Hyperelliptic integrals and mirrors of the Johnson–Kollár del Pezzo surfaces

For all integers k > 0 k>0 , we prove that the hypergeometric function \[ I ^ k ( α ) = ∑ j = 0 ∞ ( ( 8 k + 4 ) j ) ! j ! ( 2 j ) ! ( ( 2 k + 1 ) j ) ! 2 ( ( 4 k + 1 ) j ) ! α j \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus 3 k + 1 3k+1 . We prove that the function I ^ k \widehat {I}_k is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X 8 k + 4 ⊂ P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1) . Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that | − K X | = | O X ( 1 ) | = ∅ |-K_X|=|\mathcal {O}_X (1)|=\varnothing . This means that there is no way to form a Calabi–Yau pair ( X , D ) (X,D) out of X X and hence there is no known mirror construction for X X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.