Geometry & Topology

Wall-crossings in toric {G}romov–{W}itten theory {I}: crepant examples

Tom Coates, Hiroshi Iritani, and Hsian-Hua Tseng

Full-text: Open access


Let X be a Gorenstein orbifold with projective coarse moduli space X and let Y be a crepant resolution of X. We state a conjecture relating the genus-zero Gromov–Witten invariants of X to those of Y, which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan–Graber, and prove our conjecture when X=(1,1,2) and X=(1,1,1,3). As a consequence, we see that the original form of the Bryan–Graber Conjecture holds for (1,1,2) but is probably false for (1,1,1,3). Our methods are based on mirror symmetry for toric orbifolds.

Article information

Geom. Topol., Volume 13, Number 5 (2009), 2675-2744.

Received: 4 December 2006
Revised: 21 October 2008
Accepted: 25 May 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 83E30: String and superstring theories [See also 81T30]

quantum cohomology crepant resolution Gromov–Witten invariants mirror symmetry variation of semi-infinite Hodge structure Crepant Resolution Conjecture


Coates, Tom; Iritani, Hiroshi; Tseng, Hsian-Hua. Wall-crossings in toric {G}romov–{W}itten theory {I}: crepant examples. Geom. Topol. 13 (2009), no. 5, 2675--2744. doi:10.2140/gt.2009.13.2675.

Export citation


  • D Abramovich, T Graber, A Vistoli, Algebraic orbifold quantum products, from: “Orbifolds in mathematics and physics (Madison, WI, 2001)”, (A Adem, J Morava, Y Ruan, editors), Contemp. Math. 310, Amer. Math. Soc. (2002) 1–24
  • D Abramovich, T Graber, A Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008) 1337–1398
  • M Aganagic, V Bouchard, A Klemm, Topological strings and (almost) modular forms, Comm. Math. Phys. 277 (2008) 771–819
  • M Audin, Torus actions on symplectic manifolds, revised edition, Progress in Math. 93, Birkhäuser Verlag, Basel (2004)
  • S Barannikov, Semi-infinite Hodge structures and mirror symmetry for projective spaces
  • S Barannikov, Quantum periods. I. Semi-infinite variations of Hodge structures, Internat. Math. Res. Notices (2001) 1243–1264
  • L A Borisov, R P Horja, Mellin–Barnes integrals as Fourier–Mukai transforms, Adv. Math. 207 (2006) 876–927
  • J Bryan, T Graber, The crepant resolution conjecture, from: “Algebraic geometry–-Seattle 2005. Part 1”, (D Abramovich, A Bertram, L Katzarkov, R Pandharipande, M Thaddeus, editors), Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 23–42
  • P Candelas, X C de la Ossa, P S Green, L Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991) 21–74
  • S Cecotti, C Vafa, On classification of $N=2$ supersymmetric theories, Comm. Math. Phys. 158 (1993) 569–644
  • W Chen, Y Ruan, Orbifold Gromov–Witten theory, from: “Orbifolds in mathematics and physics (Madison, WI, 2001)”, (A Adem, J Morava, Y Ruan, editors), Contemp. Math. 310, Amer. Math. Soc. (2002) 25–85
  • W Chen, Y Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004) 1–31
  • T Coates, On the crepant resolution conjecture in the local case, Comm. Math. Phys. 287 (2009) 1071–1108 A longer version with more examples: Wall-crossings in toric Gromov–Witten theory II: Local examples
  • T Coates, A Corti, H Iritani, H-H Tseng, Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009) 377–438
  • T Coates, A Corti, Y-P Lee, H-H Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009) 139–193
  • T Coates, A B Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. $(2)$ 165 (2007) 15–53
  • T Coates, Y Ruan, Quantum cohomology and crepant resolutions: A conjecture
  • D A Cox, S Katz, Mirror symmetry and algebraic geometry, Math. Surveys and Monogr. 68, Amer. Math. Soc. (1999)
  • A Douai, C Sabbah, Gauss–Manin systems, Brieskorn lattices and Frobenius structures. I, from: “Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002)”, Ann. Inst. Fourier $($Grenoble$)$ 53 (2003) 1055–1116
  • B Dubrovin, Geometry and integrability of topological-antitopological fusion, Comm. Math. Phys. 152 (1993) 539–564
  • B Dubrovin, Geometry of $2$D topological field theories, from: “Integrable systems and quantum groups (Montecatini Terme, 1993)”, (M Francaviglia, S Greco, editors), Lecture Notes in Math. 1620, Springer, Berlin (1996) 120–348
  • J Fernandez, Hodge structures for orbifold cohomology, Proc. Amer. Math. Soc. 134 (2006) 2511–2520
  • W Fulton, R Pandharipande, Notes on stable maps and quantum cohomology, from: “Algebraic geometry–-Santa Cruz 1995”, (J Kollár, R Lazarsfeld, D R Morrison, editors), Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 45–96
  • A B Givental, Homological geometry and mirror symmetry, from: “Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)”, Birkhäuser, Basel (1995) 472–480
  • A B Givental, Homological geometry. I. Projective hypersurfaces, Selecta Math. $($N.S.$)$ 1 (1995) 325–345
  • A B Givental, A mirror theorem for toric complete intersections, from: “Topological field theory, primitive forms and related topics (Kyoto, 1996)”, (M Kashiwara, A Matsuo, K Saito, I Satake, editors), Progr. Math. 160, Birkhäuser, Boston (1998) 141–175
  • A B Givental, 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
  • A B Givental, Symplectic geometry of Frobenius structures, from: “Frobenius manifolds”, (C Hertling, M Marcolli, editors), Aspects Math. E36, Vieweg, Wiesbaden (2004) 91–112
  • M A Guest, Quantum cohomology via $D$–modules, Topology 44 (2005) 263–281
  • C Hertling, $tt\sp *$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003) 77–161
  • C Hertling, Y Manin, Weak Frobenius manifolds, Internat. Math. Res. Notices (1999) 277–286
  • C Hertling, Y Manin, Unfoldings of meromorphic connections and a construction of Frobenius manifolds, from: “Frobenius manifolds”, (C Hertling, M Marcolli, editors), Aspects Math. E36, Vieweg, Wiesbaden (2004) 113–144
  • K Hori, S Katz, A Klemm, R Pandharipande, R Thomas, C Vafa, R Vakil, E Zaslow, Mirror symmetry, Clay Math. Monogr. 1, Amer. Math. Soc. (2003)
  • K Hori, C Vafa, Mirror symmetry
  • P R Horja, Hypergeometric functions and mirror symmetry in toric varieties
  • H Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, to appear in Adv. Math A longer version containing a discussion on real structures: Real and integral structures in quantum cohomology I: Toric orbifolds arxiv:0712.2204
  • H Iritani, Ruan's conjecture and integral structures in quantum cohomology
  • H Iritani, Wall-crossings in toric Gromov–Witten theory III, in preparation
  • H Iritani, Quantum $D$–modules and generalized mirror transformations, Topology 47 (2008) 225–276
  • M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525–562
  • B H Lian, K Liu, S-T Yau, Mirror principle. II, Asian J. Math. 3 (1999) 109–146
  • E Lupercio, M Poddar, The global McKay–Ruan correspondence via motivic integration, Bull. London Math. Soc. 36 (2004) 509–515
  • Y I Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Coll. Publ. 47, Amer. Math. Soc. (1999)
  • A Pressley, G Segal, Loop groups, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford Univ. Press, New York (1986)
  • M A Rose, A reconstruction theorem for genus zero Gromov–Witten invariants of stacks, Amer. J. Math. 130 (2008) 1427–1443
  • Y Ruan, private communication
  • K Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983) 1231–1264
  • M Saito, On the structure of Brieskorn lattice, Ann. Inst. Fourier $($Grenoble$)$ 39 (1989) 27–72
  • W Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973) 211–319
  • E Witten, Quantum background independence in string theory
  • T Yasuda, Twisted jets, motivic measures and orbifold cohomology, Compos. Math. 140 (2004) 396–422