Log Topological Vertex Project

Define the Logarithmic \(\mathbb{C}^3\) Target: Formulate the affine patch \(X = \mathbb{C}^3\) as a log scheme with divisorial log structure given by the coordinate planes \(D = \{z_1z_2z_3 = 0\}\). Its tropicalization \(\Sigma(X)\) is the positive octant \(\mathbb{R}_{\ge 0}^3\).
Translate Boundary Conditions to Tropical Types: Replace the winding numbers (partitions \(\vec{\mu}\)) with tropical global types \(\tau = (G, g, \sigma, u)\). A map into the “vertex” will correspond to a tropical curve in \(\mathbb{R}_{\ge 0}^3\) with legs going to infinity along the three axes. The contact orders \(u_L \in \mathbb{Z}\) of these legs will match the parts of the partitions in \(\vec{\mu}\). Negative values in \(u_L\) correspond to the punctures.
Construct the Moduli Space: Form the moduli stack of basic punctured maps \(\mathfrak{M}(X/W, \tau)\). Because of the punctures, the base will be an idealized log scheme \((W, \mathcal{M}_W, \mathcal{K}_W)\).
Develop Idealized Equivariant Localization: The \(T = (\mathbb{C}^*)^3\) action on \(\mathbb{C}^3\) lifts to \(\mathfrak{M}(X/B, \tau)\). You must generalize the Atiyah-Bott localization formula to accommodate the idealized structure \(\mathcal{K}_W\). Extract the moving and fixed parts of the perfect relative obstruction theory \(\mathbb{G} \to \mathbb{L}_{\mathcal{M}/\mathfrak{M}^{ev}}\).
Evaluate the Logarithmic Vertex: Compute the localized virtual class and show that it identically recovers the three-partition Hodge integrals \(G_{\vec{\mu}}^\bullet(\lambda; w)\).
Apply the Splitting Theorem: Finally, use Siebert et al.’s splitting morphism \(\delta^{ev}\) (Theorem 5.19) along the evaluation stacks to mathematically rigorously glue the logarithmic vertices together to form the global GW invariant for any toric Calabi-Yau, completely avoiding the formal scheme gluing algorithms of Li et al.