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An integral structure in quantum cohomology and mirror symmetry for toric orbifolds. (English) Zbl 1190.14054
Consider a weak Fano projective toric orbifold \(\mathcal X\). The author introduces a \(\widehat \varGamma\)-integral structure on the quantum \(D\)-module of \(\mathcal X\), that is an integral structure on the space of flat sections of Dubrovin’s connection for \(\mathcal X\) given by a class \[ \widehat \varGamma(T\mathcal X)=\prod_{i=1}^{\dim \mathcal X}\varGamma(1+\delta_i), \] where \(\delta_i\)’s are Chern roots of \(\mathcal X\). The main theorem (Theorem 4.11) states that under some assumptions this integral structure corresponds, modulo Mirror Conjecture, to the natural integral local system on the mirror B-model \(D\)-module under the mirror isomorphism. In particular this holds for toric manifolds as assumptions are proven to hold. By assuming the existence of an integral structure, the author gives a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [in: AMS special session, San Francisco, CA, USA, May 3–4, 2003. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 403, 117–126 (2006; Zbl 1105.14078)].

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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