Abstract:

Seidel in his paper [1] shows that the monodromy around a critical point of a Lefschetz fibration of an exact symplectic manifold is isotopic to the Dehn twist along a Lagrangian sphere. We follow his approach [2] to study the Dehn twist operation on Lagrangian submanifolds in the suspension of a symplectic manifold X and show that the Dehn twist on X is compatible with that on the suspension of X. Combined with [3], the categorical Dehn twist on the Fukaya category is compatible with local mirror symmetry and the mutation operation of sheaves of its mirror.

Ref:

[1] P. Seidel, Long exact sequence of symplectic Floer homology

[2] P. Seidel, Fukaya categories and Picard Lefschetz theory

[3] T. Bridgeland, T-structures on some local Calabi-Yau varieties