A mirror theorem for toric stacks

2013/10/31 by Tom Coates, Alessio Corti, Hiroshi Iritani +2 · 89 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Algebraic geometry #Algebraic structures and combinatorial models #Cohomology #Equivariant cohomology #Genus #Hypergeometric function #Mathematics #Orbifold #Pure mathematics #Quantum cohomology #Toric variety #Zero (linguistics) #math.AG #msc:14A20 #msc:14N35 #msc:53D45 #msc:83E30

paper · pdf · doi:10.1112/s0010437x15007356

published in Compositio Mathematica 151(10), 1878-1912 (Cambridge University Press) · 35 pages. v2: key references added. v3: errors corrected; formal setup changed; proofs simplified, clarified, and shortened; references added. v4: references updated. v5: references updated

arxiv created 2014/10/02 · openalex publication_date 2015/06/01 · arxiv updated 2015/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove a Givental-style mirror theorem for toric Deligne–Mumford stacks X . This determines the genus-zero Gromov–Witten invariants of X in terms of an explicit hypergeometric function, called the I -function, that takes values in the Chen–Ruan orbifold cohomology of X .

Citations

• Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties
• Open orbifold Gromov-Witten invariants of [ℂ³/ℤ_n] : localization and mirror symmetry
• Wall-crossings in toric Gromov–Witten theory I: crepant examples
• Orbifold quantum Riemann–Roch, Lefschetz and Serre
• The orbifold Chow ring of toric Deligne-Mumford stacks
• Orbifold Quasimap Theory

Cited by

• On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities
• The quantum orbifold cohomology of toric stack bundles