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

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.

• The paper proves the Conifold Gap Conjecture by aligning higher genus GW potentials with the asymptotic expansion of the GUE for local P².
• It employs a statistical mechanical model and Eynard–Orantin topological recursion to reconstruct higher genus generating series.
• The results bridge string theory and algebraic geometry, opening new research avenues in mirror symmetry and Calabi–Yau manifolds.

Conifold Gap and All-Genus Mirror Symmetry for Local $\mathbb{P}^2$

Introduction

The paper "Conifold Gap and All-Genus Mirror Symmetry for Local $\mathbb{P}^2$" (2509.19298) investigates the Conifold Gap Conjecture and the all-genus mirror symmetry for the local projective plane. The main result of the work provides a proof for the Conifold Gap Conjecture specifically for the local $\mathbb{P}^2$, which is the total space of the canonical bundle over the projective plane. The study is embedded in the complex theory of Gromov–Witten (GW) invariants of Calabi–Yau threefolds and relates these to statistical mechanical models and topological recursion, revealing new connections in string theory and algebraic geometry.

Background and Methodology

The Conifold Gap Conjecture

The Conifold Gap Conjecture posits that the polar part of the Gromov–Witten potential of a Calabi–Yau threefold in proximity to its conifold locus assumes a universal form described using the logarithm of the Barnes G-function. The proof for the local projective plane is accomplished through a series of intricate connections between the conifold GW potentials and a certain statistical mechanical model. The proof relies on showing that these GW potentials match the asymptotic expansion of a particular large rank perturbation of the Gaussian Unitary Ensemble (GUE).

Statistical Mechanical Analogy and Topological Recursion

A key methodological point in the paper is the introduction of a statistical mechanical model: an ensemble of repulsive particles on a positive half-line whose thermodynamic limit elucidates the higher genus GW generating series of local $\mathbb{P}^2$. Within this framework, the authors deploy Eynard–Orantin topological recursion to reconstruct the series expansions necessary for proving the conjecture, leveraging connections to random matrix theory and spectral data from string theory.

Results

A significant result demonstrated in the paper is that the higher genus generating series for local $\mathbb{P}^2$ align with the asymptotic expansion of the GUE, comprehensively proving the Conifold Gap Conjecture for this geometric setting. This finding hinges critically on connecting the topological recursion yields on an elliptic curve with the mirror symmetry transformations governed by the modular curve $Y_1(3)$.

Implications and Future Directions

The findings have profound implications for theoretical physics and mathematicians, enhancing our understanding of mirror symmetry in non-trivial Calabi–Yau settings. The methodology and results open up potential new avenues for research into higher genus potentials for other Calabi–Yau manifolds, suggesting that similar techniques could potentially be applied in more complex geometric scenarios. Furthermore, the successful alignment of statistical mechanical models with string theory predictions represents a significant conceptual breakthrough, suggesting future exploration of other conjectures where statistical physics may illuminate geometric insights.

Conclusion

The paper provides a rigorous proof of the Conifold Gap Conjecture for local $\mathbb{P}^2$, connecting disparate mathematical domains and contributing significantly to the field of complex geometry and string theory. The introduction of statistical mechanical models and the use of topological recursion to solve longstanding conjectures highlight innovative approaches that can broaden our understanding of mirror symmetry across multiple dimensions and settings.