HMS for punctured tori and categorical mapping class group actions

Let X-n be a cycle of n projective lines, and T-n a symplectic torus with n punctures. In this paper we review results of [STZ] and [Si], which establish a version of homological mirror symmetry relating X-n and T-n, and define on D-b(Coh(X-n)) an action of the pure mapping class group of T-n.