Open Gromov–Witten sum-formula conjecture for Lagrangian real projective 3-spaces

Let XX be an almost Calabi–Yau 3-fold and let LRP3XL\cong \mathbb{R}P^3\subset X be an embedded Lagrangian. For odd classes βΠ2(X,L)\beta\in\Pi_2(X,L), the open Gromov–Witten invariants NβsymN_{\beta}^{sym} are considered in relation to an almost Calabi–Yau XoutX_{out} constructed from XX and homology classes Fβ,kH2(Xout,Z)F_{\beta',k}\in H_2(X_{out},\mathbb{Z}) constructed from β\beta'. Open Gromov–Witten sum-formula conjecture. For odd classes βΠ2(X,L)\beta\in\Pi_2(X,L), the invariants NβsymN_{\beta}^{sym} can be expressed as a function of ordinary Gromov–Witten invariants of XoutX_{out}, using the homology classes Fβ,kH2(Xout,Z)F_{\beta',k}\in H_2(X_{out},\mathbb{Z}). The proposed relation would connect open invariants for Lagrangian real projective 3-spaces with ordinary invariants of a symplectically cut-out manifold, but the source does not provide a formula or establish the claim.

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Primary source

Mohammad Farajzadeh Tehrani, “Open GW theory on symplectic manifolds and symplectic cutting”, arXiv:1003.4325 (2012).

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