Categorical enumerative invariants for Fukaya categories equal Gromov–Witten invariants
Establish that for a compact symplectic manifold X with Fukaya category C = Fuk(X), there exists a canonical splitting s of the noncommutative Hodge filtration and an isomorphism HH^*(C) ≅ H^*(X, Λ), and prove that for any classes γ_i ∈ H^*(X, Λ) the coefficient of γ_1 u^{k_1} ⋯ γ_n u^{k_n} in the generating function 𝒟^{C,s} ∈ Sym(HH^*(C)⟦u⟧) equals the Gromov–Witten invariant ∫_{𝑀̄_{g,n}(X)} ψ^{k_1} ev_*(γ_1) ⋯ ψ^{k_n} ev_*(γ_n).
References
The following conjecture, likely going back to Kontsevich, see also Costello , phrases this expectations: \begin{Conj_nn} Let $X$ be a compact symplectic manifold\footnote{We write $\Lambda$ for its Novikov ring.} and denote $\mathcal{C}=Fuk(X)$ its Fukaya category. Then there exist a canonical splitting s and an isomorphism $HH*(\mathcal{C})\cong H*(X,\Lambda)$ and for $\gamma_i\in H*(X,\Lambda)$ the coefficient of $\gamma_1u{k_1}\cdots\gamma_nu{k_n}$ of $\mathcal{D}{\mathcal{C},s}\in Sym\left(HH*(\mathcal{C})\llbracket u\rrbracket\right)$ is the GW-invariant $$\int_{\bar{\mathcal{M}{g,n}(X)}\psi{k_1}ev(\gamma_1)\dots \psi{k_n}ev_(\gamma_n).$$ \end{Conj_nn}