Quantum Cohomology and Periods

Abstract

In a previous paper, the author introduced a Z-structure in quantum cohomology defined by the K-theory and the Gamma class and showed that it is compatible with mirror symmetry for toric orbifolds. Applying the quantum Lefschetz principle to the previous results, we find an explicit relationship between solutions to the quantum differential equation for toric complete intersections and the periods (or oscillatory integrals) of their mirrors. We describe in detail the mirror isomorphism of variations of Z-Hodge structure for a mirror pair of Calabi-Yau hypersurfaces (Batyrev's mirror).

39 pages, v2: improved the explanation on multi-generation, quintic example added, v3: one of the main theorem (Theorem 6.9) improved, references added

References in corpus (7)

• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• On the $K$-theory of smooth toric DM stacks
• tt*-geometry in quantum cohomology
• Period- and mirror-maps for the quartic K3
• Quantum cohomology via D-modules
• On the Grothendieck groups of toric stacks
• Local B-model and Mixed Hodge Structure

Cited by in corpus (27)

• Infinite Distance Networks in Field Space and Charge Orbits
• Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary
• A Mirror Theorem for Toric Stacks
• The Crepant Transformation Conjecture for Toric Complete Intersections
• Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
• Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies
• Polarized relations on horizontal SL(2)s
• Twistor property of GKZ-hypergeometric systems
• An analogue of Dubrovin's conjecture
• The Quantum Lefschetz Principle for Vector Bundles as a Map Between Givental Cones
• On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities
• Gromov-Witten invariants and localization
• Residue mirror symmetry for Grassmannians
• Gromov-Witten Invariants of Local P^2 and Modular Forms
• Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds
• A Mirror Theorem for the Mirror Quintic
• A Fock Sheaf For Givental Quantization
• Narrow quantum D-modules and quantum Serre duality
• Mirror symmetry and K3 surfaces
• The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods
• Modularity of Landau-Ginzburg models
• Asymptotics of the banana Feynman amplitudes at the large complex structure limit
• Apéry extensions
• A note on exceptional unimodal singularities and K3 surfaces
• On the better behaved version of the GKZ hypergeometric system
• Quintic periods and stability conditions via homological mirror symmetry
• Central charges in local mirror symmetry via hypergeometric duality