Gamma conjecture I for del Pezzo surfaces

TL;DR

This paper proves Gamma conjecture I and Conjecture O for all two-dimensional Fano manifolds, including del Pezzo surfaces, using mirror symmetry techniques and eigenvalue analysis of Gromov-Witten invariants.

Contribution

It establishes the validity of both conjectures for all two-dimensional Fano manifolds, extending previous partial results.

Findings

Conjecture O holds for all del Pezzo surfaces.

Gamma conjecture I is verified for all del Pezzo surfaces.

A generalized Perron-Frobenius theorem is developed for eigenvalue analysis.

Abstract

Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture $\mathcal{O}$ by deriving a generalized Perron-Frobenius theorem on eigenvalues of real matrices and a vanishing result of certain Gromov-Witten invariants for del Pezzo surfaces. We prove Gamma conjecture I by applying mirror techniques proposed by Galkin-Iritani together with the study of Gamma conjecture I for weighted projective spaces.