[Paper Review] Apéry constants of homogeneous varieties

Sergey Galkin|arXiv (Cornell University)|Apr 15, 2016

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper introduces Apéry constants and the Apéry class for Fano manifolds, particularly homogeneous varieties, as limits of coefficient ratios in solutions to the quantum differential equation. It establishes that these constants are polynomials in Riemann zeta values ζ(k) for natural numbers k, providing strong numerical evidence for a conjecture linking quantum cohomology to special values of zeta functions.

ABSTRACT

For Fano manifolds we define Apéry constants and Apéry class as particular limits of ratios of coefficients of solutions of the quantum differential equation. We do numerical computations in case of homogeneous varieties. These numbers are identified to be polynomials in the values of Riemann zeta-function with natural arguments.

Motivation & Objective

• To define and compute Apéry constants for Fano manifolds, especially homogeneous varieties, as limits of coefficient ratios in solutions to the quantum differential equation.
• To investigate the appearance of special values of the Riemann zeta function, ζ(k), in the asymptotic behavior of quantum cohomology solutions.
• To test a conjecture that Apéry constants arise as evaluations of homogeneous polynomials in zeta values under a map sending c₁ to Euler’s constant and cₖ to ζ(k).
• To explore the structure of Lefschetz decompositions in cohomology and their impact on convergence speeds of Apéry-type approximations.
• To examine whether Apéry limits can be interpreted as convolutions or square roots of geometric invariants, particularly in the context of Severi and Grassmannian varieties.

Proposed method

• Define Apéry constants as the limit limₙ→∞ aₙ^(γ)/aₙ^(0) for solutions A₀ and Aγ of the quantum D-module associated with the primitive class 1 and a coprime class γ.
• Use Givental’s J-series and the quantum D-module structure to generate holomorphic solutions via Newton’s method and matrix recursion on the quantum multiplication operator H⋆.
• Apply the Peterson version of the Quantum Chevalley formula to explicitly compute the action of H⋆ on cohomology for homogeneous spaces G/P, using computer algebra systems like LiE and PARI/GP.
• Perform numerical computations on homogeneous Fano varieties (e.g., Grassmannians, Lagrangian Grassmannians, orthogonal Grassmannians, and Severi varieties) to extract Apéry constants.
• Use the expansion logΓ(1−x) = Cx + ∑(k≥2) ζ(k)/k xᵏ to relate Γ-function coefficients to zeta values and infer the appearance of ζ(k) in Apéry constants.
• Analyze the Lefschetz decomposition of cohomology into blocks and study the asymptotic behavior of solution coefficients across different cohomological degrees.

Experimental results

Research questions

• RQ1 Are Apéry constants for homogeneous Fano varieties expressible as polynomials in Riemann zeta values ζ(k) for natural k?
• RQ2 Does the Apéry constant associated with a coprimitive cohomology class γ correspond to the evaluation of a homogeneous polynomial fγ ∈ R^(n) under a map sending c₁ to Euler’s constant and cₖ to ζ(k)?
• RQ3 How do the convergence speeds of Apéry-type approximations relate to geometric invariants such as the defect or dimension of the variety?
• RQ4 Can the Apéry constant be interpreted as a limit of ratios of components in the nilpotent cohomology ring, even when M₀ and M₁ do not commute?
• RQ5 What is the role of Lefschetz decomposition blocks in determining the asymptotic behavior of quantum cohomology solution coefficients?

Key findings

• Apéry constants for homogeneous Fano varieties are found to be polynomials in ζ(k) for natural numbers k, with no appearance of Euler’s constant C in the final results.
• For fourfolds, the monodromy of the quantum differential equation is expressed in terms of ζ(3), ζ(2k), and characteristic numbers of the anti-canonical section, supporting the emergence of zeta values in geometric invariants.
• In the case of Severi varieties (e.g., Grassmannians, orthogonal Grassmannians, and the Cartan variety E(6,6)), the Apéry number for the middle cohomology block is zero, indicating a degenerate approximation behavior.
• Numerical experiments suggest that Aₙ^(γ) ≈ Apéry(γ) · Aₙ^(0) as n → ∞, supporting the conjecture that the ratio of solution components converges to the Apéry constant in the nilpotent cohomology ring.
• The recursion structure of the quantum D-module implies that the asymptotic growth of solution coefficients is governed by Γ(1 + M₀), linking the Apéry limit to special functions and zeta values via the expansion of logΓ.
• For varieties with multiple Lefschetz blocks, the convergence speed of Apéry approximations appears better than for general varieties, suggesting a connection to geometric defect or symmetry.