Real and integral structures in quantum cohomology I: toric orbifolds
arXiv:0712.2204
Abstract
We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. Secondly, we use mirror symmetry to calculate the "most natural" integral structure in quantum cohomology of toric orbifolds. We show that the integral structure pulled back from the singularity B-model is described only in terms of topological data in the A-model; K-group and a characteristic class. Using integral structures, we give a natural explanation why the quantum parameter should specialize to a root of unity in Ruan's crepant resolution conjecture.
66 pages, v2: many minor changes, v3: many changes, references added, the results on the integral structure and mirror symmetry (except for real structures) of this preprint have been revised in the paper arXiv:0903.1463
References in corpus (13)
- A New Cohomology Theory for Orbifold
- Geometry and Integrability of Topological-Antitopological Fusion
- Geometry and analytic theory of Frobenius manifolds
- Hypergeometric functions and mirror symmetry in toric varieties
- Hypergeometric periods for a tame polynomial
- Fourier-Laplace transform of a variation of polarized complex Hodge structure
- Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
- Rational curves on hypersurfaces [after A. Givental]
- Mixed twistor structures
- Quantum Cohomology and Crepant Resolutions: A Conjecture
- T-Duality and Homological Mirror Symmetry of Toric Varieties
- The Crepant Resolution Conjecture for Type A Surface Singularities
- Root Systems and the Quantum Cohomology of ADE resolutions
Cited by in corpus (21)
- An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
- Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples
- Perturbative Corrections to Kahler Moduli Spaces
- Special Points of Inflation in Flux Compactifications
- Quantum periods of Calabi-Yau fourfolds
- On the Crepant Resolution Conjecture in the Local Case
- Exact Partition Functions on RP2 and Orientifolds
- Lagrangian Floer theory on compact toric manifolds: survey
- tt*-geometry in quantum cohomology
- Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples
- Curvature of classifying spaces for Brieskorn lattices
- Gromov-Witten invariants and localization
- On SYZ mirror transformations
- Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
- The tt* structure of the quantum cohomology of CP^1 from the viewpoint of differential geometry
- Deresonating a Tate period
- Apéry constants of homogeneous varieties
- Solutions of the tt*-Toda Equations and Quantum Cohomology of Flag Manifolds
- Gamma structures and Gauss's contiguity
- Asymptotics of the banana Feynman amplitudes at the large complex structure limit
- Limits of families of Brieskorn lattices and compactified classifying spaces