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Math Notes

University Mathematics

TheoremComplete

PSS Isomorphism

Theorem8.2Piunikhin-Salamon-Schwarz (PSS) Isomorphism

For a closed symplectic manifold (M,ω)(M, \omega), there is a canonical ring isomorphism Φ:QH∗(M)→∼HF∗(M)\Phi: QH^*(M) \xrightarrow{\sim} HF^*(M) between the quantum cohomology ring and the Hamiltonian Floer cohomology ring (both defined over the Novikov ring). The isomorphism is constructed by counting "spiked discs" — configurations interpolating between Morse flow lines and Floer cylinders.

Proof

Define Φ:CM∗(f)→CF∗(H)\Phi: CM^*(f) \to CF^*(H) by counting solutions to a mixed equation on the half-cylinder [0,∞)×S1[0, \infty) \times S^1: for s≫0s \gg 0, the equation is the Floer equation ∂su+J(∂tu−XH)=0\partial_s u + J(\partial_t u - X_H) = 0; for s≤0s \leq 0, it reduces to the gradient flow equation ∂su+∇f(u)=0\partial_s u + \nabla f(u) = 0. The transition is achieved by a smooth interpolation.

The map Φ\Phi is shown to be a chain map by analyzing the boundary of 1-dimensional moduli spaces. Invertibility follows by constructing a reverse map Ψ:CF∗→CM∗\Psi: CF^* \to CM^* using the reversed spiked disc equation and showing Φ∘Ψ∼id\Phi \circ \Psi \sim \mathrm{id} and Ψ∘Φ∼id\Psi \circ \Phi \sim \mathrm{id} via chain homotopies. □\square

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ExamplePSS for $\mathbb{CP}^n$

For CPn\mathbb{CP}^n, the PSS isomorphism identifies the quantum product H∗H∗⋯∗H=q⋅1H * H * \cdots * H = q \cdot \mathbf{1} (in QH∗QH^*) with the composition of Floer continuation maps, providing a Floer-theoretic interpretation of the quantum relation Hn+1=qH^{n+1} = q.

RemarkApplications

The PSS isomorphism enables: (1) computation of Floer homology via quantum cohomology; (2) construction of spectral invariants by filtering through the PSS map; (3) proof that the pair-of-pants product in Floer theory equals the quantum product. It is the bridge between the enumerative world of Gromov-Witten theory and the dynamical world of Hamiltonian Floer theory.