Projective twists in A-infinity categories

Abstract: Given a Lagrangian V \cong CP^n in a symplectic manifold (M,\omega), there is an associated symplectomorphism \phi_V of M. We define the notion of a CP^n-object in an A-infinity-category A and use this to construct algebraically an A-infinity-functor \Phi_V and prove that it induces an autoequivalence of the derived category DA. We conjecture that \Phi_V corresponds to the action of \phi_V and prove this in the lowest dimension n=1. The construction is designed to be mirror to a construction of Huybrechts and Thomas.