# How much further could the PSS morphism be pushed?

Given a closed symplectic manifold $$(X, \omega)$$, the well known Piunikhin-Salamon-Schwarz morphism identifies the quantum cohomology $$QH(X, \omega)$$ with the Hamiltonian Floer cohomology $$HF(X,H)$$ for a non-degenerate Hamiltonian $$H \in C^{\infty}(S^1 \times X, \mathbb{R})$$. Moreover, the PSS morphism respects the product structures.

The quantum product defined over $$QH(X, \omega)$$ uses the genus $$0$$ three-pointed Gromov-Witten invariants. By considering curves in $$X$$ with arbitrary genera and marked points, and cohomology classes in the Deligne-Mumford space $$\mathcal{M}_{g,n}$$, the cohomology group $$H^{*}(X, \mathbb{Q})$$ actually has a structure of cohomological field theory, see for instance this survey.

Treating the marked points on the curves as small loops, one might wonder that there should be a Hamiltonian version of the cohomological field theory, and a generalization of the PSS morphism might identify these two models. Does such a thing exist?