Wall-crossings in toric Gromov–Witten theory I: crepant examples

Coates, Tom¹ ; Iritani, Hiroshi² ; Tseng, Hsian-Hua³

¹ Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
² Faculty of Mathematics, Kyushu University, 6-10-1, Hakozaki, Higashiku, Fukuoka, 812-8581, Japan
³ Department of Mathematics, University of Wisconsin–Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, USA

Geometry & topology, Tome 13 (2009) no. 5, pp. 2675-2744.

Résumé

Let $\mathcal{X}$ 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 $\mathcal{X}$ to those of $Y$, which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan–Graber, and prove our conjecture when $\mathcal{X}=\mathbb{P}\left(1,1,2\right)$ and $\mathcal{X}=\mathbb{P}\left(1,1,1,3\right)$. As a consequence, we see that the original form of the Bryan–Graber Conjecture holds for $\mathbb{P}\left(1,1,2\right)$ but is probably false for $\mathbb{P}\left(1,1,1,3\right)$. Our methods are based on mirror symmetry for toric orbifolds.

Détail

DOI : 10.2140/gt.2009.13.2675

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

Bibliographie

[1] D Abramovich, T Graber, A Vistoli, Algebraic orbifold quantum products, from: "Orbifolds in mathematics and physics (Madison, WI, 2001)" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 1

[2] D Abramovich, T Graber, A Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008) 1337

[3] M Aganagic, V Bouchard, A Klemm, Topological strings and (almost) modular forms, Comm. Math. Phys. 277 (2008) 771

[4] M Audin, Torus actions on symplectic manifolds, Progress in Math. 93, Birkhäuser Verlag (2004)

[5] S Barannikov, Semi-infinite Hodge structures and mirror symmetry for projective spaces

[6] S Barannikov, Quantum periods. I. Semi-infinite variations of Hodge structures, Internat. Math. Res. Notices (2001) 1243

[7] L A Borisov, R P Horja, Mellin–Barnes integrals as Fourier–Mukai transforms, Adv. Math. 207 (2006) 876

[8] J Bryan, T Graber, The crepant resolution conjecture, from: "Algebraic geometry—Seattle 2005. Part 1" (editors D Abramovich, A Bertram, L Katzarkov, R Pandharipande, M Thaddeus), Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 23

[9] 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

[10] S Cecotti, C Vafa, On classification of $N=2$ supersymmetric theories, Comm. Math. Phys. 158 (1993) 569

[11] W Chen, Y Ruan, Orbifold Gromov–Witten theory, from: "Orbifolds in mathematics and physics (Madison, WI, 2001)" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 25

[12] W Chen, Y Ruan, A new cohomology theory of orbifold, Comm. Math. Phys. 248 (2004) 1

[13] T Coates, On the crepant resolution conjecture in the local case, Comm. Math. Phys. 287 (2009) 1071

[14] T Coates, A Corti, H Iritani, H H Tseng, Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009) 377

[15] T Coates, A Corti, Y P Lee, H H Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009) 139

[16] T Coates, A B Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. $(2)$ 165 (2007) 15

[17] T Coates, Y Ruan, Quantum cohomology and crepant resolutions: A conjecture

[18] D A Cox, S Katz, Mirror symmetry and algebraic geometry, Math. Surveys and Monogr. 68, Amer. Math. Soc. (1999)

[19] 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

[20] B Dubrovin, Geometry and integrability of topological-antitopological fusion, Comm. Math. Phys. 152 (1993) 539

[21] B Dubrovin, Geometry of $2$D topological field theories, from: "Integrable systems and quantum groups (Montecatini Terme, 1993)" (editors M Francaviglia, S Greco), Lecture Notes in Math. 1620, Springer (1996) 120

[22] J Fernandez, Hodge structures for orbifold cohomology, Proc. Amer. Math. Soc. 134 (2006) 2511

[23] W Fulton, R Pandharipande, Notes on stable maps and quantum cohomology, from: "Algebraic geometry—Santa Cruz 1995" (editors J Kollár, R Lazarsfeld, D R Morrison), Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 45

[24] A B Givental, Homological geometry and mirror symmetry, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994)", Birkhäuser (1995) 472

[25] A B Givental, Homological geometry. I. Projective hypersurfaces, Selecta Math. $($N.S.$)$ 1 (1995) 325

[26] A B Givental, A mirror theorem for toric complete intersections, from: "Topological field theory, primitive forms and related topics (Kyoto, 1996)" (editors M Kashiwara, A Matsuo, K Saito, I Satake), Progr. Math. 160, Birkhäuser (1998) 141

[27] A B Givental, Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001) 551, 645

[28] A B Givental, Symplectic geometry of Frobenius structures, from: "Frobenius manifolds" (editors C Hertling, M Marcolli), Aspects Math. E36, Vieweg (2004) 91

[29] M A Guest, Quantum cohomology via $D$–modules, Topology 44 (2005) 263

[30] C Hertling, $tt^{*}$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003) 77

[31] C Hertling, Y Manin, Weak Frobenius manifolds, Internat. Math. Res. Notices (1999) 277

[32] C Hertling, Y Manin, Unfoldings of meromorphic connections and a construction of Frobenius manifolds, from: "Frobenius manifolds" (editors C Hertling, M Marcolli), Aspects Math. E36, Vieweg (2004) 113

[33] 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)

[34] K Hori, C Vafa, Mirror symmetry

[35] P R Horja, Hypergeometric functions and mirror symmetry in toric varieties

[36] H Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, to appear in Adv. Math

[37] H Iritani, Ruan's conjecture and integral structures in quantum cohomology

[38] H Iritani, Wall-crossings in toric Gromov–Witten theory III, in preparation

[39] H Iritani, Quantum $D$–modules and generalized mirror transformations, Topology 47 (2008) 225

[40] M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525

[41] B H Lian, K Liu, S T Yau, Mirror principle. II, Asian J. Math. 3 (1999) 109

[42] E Lupercio, M Poddar, The global McKay–Ruan correspondence via motivic integration, Bull. London Math. Soc. 36 (2004) 509

[43] Y I Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Coll. Publ. 47, Amer. Math. Soc. (1999)

[44] A Pressley, G Segal, Loop groups, Oxford Math. Monogr., Oxford Science Publ., The Clarendon Press, Oxford Univ. Press (1986)

[45] M A Rose, A reconstruction theorem for genus zero Gromov–Witten invariants of stacks, Amer. J. Math. 130 (2008) 1427

[46] Y Ruan, private communication

[47] K Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. 19 (1983) 1231

[48] M Saito, On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble) 39 (1989) 27

[49] W Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973) 211

[50] E Witten

[51] T Yasuda, Twisted jets, motivic measures and orbifold cohomology, Compos. Math. 140 (2004) 396