On Equivalence of Two Constructions of Invariants of Lagrangian Submanifolds

Let M be a compact smooth manifold. Its cotangent bundle T ∗M carries a natural symplectic structure associated to a Liouville form θ = pdq. For a given compactly supported Hamiltonian function H : T ∗M → R and a closed submanifold N ⊂ M Oh [30, 27] defined a symplectic invariants of certain Lagrangian submanifolds in T ∗M in a following way. Let ν∗N ⊂ T ∗M be a conormal bundle of N . Denote by HF λ ∗ (H,N ;M) the Floer homology groups generated by Hamiltonian orbits γ starting at the zero section and ending at ν∗N such that AH(γ) := ∫ γ pdq−Hdt ≤ λ (see, e.g., [30]). In particular, for λ = ∞ we write HF∗(H,N ;M) := HF∞ ∗ (H,N ;M). These groups are known to be isomorphic to H∗(N) [31]. We denote the corresponding isomorphism by F . For a ∈ H∗(N) one defines ρ(a,H : N) := inf{λ | FH(a) ∈ Im(j ∗ ) ⊂ HF∗(H,N ;M)}, (1)