Conifold Gap Conjecture

• Conifold Gap Conjecture is a foundational idea in enumerative geometry and mathematical physics that predicts a universal gap in the Gromov-Witten free energy near the conifold point.
• It connects the asymptotic behavior of the Barnes G-function with matrix model methods, mirror symmetry, and integrable hierarchies, enabling precise predictions for singular structures.
• The conjecture’s resolution clarifies holomorphic anomaly equations and moduli connectivity in Calabi–Yau manifolds, offering actionable insights for advanced enumerative and string theory research.

The Conifold Gap Conjecture is a foundational assertion in enumerative geometry and mathematical physics that stipulates a universal gap in the singular part of the higher genus Gromov-Witten free energy for Calabi-Yau threefolds at the conifold locus of the moduli space. This conjecture precisely identifies the polar part of the free energy as governed by the asymptotics of the Barnes G-function, with all subleading singular terms vanishing and only regular contributions remaining. The phenomenon encapsulated by the Conifold Gap Conjecture thereby links the geometry of Calabi-Yau singularities, integrable hierarchies, matrix models, BPS state counting, mirror symmetry, and the holomorphic anomaly equations of BCOV theory.

1. Statement of the Conjecture and Universal Polar Structure

The Conifold Gap Conjecture predicts that, for Calabi-Yau threefolds and especially local geometries such as the total space of the canonical bundle over $\mathbb{P}^2$ ("local $\mathbb{P}^2$"), the all-genus Gromov-Witten free energies $GW_{g}^{(CF)}$ near the conifold locus $t_{CF} \to 0$ have a Laurent expansion whose leading polar part is universal: $$GW_{g}^{(CF)} = \frac{3^{g-1} B_{2g}}{2g(2g-2)}\, t_{CF}^{2-2g} + O(1),$$ for $g \geq 2$, with $B_{2g}$ the $2g$th Bernoulli number. All terms beyond this polar contribution are regular, i.e., analytic at $t_{CF}=0$ (Brini, 23 Sep 2025). This universal gap coincides with the leading singularity in the expansion of the logarithm of the Barnes G-function,

$$\log G(N+1) \sim \sum_{g=2}^{\infty} \frac{B_{2g}}{2g(2g-2)} N^{2-2g},$$

thereby establishing a direct correspondence between the asymptotic behavior of Gromov-Witten potentials and special functions in statistical mechanics (Brini, 23 Sep 2025).

2. Matrix Model Realization and Statistical Mechanics

The proof for local $\mathbb{P}^2$ involves the construction of a mean-field matrix model—essentially a generalised Coulomb gas of repulsive particles on the positive half-line, interacting through a strictly convex two-body potential. The partition function $Z^{(\Phi, \Lambda)}$ of this ensemble admits a topological expansion: $$\log Z^{(\Phi, \Lambda)} \sim -\frac{1}{12}\log N + \sum_{g=0}^\infty \left(\frac{N}{t}\right)^{2-2g} F_g^{(\Phi, \Lambda)},$$ where the leading large-$N$ behavior and the subleading topological corrections correspond to the equilibrium measure and loop equations of the Coulomb gas (Brini, 23 Sep 2025).

The result is that, for potentials in the "analytic strictly convex pair" class, the expansion of the matrix model's free energy reproduces the universal singularity given by the Barnes G-function. The mapping between the conifold Gromov-Witten series and the matrix model expansion is performed after a specific change of variables linking the ’t Hooft coupling and the mirror moduli coordinate (Brini, 23 Sep 2025).

3. Integration of BCOV Holomorphic Anomaly and Mirror Symmetry

The Barnes G-function universality fixes the "holomorphic ambiguity" in the all-genus integration of the BCOV holomorphic anomaly equations. Direct integration, utilizing the gap behavior of the conifold free energies, yields the all-genus mirror principle for local $\mathbb{P}^2$; specifically, the topological recursions on the mirror elliptic curve (with appropriate Torelli marking) produce free energies that match those computed via the statistical mechanical ensemble (Brini, 23 Sep 2025).

This resolution sharply distinguishes singularity-generic contributions (regular at $t_{CF}=0$) from singular ones, with the latter entirely captured by the universal Barnes G-function expansion. The interplay guarantees a unique solution to the anomaly equations in terms of mirror geometry, thereby answering longstanding conjectures regarding enumerative invariants and their moduli dependence.

4. Connections to Integrable Hierarchies and Special Functions

The matrix model representation connects the Gromov-Witten theory with integrable hierarchies, particularly the Ablowitz-Ladik hierarchy (Alim et al., 2021). The conifold gap property is encoded in a discrete difference equation satisfied by the GW potentials: $$F(X, t+x) - 2F(X, t) + F(X, t-x) = \log(1 - Q) \quad \text{where}\ Q = \exp(2\pi i t),$$ which governs the absence of intermediate terms in the expansion and reflects the integrable structure underlying the enumerative theory. Distinguished solutions to these equations are identified with tau functions arising in wall-crossing phenomena of Donaldson-Thomas invariants, further reinforcing the universality of the gap (Alim et al., 2021).

5. Representation via Tropical and SYZ Mirror Geometry

The gap structure finds geometrical embodiment in SYZ mirror symmetry and tropical geometry. Explicit analysis of Lagrangian torus fibrations and their duals via family Floer theory reveals that singular T-duality fibers correspond to codimension-2 missing points in the mirror cluster variety, filling the "gap" in moduli space (Yuan, 2022). The matching between the discriminant loci in both the A-model and mirror B-model settings captures the precise nature of the missing singularities and their role in the mirror correspondence, thereby geometrically encoding the conifold gap phenomenon.

6. Physical Interpretation and BPS State Counting

In M-theory and topological string theory, the gap is reflected in the BPS spectrum: for the conifold, a unique genus-zero Gopakumar–Vafa invariant appears ($n_1 = 1$), with all lower-degree states absent (Collinucci et al., 2022). More complicated flop transitions exhibit higher-degree BPS bound states, demonstrating that the gap is a special limit. The gap's physical manifestation is thus a sparseness in the BPS spectra, directly encoded in the enumerative geometry (Collinucci et al., 2022).

In resolved conifold settings, Ooguri–Vafa integrality and mirror curve constructions further support the discrete BPS-counting structure in explicit closed formulae, verifying that vanishing low-degree contributions correspond to the predicted gap (Zhou, 2010).

7. Consequences for Moduli Space Connectivity and Enumerative Predictions

Existence of the universal conifold gap supports the prediction that the landscape of Calabi–Yau moduli spaces is highly connected via conifold transitions (Xu, 2012). The gap behavior ensures that singularities can be traversed in moduli space without encountering intermediate states, leading to phase transitions between distinct topological and geometric configurations. This underpins both the categorical and geometric perspectives on conifold transitions in string theory, F-theory, and algebraic geometry, with ramifications for mirror symmetry, wall-crossing phenomena, and the computation of Gromov-Witten invariants.

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The Conifold Gap Conjecture, now proven for local $\mathbb{P}^2$ (Brini, 23 Sep 2025), is thus central in the mathematical physics of Calabi–Yau manifolds, providing universal predictions for the structure of Gromov-Witten potentials, their singularities, and their relation to special functions, integrable systems, and moduli connectivity. Its resolution involves deep interactions between enumerative geometry, mirror symmetry, statistical mechanics, and quantum field theory.