An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

By H. Iritani. 2009

292 citations recorded by the source corpus.

Abstract We introduce an integral structure in orbifold quantum cohomology associated to the K -group and the Γ ˆ -class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau–Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan's crepant resolution conjecture [Yongbin Ruan, The cohomology ring of crepant resolutions of orbifolds, in: Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 117–126].

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1d432badafbd2e59b1279a74022a84eaf133e654

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Autoequivalences of derived categories via geometric invariant theory (2013, unknown): “Thus mirror symmetry predicts that the automorphism Tγ ∈ Aut(Db(V )) corresponds to the the monodromy of quantum connection around γ under the twisted Chern character ch defined in [16].”
The Crepant Transformation Conjecture for Toric Complete Intersections (2014, methodology): “In the papers [51,71], the toric quantum connection is described in terms of GKZ-type differential equations through mirror symmetry.”
Orbifold quasimap theory (2014, unknown): “3 of [20] after the identification of H∗ T (ĪμX) with H ∗ T (IμX) by the pullback ̟∗.”
Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures (2014, methodology): “The Gamma Conjectures for Pn were implicit but essentially shown in the work of Dubrovin [19]; they also follow from mirror symmetry computations in [44, 45, 50].”
Wall-crossing in genus zero quasimap theory and mirror maps (2013, methodology): “be used by the authors in a forthcoming work to obtain similar ε-wallcrossing formulas for such targets. In particular, we will obtain orbifold mirror theorems more general than the ones announced in [I] for toric orbifolds and complete intersections in them. In a diﬀerent direction, we expect wall-”
Yangians, quantum loop algebras and abelian difference equations (2013, background): “Their relevance seem to be new, although they are very closely related to the Γ -factors which appear in quantum cohomology [34,45].”