Gamma conjecture II for quadrics

TL;DR

This paper proves the Gamma conjecture II for smooth quadric hypersurfaces by establishing convergence of quantum cohomology, deriving explicit formulas for spinor bundles, and analyzing asymptotic behaviors of flat sections.

Contribution

It provides a criterion for Gamma II in Fano manifolds with semisimple quantum cohomology and verifies the conjecture for quadrics through explicit calculations and asymptotic analysis.

Findings

Convergence of full quantum cohomology for quadrics.

Explicit formulas for Chern characters of spinor bundles.

Verification of Gamma conjecture II for quadrics.

Abstract

The Gamma conjecture II for the quantum cohomology of a Fano manifold $F$, proposed by Galkin, Golyshev and Iritani, describes the asymptotic behavior of the flat sections of the Dubrovin connection near the irregular singularities, in terms of a full exceptional collection, if there exists, of $\mathcal D^b(F)$ and the $\widehat{\Gamma}$-integral structure. In this paper, for the smooth quadric hypersurfaces we prove the convergence of the full quantum cohomology and the Gamma conjecture II. For the proof, we first give a criterion on Gamma II for Fano manifolds with semisimple quantum cohomology, by Dubrovin's theorem of analytic continuations of semisimple Frobenius manifolds. Then we work out a closed formula of the Chern characters of spinor bundles on quadrics. By the deformation-invariance of Gromov-Witten invariants we show that the full quantum cohomology can be reconstructed by its ambient part, and use this to obtain estimations. Finally we complete the proof of Gamma II for quadrics by explicit asymptotic expansions of flat sections corresponding to Kapranov's exceptional collections and an application of our criterion.