor from the Deligne groupoid to the category of triangulated equivalences. This gives rise to a canonical representation of the fundamental group, which under suitable assumptions recovers the braid group, acting on $D^b(\mathfrak{M}_{\eta})$.