Quantum Cohomology and Crepant Resolutions: A Conjecture

[Cohomologie Quantique et Résolutions Crépantes : Une Conjecture]

Coates, Tom ; Ruan, Yongbin

Annales de l'Institut Fourier, Tome 63 (2013), p. 431-478 / Harvested from Numdam

Résumé

Nous présentons une conjecture développée par Coates-Iritani-Tseng et Ruan, qui relie la cohomologie quantique d’un orbifold de Gorenstein $\mathcal{X}$ à la cohomologie quantique d’une résolution crépante $\displaystyle Y$ de $\mathcal{X}$. Nous explorons quelque conséquences de cette conjecture et montrons qu’elle implique des versions de la Conjecture de la Résolution Crépante Cohmologique et des Conjectures de la Résolution Crépante de Ruan et Bryan-Graber. Nous donnons aussi une version «  quantisée  » de la conjecture, qui détermine les invariants de Gromov-Witten de genre supérieur de $\mathcal{X}$ à partir de ceux de $\displaystyle Y$.

Abstract

We give an expository account of a conjecture, developed by Coates–Iritani–Tseng and Ruan, which relates the quantum cohomology of a Gorenstein orbifold $\mathcal{X}$ to the quantum cohomology of a crepant resolution $Y$ of $\mathcal{X}$. We explore some consequences of this conjecture, showing that it implies versions of both the Cohomological Crepant Resolution Conjecture and of the Crepant Resolution Conjectures of Ruan and Bryan–Graber. We also give a ‘quantized’ version of the conjecture, which determines higher-genus Gromov–Witten invariants of $\mathcal{X}$ from those of $Y$.

Détail

Publié le : 2013-01-01

MR 3112518 | Zbl 1275.53083

DOI : https://doi.org/10.5802/aif.2766
Classification:  53D45,  14N35,  83E30
Mots clés: Cohomologie quantique, orbifold, résolution crépante, les invariants de Gromov-Witten.

Bibliographie

[1] Abramovich, Dan; Graber, Tom; Vistoli, Angelo Algebraic orbifold quantum products, Orbifolds in mathematics and physics, Amer. Math. Soc., Providence, RI, Madison, WI, 2001 (Contemp. Math.) Tome 310 (2002), pp. 1-24 | MR 1950940 | Zbl 1067.14055

[2] Abramovich, Dan; Graber, Tom; Vistoli, Angelo Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math., Tome 130 (2008), pp. 1337-1398 | Article | MR 2450211 | Zbl 1193.14070

[3] Aganagic, Mina; Bouchard, Vincent; Klemm, Albrecht Topological strings and (almost) modular forms, Comm. Math. Phys., Tome 277 (2008), pp. 771-819 | Article | MR 2365453 | Zbl 1165.81037

[4] Aspinwall, Paul S.; Greene, Brian R.; Morrison, David R. Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nuclear Phys. B, Tome 416 (1994), pp. 414-480 | Article | MR 1274435 | Zbl 0899.32006

[5] Barannikov, Serguei Quantum periods. I. Semi-infinite variations of Hodge structures, Internat. Math. Res. Notices (2001), pp. 1243-1264 | Article | MR 1866443 | Zbl 1074.14510

[6] Beĭlinson, A. A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Analysis and topology on singular spaces, I, Soc. Math. France, Luminy, 1981 (Astérisque) Tome 100 (1982), pp. 5-171 | MR 751966 | Zbl 0536.14011

[7] Bryan, Jim; Graber, Tom The crepant resolution conjecture, Algebraic geometry—Seattle 2005. Part 1, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 80 (2009), pp. 23-42 | MR 2483931 | Zbl 1198.14053

[8] Bryan, Jim; Graber, Tom; Pandharipande, Rahul The orbifold quantum cohomology of $\mathbb{C}^2/Z_3$ and Hurwitz-Hodge integrals, J. Algebraic Geom., Tome 17 (2008), pp. 1-28 | Article | MR 2357679 | Zbl 1129.14075

[9] Chen, Weimin; Ruan, Yongbin Orbifold Gromov–Witten theory, Orbifolds in mathematics and physics, Amer. Math. Soc., Providence, RI, Madison, WI, 2001 (Contemp. Math.) Tome 310 (2002), pp. 25-85 | MR 1950941 | Zbl 1091.53058

[10] Chen, Weimin; Ruan, Yongbin A new cohomology theory of orbifold, Comm. Math. Phys., Tome 248 (2004), pp. 1-31 | Article | MR 2104605 | Zbl 1063.53091

[11] Coates, Tom Givental’s Lagrangian cone and $S^1$-equivariant Gromov-Witten theory, Math. Res. Lett., Tome 15 (2008), pp. 15-31 | Article | MR 2367170 | Zbl 1169.14037

[12] Coates, Tom On the crepant resolution conjecture in the local case, Comm. Math. Phys., Tome 287 (2009), pp. 1071-1108 | Article | MR 2486673 | Zbl 1200.53081

[13] Coates, Tom; Givental, Alexander Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2), Tome 165 (2007), pp. 15-53 | Article | MR 2276766 | Zbl 1189.14063

[14] Coates, Tom; Iritani, Hiroshi; Tseng, Hsian-Hua Wall-crossings in toric Gromov-Witten theory. I. Crepant examples, Geom. Topol., Tome 13 (2009), pp. 2675-2744 | Article | MR 2529944 | Zbl 1184.53086

[15] Coates, Tom; Lee, Yuan-Pin; Corti, Alessio; Tseng, Hsian-Hua The quantum orbifold cohomology of weighted projective spaces, Acta Math., Tome 202 (2009), pp. 139-193 | Article | MR 2506749 | Zbl 1213.53106

[16] Cox, David A.; Katz, Sheldon Mirror symmetry and algebraic geometry, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 68 (1999) | MR 1677117 | Zbl 0951.14026

[17] Dubrovin, Boris Geometry of $2$D topological field theories, Integrable systems and quantum groups, Springer, Berlin, Montecatini Terme, 1993 (Lecture Notes in Math.) Tome 1620 (1996), pp. 120-348 | MR 1397274 | Zbl 0841.58065

[18] Faber, Carel; Shadrin, Sergey; Zvonkine, Dimitri Tautological relations and the r-spin Witten conjecture (preprint, available arXiv:math/0612510) | Numdam | MR 2722511

[19] Fulton, W.; Pandharipande, R. Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 62 (1997), pp. 45-96 | MR 1492534 | Zbl 0898.14018

[20] Givental, Alexander B. Homological geometry. I. Projective hypersurfaces, Selecta Math. (N.S.), Tome 1 (1995), pp. 325-345 | Article | MR 1354600 | Zbl 0920.14028

[21] Givental, Alexander B. Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J., Tome 1 (2001), p. 551-568, 645 | MR 1901075 | Zbl 1008.53072

[22] Givental, Alexander B. Symplectic geometry of Frobenius structures, Frobenius manifolds, Vieweg, Wiesbaden (Aspects Math., E36) (2004), pp. 91-112 | MR 2115767 | Zbl 1075.53091

[23] Hertling, C. Frobenius manifolds and moduli spaces for singularities, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 151 (2002) | MR 1924259 | Zbl 1023.14018

[24] Hertling, C.; Manin, Yu. Weak Frobenius manifolds, Internat. Math. Res. Notices (1999), pp. 277-286 | Article | Zbl 0960.58003

[25] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. of Math. (2), Tome 145 (1997), pp. 193-213 | Article | MR 1432041 | Zbl 0881.14018

[26] Lee, Y.-P. Invariance of tautological equations. I. Conjectures and applications, J. Eur. Math. Soc. (JEMS), Tome 10 (2008), pp. 399-413 | Article | MR 2390329 | Zbl 1170.14021

[27] Lee, Y.-P. Invariance of tautological equations. II. Gromov-Witten theory, J. Amer. Math. Soc., Tome 22 (2009), pp. 331-352 | Article | MR 2476776 | Zbl 1206.14078

[28] Manin, Yuri I. Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society, Providence, RI, American Mathematical Society Colloquium Publications, Tome 47 (1999) | MR 1702284 | Zbl 0952.14032

[29] Milanov, Todor E. The equivariant Gromov-Witten theory of $\mathbb{C}{\rm P}^1$ and integrable hierarchies, Int. Math. Res. Not. IMRN (2008) (Art. ID rnn 073, 21) | MR 2439568 | Zbl 1146.53067

[30] Pan, Jianzhong; Ruan, Yongbin; Yin, Xiaoqin Gerbes and twisted orbifold quantum cohomology, Sci. China Ser. A, Tome 51 (2008), pp. 995-1016 | Article | MR 2410979 | Zbl 1146.53069

[31] Perroni, Fabio Chen-Ruan cohomology of $ADE$ singularities, Internat. J. Math., Tome 18 (2007), pp. 1009-1059 | Article | MR 2360646 | Zbl 1149.14010

[32] Ruan, Yongbin The cohomology ring of crepant resolutions of orbifolds, Gromov-Witten theory of spin curves and orbifolds, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 403 (2006), pp. 117-126 | MR 2234886 | Zbl 1105.14078

[33] Ruan, Yongbin (unpublished)

[34] Tseng, Hsian-Hua Orbifold quantum Riemann-Roch, Lefschetz and Serre, Geom. Topol., Tome 14 (2010), pp. 1-81 | Article | MR 2578300 | Zbl 1178.14058

[35] Witten, Edward Quantum Background Independence In String Theory (arXiv:hep-th/9306122)