Equality of the hybrid Floer–Morse and holomorphic curve-counting maps

Let $C$ be the underlying curve, let $\Sigma$ be a real exact Lagrangian in $T^*C$, and let $q_1,\dots,q_\kappa$ be points of $C$. The maps

$$\mathcal{F}_{\operatorname{hol}}\colon HW(\sqcup_{i=1}^\kappa T^*_{q_i}C)\to \operatorname{Mat}(N^{\kappa},\operatorname{BSk}_\kappa(\Sigma))$$

and

$$\mathcal{F}_{\operatorname{Mor}}\colon HW(\sqcup_{i=1}^\kappa T^*_{q_i}C)\to \operatorname{Mat}(N^{\kappa},\operatorname{BSk}_\kappa(\Sigma))$$

are respectively defined by holomorphic curve counts and by counting holomorphic curves coupled with folded Morse flow trees. Equality of the hybrid Floer–Morse and holomorphic curve-counting maps.

$$\mathcal{F}_{\operatorname{Mor}}=\mathcal{F}_{\operatorname{hol}}.$$

This identifies the hybrid Floer–Morse construction with the holomorphic curve-counting construction and is intended to make the connection with the spectral network associated to the spectral curve more transparent. The supplied text does not establish whether this equality is proved or remains conjectural.