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

Andrea Brini

TL;DR

The paper proves the Conifold Gap Conjecture for the local Calabi–Yau threefold $X=K_{\,\mathbb{P}^2}$ and uses a novel mean-field Coulomb-gas model to connect higher-genus Gromov–Witten potentials with the large-$N$ limit of random matrix theory. By solving a sequence of Riemann–Hilbert problems and implementing the Eynard–Orantin topological recursion on an elliptic spectral curve tied to the Hori–Iqbal–Vafa mirror, the authors link the conifold polar data to the GUE free energy and establish all-genus mirror symmetry via the BCOV holomorphic anomaly equations. The main result is that for each genus $g\ge 2$, the conifold potential satisfies $\mathrm{GW}^{\rm CF}_g = \frac{3^{g-1} B_{2g}}{2g(2g-2)} t_{\rm CF}^{2-2g} + \mathcal{O}(1)$, with the gap structure arising from modular and symmetric-data constraints. Consequently, the all-genus mirror map is realized through direct integration of the BCOV equations, providing a rigorous bridge between matrix-model asymptotics, topological recursion, and local mirror symmetry with significant implications for explicit higher-genus GW computations.

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.

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

TL;DR

The paper proves the Conifold Gap Conjecture for the local Calabi–Yau threefold and uses a novel mean-field Coulomb-gas model to connect higher-genus Gromov–Witten potentials with the large- limit of random matrix theory. By solving a sequence of Riemann–Hilbert problems and implementing the Eynard–Orantin topological recursion on an elliptic spectral curve tied to the Hori–Iqbal–Vafa mirror, the authors link the conifold polar data to the GUE free energy and establish all-genus mirror symmetry via the BCOV holomorphic anomaly equations. The main result is that for each genus , the conifold potential satisfies , with the gap structure arising from modular and symmetric-data constraints. Consequently, the all-genus mirror map is realized through direct integration of the BCOV equations, providing a rigorous bridge between matrix-model asymptotics, topological recursion, and local mirror symmetry with significant implications for explicit higher-genus GW computations.

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 -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 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 through the direct integration of the BCOV holomorphic anomaly equations.

Paper Structure

This paper contains 28 sections, 21 theorems, 209 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Local $\mathbb{P}^2$
  4. Strategy of the proof
  5. Relation to other work
  6. Organisation of the paper
  7. Review of mean-field models at large $N$
  8. Analytic strictly convex pairs
  9. Equilibrium measures
  10. Asymptotic expansions at $N=\infty$
  11. The Eynard--Orantin recursion
  12. Equivariant correlators
  13. The conifold model
  14. Strict convexity
  15. $\mu_3$-equivariant correlators
  16. ...and 13 more sections

Key Result

Theorem 1

The Conifold Gap Conjecture holds for $X=K_{\mathbb{P}^2}$ in all genera.

Figures (5)

  • Figure 1: The graph of $\Phi(x)$ (in blue) and of $\frac{2 \pi}{3\sqrt{3}} |\log x|+\phi_\infty$, superimposed (in orange).
  • Figure 2: The cylinder given by the Riemann sphere with the sides of the intervals $\mathfrak{I_{\pm}}$ cut open (left), and the topological genus one surface obtained upon identifying their opposite sides (right).
  • Figure 3: The fundamental domain $\Delta$ of $T\simeq \mathbb{R}^2/M$ (left) and the complex plane with cuts $\mathfrak{I}_\pm$ for $\mathcal{R}_{0,1}$ (right). .
  • Figure 4: The boundary rectangles of $\Delta_-$ (left, red) and $\Delta^{[\epsilon]}_-$ (left, green) and their images $\mathcal{J}_0$ (right, red) and $\mathcal{J}_\epsilon$ (right, green) under $x(u)$.
  • Figure 5: The fundamental domain for $\Gamma_1(3)$. The cusps at $\tau_{\rm LR}=\mathtt{i} \infty$ and $\tau_{\rm LR}=0$ ($\tau_{\rm CF}=\mathtt{i} \infty$) correspond to the large radius $(y=0)$ and conifold point $(y=-1/27)$ respectively. The cubic elliptic point at $\tau_{\rm LR}=\varphi$ corresponds to the orbifold point $y=\infty$.

Theorems & Definitions (34)

  • Conjecture : Conifold Gap property for $K_{\mathbb{P}^2}$
  • Theorem : =Corollary \ref{['cor:cg']}
  • Definition 2.1
  • Proposition 2.2: Borot:2013lpaMR3265167Borot:2013pda
  • Proposition 2.3: Off-criticality, Borot:2013pda
  • Proposition 2.4: Borot:2013pdaBorot:2013lpa
  • Proposition 2.5: Gap property
  • Proposition 2.6: Eynard--Orantin topological recursion, Eynard:2007hfBorot:2013lpa
  • Proposition 2.7
  • Proposition 2.8
  • ...and 24 more