On the positivity and integrality of coefficients of mirror maps

1. Problem Statement and Context

The paper addresses a fundamental phenomenon in mirror symmetry, specifically concerning the integrality and positivity of coefficients in "mirror maps."

• Background: In genus-zero enumerative mirror symmetry (e.g., for the quintic threefold), there is a relationship between the generating function of Gromov–Witten invariants ($f(Q)$) and the Yukawa coupling of the mirror family ($g(q)$). The "mirror map" $Q(q)$ is the coordinate transformation linking these two.
• The Phenomenon: It has been observed (and partially proven) that the coefficients of these mirror maps are often integers. Furthermore, for specific cases like the quintic, the coefficients are not just integers but positive integers.
• The Gap: Previous works focused on a specific formulation of the mirror map (often called the "naive" map) or relied on specific combinatorial criteria (like nef partitions) where the "naive" and "true" maps coincide. There was no unified framework explaining the integrality of the "true" mirror map (which appears in rigorous mirror symmetry theorems) for general Calabi–Yau complete intersections in Fano toric varieties, nor a clear distinction between the two formulations.

2. Methodology

The authors develop a combinatorial framework to generalize the construction of mirror maps beyond the classical quintic case.

• Combinatorial Data: They define mirror maps using a set of vectors $(v_{ij})$ in $\mathbb{Z}^d$ indexed by a set $I$. These vectors define a kernel $K$ of a linear map $V: \mathbb{Z}^I \to \mathbb{Z}^d$.
• Two Definitions of Mirror Maps:
  1. Naive Mirror Map ($\psi^n_{ij}$): Defined as $\exp(\phi_{ij}/\phi_0)$, where $\phi_0$ and $\phi_{ij}$ are power series constructed from multinomial coefficients (combinations) and harmonic sums over the monoid $K_0$ (non-negative elements of the kernel).
  2. True Mirror Map ($\psi^t_{ij}$): Defined as $\exp((\phi_{ij} + \tau_{ij})/\phi_0)$. This includes an additional correction term $\tau_{ij}$, which is a sum over a different monoid $K_{ij}$ involving negative indices. This term arises from the rigorous derivation in mirror symmetry theorems.
• The "Fano" Condition: The authors introduce a geometric criterion called Fano data. A set of vectors is Fano if the origin is the unique interior lattice point of the Minkowski sum of the convex hulls of the vectors (and the origin).
• Connection to Delaygue's Criterion: The authors translate their setup into the framework of Delaygue (who studied integrality of hypergeometric series) to leverage existing theorems. They prove an equivalence between their "Fano" condition and Delaygue's integrality criterion under specific isomorphism conditions.
• Computer Verification: Extensive computational checks were performed using SageMath to verify conjectures for thousands of examples, including vertices of reflexive polytopes in dimensions 2 and 3, and random high-dimensional examples.

3. Key Contributions and Conjectures

The paper proposes two main conjectures distinguishing the behavior of the naive and true mirror maps:

Conjecture A (Naive Mirror Map)

If the data $(v_{ij})$ is Fano:

1. Integrality: The naive mirror map $\psi^n_{ij}$ has integer coefficients.
2. Positivity: The logarithm of the naive mirror map, $\log \psi^n_{ij} = \phi_{ij}/\phi_0$, has non-negative (positive) coefficients.

• Significance: This implies $\psi^n_{ij}$ has positive integer coefficients. This generalizes the known positivity of the quintic mirror map.

Conjecture B (True Mirror Map)

If the data $(v_{ij})$ is Fano:

1. Integrality: The true mirror map $\psi^t_{ij}$ has integer coefficients.

• Significance: This addresses the integrality of the map actually used in mirror symmetry theorems. The authors note that $\psi^t_{ij}$ does not necessarily have positive coefficients (it can have negative coefficients), distinguishing it from the naive map.

Key Theoretical Results

• Theorem 1.10: Proves Conjecture A(1) (Integrality of the naive map) under the condition that the monoid $K_0$ is isomorphic to $\mathbb{N}^r$. This covers a broad class of cases where the rank of the kernel is small relative to the dimension.
• Theorem 1.14: Proves Conjecture A(2) (Positivity of the log of the naive map) for the case where the rank of $K$ is 1.
• Lemma 1.8: Establishes that if the origin does not lie in the interior of the convex hull of the vectors excluding a specific pair, then the correction term $\tau_{ij}$ vanishes. In this case, the naive and true maps are identical, and Conjecture A(1) implies Conjecture B.
• Equivalence to Delaygue: The authors prove that their "Fano" hypothesis is equivalent to Delaygue's criterion for integrality in the case where $K_0 \cong \mathbb{N}^r$.

4. Results and Findings

• Distinction between Maps: The paper clarifies that the "naive" and "true" mirror maps are distinct in general. They coincide only under specific combinatorial conditions (e.g., the quintic threefold).
• Integrality vs. Positivity:
  • The naive map is conjectured to always have positive integer coefficients.
  • The true map is conjectured to have integer coefficients, but not necessarily positive.
  • Example 1.9 demonstrates a case where the true mirror map has a negative coefficient, proving that the positivity conjecture does not extend to the true map.
• Computational Evidence:
  • Verified Conjectures A and B for all 16 classes of 2D reflexive polytopes.
  • Verified for 4319 classes of 3D reflexive polytopes (checking the first 25–50 terms).
  • Verified for random high-dimensional examples and cases with $p > 1$ (multiple parameters).
• Geometric Interpretation: The authors pose a question regarding the geometric meaning of the naive mirror map. They speculate it might relate to relative Gromov–Witten invariants or curves inside specific divisors, though this remains an open problem.

5. Significance

• Unification: The paper provides a unified combinatorial framework that encompasses the classical Batyrev–Borisov construction and generalizes it to arbitrary Calabi–Yau complete intersections in Fano toric varieties.
• Clarification of "Integrality": It resolves the confusion in the literature by distinguishing between the "naive" map (often used in physics heuristics) and the "true" map (mathematically rigorous). It shows that while both are integral, only the naive map exhibits the strong property of positivity.
• New Proofs: It provides new proofs for integrality results previously known only for specific cases (like prime dimensions or rank 1), extending them to broader classes of toric varieties via the Fano condition.
• Foundation for Future Work: By identifying the precise conditions under which the naive map is positive and the true map is integral, the paper sets the stage for deeper arithmetic and geometric investigations into the structure of mirror symmetry, potentially linking these series to new enumerative invariants.

In summary, Bleau and Sheridan rigorously formalize the "positivity and integrality" phenomenon, proving it for wide classes of cases and conjecturing its validity for all Fano data, while highlighting the subtle but crucial differences between the naive and true formulations of mirror maps.