Proof of the gamma conjecture for Fano 3-folds of Picard rank 1

Abstract

We verify the (first) Gamma Conjecture, which relates the gamma class of a Fano variety to the asymptotics at infinity of the Frobenius solutions of its associated quantum differential equation, for all of the 17 deformation classes of rank one Fano 3-folds. Doing this involves computing the corresponding limits ("Frobenius limits") for the Picard-Fuchs differential equations of Apéry type associated by mirror symmetry to the Fano families, and is achieved by two methods, one combinatorial and one using the modular properties of the differential equations. The Gamma Conjecture for Fano 3-folds always contains a rational multiple of the number ζ(3). We present numerical evidence suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

The arithmetic and topology of differential equations

“…The Gamma Conjecture was proved by its authors for projective spaces, toric manifolds, and certain toric complete intersections and Grassmannians, and in [31] for all of the Fano 3-folds with ρ = 1 (some cases of which were already known previously by work of Dubrovin and others). Actually, two methods of proof were given in [31].…”

“…The Gamma Conjecture was proved by its authors for projective spaces, toric manifolds, and certain toric complete intersections and Grassmannians, and in [31] for all of the Fano 3-folds with ρ = 1 (some cases of which were already known previously by work of Dubrovin and others). Actually, two methods of proof were given in [31]. The first is combinatorial and proceeds by giving explicit formulas for the coefficients of the power-series parts of the Frobenius solution Ψ i (z), involving the harmonic numbers 1+ 1 2 +· · ·+ 1 n and the nth partial sums of ζ(k) for 2 ≤ k ≤ i, while the second is based on the modular parametrizations of the power series involved, and more specifically on the properties of Eichler integrals of weight 4 Eisenstein series.…”

“…6), which is the one related to the Apéry numbers, is called V 12 and is defined by starting from the Fano 10-fold G(10, 5) (orthogonal Grassmannian of 5-dimensional isotropic subspaces in C 10 ) and then taking generic hyperplane sections 7 times in a row to reduce the dimension to 3. We now give the description of the mirrors of these varieties as found by Golyshev, though in a somewhat modified form taken from [31]. This description is completely modular.…”

“…Our first example is classical. We take as our Fuchsian group the principal congruence subgroup Γ = Γ(2) of Γ 1 , as our modular form f the square of the Jacobi theta function ϑ 3 defined in (31), and as our modular function t the Hauptmodul λ defined in (30). Here k = 1, so ϑ 3 (τ ) 2 should satisfy a differential equation of order 2 with respect to λ(τ ), and indeed…”

“…The quantum differential equation associated to F is defined in terms of its ("small") quantum cohomology ring. We will not give the complete definitions, since they play no role for us; a short description in the rank 1 case is given in Section 4 of [31] and more detailed expositions can be found in [16] and in the references listed at the beginning of the section. Very briefly and very roughly, the quantum cohomology ring is the vector space H * (F ; Q) ⊗ Q[z] (at least in the cases with ρ = 1; if ρ > 1 then one has to take z to be a multi-variable of length ρ) equipped with an associative multiplication extending the usual cup product (the specialization to z = 0) that is defined in terms of the genus 0 Gromov-Witten invariants of F (the counting functions of rational curves in F intersecting divisors with given homology classes and having a given image in H 2 (F ; Z)).…”