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This talk begins with a brief review of the BCOV formalism for higher-genus Gromov–Witten theory, with emphasis on the holomorphic anomaly equations, polynomial structure, and boundary conditions. I will then discuss the BKMP Remodeling Conjecture, which relates the Gromov–Witten potentials of a toric Calabi–Yau threefold to the Eynard–Orantin free energies of its mirror curve. 更多阅读

The main focus is the family of non-semiprojective local curves $X_p={Tot}\!\left(\mathcal O_{\mathbb P^1}(p-1)\oplus\mathcal O_{\mathbb P^1}(-p-1)\right),\qquad p\ge 2$. 更多阅读

Using virtual localization, semisimple cohomological field theories, Givental–Teleman reconstruction, and an analysis of the quantum differential equation, we identify the higher-genus Gromov–Witten potentials of \(X_p\) with the free energies produced by topological recursion on an explicit spectral curve. We further prove that, for \(g\ge 2\), these potentials are expansions of rational functions with a controlled pole along the discriminant and polynomial numerators, thereby establishing the ansatz of Caporaso, Griguolo, Mariño, Pasquetti, and Seminara. Finally, we determine the leading behavior at the critical point and verify their conjecture that the double-scaling limit is governed by the \((3,2)\) minimal model.