Cluster Structures on Double Bott–Samelson Cells

Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of $\Delta \square \mathrm {A}_r$ . When $\mathsf {C}$ is of type $\mathrm {A}$ , the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb {F}_q$ -points we obtain rational functions that are Legendrian link invariants.