Gamma conjectures for Milnor hypersurfaces

Abstract

We prove Gamma conjectures I and II of Galkin, Golyshev and Iritani for Milnor hypersurfaces, the smooth hypersurfaces of bidegree $(1,1)$ in the product of two projective spaces. We provide a criterion for Gamma conjecture II for general Fano manifolds, generalizing the criterion by Hu and Ke. Using the mirror symmetry technique of Galkin and Iritani, we establish global Gamma-I for Milnor hypersurfaces, thereby proving Gamma conjecture I as a special case. Using it, together with the Galois action and our criterion, we verify Gamma conjecture II for Milnor hypersurfaces.