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 Gamma 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.
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Acknowledgements
Both authors are grateful to Jianxun Hu for his encouragement and support, and would like to thank Giordano Cotti and Sergey Galkin for their comments. The authors also thank Weiqiang He for pointing out typos in an early version of this paper. Ke would like to thank Xingbang Hao, Changzheng Li for helpful discussions, and especially Di Yang for sharing his knowledge on Frobenius manifolds. We are also grateful to the referee for many helpful suggestions. The authors are partially supported by NSFC (No. 11831017), and Ke is also supported by NSFC (Nos. 11890662, 11771461, 11521101, 11601534), and Hu by NSFC (No. 11701579).
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