Gamma conjecture I for del Pezzo surfaces

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.

28 pages. Comments are welcome!

References in corpus (3)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• The $\hat{\Gamma }$-genus and a regularization of an $S^1$-equivariant Euler class

Cited by in corpus (1)

• Gamma II for toric varieties from integrals on T-dual branes and homological mirror symmetry