Conifold gap and all-genus mirror symmetry for local $\mathbb{P}^2$

Andrea Brini

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arXiv · 2509.19298

Conifold gap and all-genus mirror symmetry for local $\mathbb{P}^2$

Andrea Brini

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Abstract

The Conifold Gap Conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes $G$-function. In this paper, I prove the Conifold Gap Conjecture for the local projective plane. The proof relates the higher genus conifold Gromov-Witten generating series of local $\mathbb{P}^2$ to the thermodynamics of a certain statistical mechanical ensemble of repulsive particles on the positive half-line. As a corollary, this establishes the all-genus mirror principle for local $\mathbb{P}^2$ through the direct integration of the BCOV holomorphic anomaly equations.

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Andrea Brini. 2025-09-23. Conifold gap and all-genus mirror symmetry for local $\mathbb{P}^2$. https://arxiv.org/abs/2509.19298

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