The Quantum Orbifold Cohomology of Weighted Projective Spaces

Abstract

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental's heuristic argument, which relates small quantum cohomology to S^1-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov-Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. We also obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.

LaTeX, 45 pages. Version 2: minor changes. Version 3: further typos corrected. Version 4: minor changes. Version 5: major changes to the exposition (especially in Section 3), title changed. Version 6: this is the final version; to appear in Acta Mathematica

References in corpus (5)

• Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples
• Gromov-Witten theory of Deligne-Mumford stacks
• Crepant resolutions of weighted projective spaces and quantum deformations
• Hypergeometric Equations and Weighted Projective Spaces
• The Chen-Ruan Cohomology of Weighted Projective Spaces

Cited by in corpus (7)

• On the Crepant Resolution Conjecture in the Local Case
• Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples
• Crepant resolutions of weighted projective spaces and quantum deformations
• Examples of limits of Frobenius (type) structures: the singularity case
• The Full Orbifold $K$-theory of Abelian Symplectic Quotients
• Torsion in the full orbifold K-theory of abelian symplectic quotients
• Mirror fibrations and root stacks of weighted projective spaces