Anti-symplectic Involution and Maslov Indices

We carry out some first steps in setting up a theory for Lagrangian Floer theory, mimicking Seidel’s construction for Hamiltonian Floer homology [7], for the subgroup HamL(M,ω) of Ham(M,ω) which preserves the Lagrangian L. When the symplectic manifold M has anti-symplectic involution c and L is the fixed Lagrangian submanifold, we consider the subgroup Hamc(M,ω) which commute with c. In the later case, we relate the critical points of the two Floer theories and their Maslov indices.