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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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