Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-10T04:16:55.855Z Has data issue: false hasContentIssue false
  • English
  • Français

A mirror theorem for toric stacks

Part of: Surfaces and higher-dimensional varieties Symplectic geometry, contact geometry Special varieties Algebraic geometry: Foundations Projective and enumerative geometry

Published online by Cambridge University Press:  01 June 2015

Tom Coates ,
Alessio Corti ,
Hiroshi Iritani  and
Hsian-Hua Tseng
Tom Coates
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK email t.coates@imperial.ac.uk
Alessio Corti
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK email a.corti@imperial.ac.uk
Hiroshi Iritani
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan email iritani@math.kyoto-u.ac.jp
Hsian-Hua Tseng
Affiliation:
Department of Mathematics, Ohio State University, 100 Math Tower, 231 West 18th Ave., Columbus OH 43210, USA email hhtseng@math.ohio-state.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Keywords

Gromov–Witten theorytoric Deligne–Mumford stacksorbifoldsquantum cohomologymirror symmetryGivental’s symplectic formalismhypergeometric functions

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14A20: Generalizations (algebraic spaces, stacks) 14M25: Toric varieties, Newton polyhedra 14J33: Mirror symmetry 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 10 , October 2015 , pp. 1878 - 1912
Creative Commons
Creative Common License - CCCreative Common License - BY
This article is distributed with Open Access under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
Copyright
© The Authors 2015

References

Abramovich, D., Graber, T. and Vistoli, A., Algebraic orbifold quantum products, in Orbifolds in mathematics and physics (Madison, WI, 2001), Contemporary Mathematics, vol. 310 (American Mathematical Society, Providence, RI, 2002), 1–24; MR 1950940 (2004c:14104).Google Scholar
Abramovich, D., Graber, T. and Vistoli, A., Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008), 1337–1398; MR 2450211 (2009k:14108).CrossRefGoogle Scholar
Bayer, A. and Cadman, C., Quantum cohomology of [ℂN∕𝜇r], Compositio Math. 146 (2010), 1291–1322; MR 2684301 (2012d:14095).CrossRefGoogle Scholar
Borisov, L. A., Chen, L. and Smith, G. G., The orbifold Chow ring of toric Deligne–Mumford stacks, J. Amer. Math. Soc. 18 (2005), 193–215 (electronic); MR 2114820 (2006a:14091).CrossRefGoogle Scholar
Brini, A. and Cavalieri, R., Open orbifold Gromov–Witten invariants of [ℂ3∕ℤn]: localization and mirror symmetry, Selecta Math. (N.S.) 17 (2011), 879–933; MR 2861610 (2012i:14068).CrossRefGoogle Scholar
Brown, J., Gromov–Witten invariants of toric fibrations, Int. Math. Res. Not. IMRN 2014(19) (2014), 5437–5482; MR 3267376.CrossRefGoogle Scholar
Bryan, J. and Graber, T., The crepant resolution conjecture, in Algebraic geometry—Seattle 2005. Part 1, Proceedings of Symposia in Pure Mathematics, vol. 80 (American Mathematical Society, Providence, RI, 2009), 23–42; MR 2483931 (2009m:14083).Google Scholar
Cadman, C. and Cavalieri, R., Gerby localization, Z 3 -Hodge integrals and the GW theory of [ℂ3∕Z 3], Amer. J. Math. 131 (2009), 1009–1046; MR 2543921 (2010e:14051).CrossRefGoogle Scholar
Chan, K., Cho, C.-H., Lau, S.-C. and Tseng, H.-H., Gross fibrations, SYZ mirror symmetry, and open Gromov–Witten invariants for toric Calabi–Yau orbifolds, Preprint (2013), arXiv:1306.0437 [math.SG].Google Scholar
Chan, K., Cho, C.-H., Lau, S.-C. and Tseng, H.-H., Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds, Comm. Math. Phys. 328 (2014), 83–130; MR 3196981.CrossRefGoogle Scholar
Chen, W. and Ruan, Y., Orbifold Gromov–Witten theory, in Orbifolds in mathematics and physics (Madison, WI, 2001), Contemporary Mathematics, vol. 310 (American Mathematical Society, Providence, RI, 2002), 25–85; MR 1950941 (2004k:53145).CrossRefGoogle Scholar
Chen, W. and Ruan, Y., A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004), 1–31; MR 2104605 (2005j:57036).CrossRefGoogle Scholar
Cheong, D., Ciocan-Fontanine, I. and Kim, B., Orbifold quasimap theory, Preprint (2014),arXiv:1405.7160 [math.AG].CrossRefGoogle Scholar
Ciocan-Fontanine, I. and Kim, B., Wall-crossing in genus zero quasimap theory and mirror maps, Algebr. Geom. 1 (2014), 400–448; MR 3272909.CrossRefGoogle Scholar
Coates, T., Corti, A., Iritani, H. and Tseng, H.-H., Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009), 377–438; MR 2510741 (2010a:14090).CrossRefGoogle Scholar
Coates, T., Corti, A., Iritani, H. and Tseng, H.-H., Some applications of the mirror theorem for toric stacks, Preprint (2014), arXiv:1401.2611 [math.AG].CrossRefGoogle Scholar
Coates, T. and Givental, A., Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53; MR 2276766 (2007k:14113).CrossRefGoogle Scholar
Coates, T., Iritani, H. and Tseng, H.-H., Wall-crossings in toric Gromov–Witten theory. I. Crepant examples, Geom. Topol. 13 (2009), 2675–2744; MR 2529944 (2010i:53173).CrossRefGoogle Scholar
Coates, T., Lee, Y.-P., Corti, A. and Tseng, H.-H., The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009), 139–193; MR 2506749 (2010f:53155).CrossRefGoogle Scholar
Corti, A., Quantum orbifold cohomology of weak Fano toric DM stacks, Talk at Institute Henri Poincaré, February 15, 2007, as part of the Workshop on Quantum Cohomology of Stacks and String Theory, February 12–16, 2007.Google Scholar
Corti, A. and Golyshev, V., Hypergeometric equations and weighted projective spaces, Sci. China Math. 54 (2011), 1577–1590; MR 2824960.CrossRefGoogle Scholar
Cox, D. A., Little, J. B. and Schenck, H. K., Toric varieties, Graduate Studies in Mathematics, vol. 124 (American Mathematical Society, Providence, RI, 2011); MR 2810322 (2012g:14094).Google Scholar
Fang, B., Liu, C.-C. M. and Tseng, H.-H., Open-closed Gromov–Witten invariants of 3-dimensional Calabi–Yau smooth toric DM stacks, Preprint (2012), arXiv:1212.6073 [math.AG].CrossRefGoogle Scholar
Fantechi, B., Mann, E. and Nironi, F., Smooth toric Deligne–Mumford stacks, J. Reine Angew. Math. 648 (2010), 201–244; MR 2774310 (2012b:14097).Google Scholar
Givental, A. B., Homological geometry. I. Projective hypersurfaces, Selecta Math. (N.S.) 1 (1995), 325–345; MR 1354600 (97c:14052).CrossRefGoogle Scholar
Givental, A. B., A mirror theorem for toric complete intersections, in Topological field theory, primitive forms and related topics (Kyoto, 1996), Progress in Mathematics, vol. 160 (Birkhäuser, Boston, 1998), 141–175; MR 1653024 (2000a:14063).CrossRefGoogle Scholar
Givental, A. B., Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551–568; 645, Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary; MR 1901075 (2003j:53138).CrossRefGoogle Scholar
Givental, A. B., Symplectic geometry of Frobenius structures, in Frobenius manifolds, Aspects Mathematics, E36 (Friedr. Vieweg, Wiesbaden, 2004), 91–112; MR 2115767 (2005m:53172).CrossRefGoogle Scholar
Gonzalez, E. and Woodward, C. T., Quantum cohomology and toric minimal model programs, Preprint (2012), arXiv:1207.3253 [math.AG].Google Scholar
Gonzalez, E. and Woodward, C. T., A wall-crossing formula for Gromov–Witten invariants under variation of GIT quotient, Preprint (2012), arXiv:1208.1727 [math.AG].Google Scholar
Guest, M. and Sakai, H., Orbifold quantum D-modules associated to weighted projective spaces, Comment. Math. Helv. 89 (2014), 273–297; MR 3225449.CrossRefGoogle Scholar
Iritani, H., Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (2006), 577–622; MR 2207760 (2007e:53118).CrossRefGoogle Scholar
Iritani, H., An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009), 1016–1079; MR 2553377 (2010j:53182).CrossRefGoogle Scholar
Iritani, H., Quantum cohomology and periods, Ann. Inst. Fourier (Grenoble) 61 (2011), 2909–2958; MR 3112512.CrossRefGoogle Scholar
Iwanari, I., The category of toric stacks, Compositio Math. 145 (2009), 718–746; MR 2507746 (2011b:14113).CrossRefGoogle Scholar
Iwanari, I., Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Publ. Res. Inst. Math. Sci. 45 (2009), 1095–1140; MR 2597130 (2011f:14084).CrossRefGoogle Scholar
Jarvis, T. J. and Kimura, T., Orbifold quantum cohomology of the classifying space of a finite group, in Orbifolds in mathematics and physics (Madison, WI, 2001), Contemporary Mathematics, vol. 310 (American Mathematical Society, Providence, RI, 2002), 123–134; MR 1950944 (2004a:14056).CrossRefGoogle Scholar
Jiang, Y., The orbifold cohomology ring of simplicial toric stack bundles, Illinois J. Math. 52 (2008), 493–514; MR 2524648 (2011c:14059).CrossRefGoogle Scholar
Jiang, Y. and Tseng, H.-H., Note on orbifold Chow ring of semi-projective toric Deligne–Mumford stacks, Comm. Anal. Geom. 16 (2008), 231–250; MR 2411474 (2009h:14092).CrossRefGoogle Scholar
Johnson, P., Equivariant GW theory of stacky curves, Comm. Math. Phys. 327 (2014), 333–386; MR 3183403.CrossRefGoogle Scholar
Johnson, P., Pandharipande, R. and Tseng, H.-H., Abelian Hurwitz–Hodge integrals, Michigan Math. J. 60 (2011), 171–198; MR 2785870 (2012c:14025).CrossRefGoogle Scholar
Lee, Y.-P. and Shoemaker, M., A mirror theorem for the mirror quintic, Geom. Topol. 18 (2014), 1437–1483; MR 3228456.CrossRefGoogle Scholar
Liu, C.-C. M., Localization in Gromov–Witten theory and orbifold Gromov–Witten theory, in Handbook of moduli. Vol. II, Advanced Lectures in Mathematics (ALM), vol. 25 (International Press, Somerville, MA, 2013), 353–425; MR 3184181.Google Scholar
Mann, E., Orbifold quantum cohomology of weighted projective spaces, J. Algebraic Geom. 17 (2008), 137–166; MR 2357682 (2008k:14106).CrossRefGoogle Scholar
Milanov, T. E. and Tseng, H.-H., The spaces of Laurent polynomials, Gromov–Witten theory of ℙ1 -orbifolds, and integrable hierarchies, J. Reine Angew. Math. 622 (2008), 189–235; MR 2433616 (2010e:14053).Google Scholar
Tseng, H.-H., Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), 1–81; MR 2578300 (2011c:14147).CrossRefGoogle Scholar
Vlassopoulos, Y., Quantum cohomology and Morse theory on the loop space of toric varieties, Preprint (2002), arXiv:math/0203083.Google Scholar
Woodward, C. T., Quantum Kirwan morphism and Gromov–Witten invariants of quotients I, Preprint (2012), arXiv:1204.1765 [math.AG].Google Scholar
Woodward, C. T., Quantum Kirwan morphism and Gromov–Witten invariants of quotients II, Preprint (2014), arXiv:1408.5864 [math.AG].CrossRefGoogle Scholar
Woodward, C. T., Quantum Kirwan morphism and Gromov–Witten invariants of quotients III, Preprint (2014), arXiv:1408.5869 [math.AG].CrossRefGoogle Scholar