Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-19T20:14:02.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted orbifold Gromov–Witten invariants

Published online by Cambridge University Press:  11 January 2016

Valentin Tonita
Valentin Tonita*
Affiliation:
Kavli IPMU University of Tokyo, Kashiwa City, Chiba 277-8583, Japan, valentin.tonita@ipmu.jp
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.

Let χ be a smooth proper Deligne–Mumford stack over ℂ. One can define twisted orbifold Gromov–Witten invariants of χ by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable maps χg,n,d, cupping them with evaluation and cotangent line classes, and then integrating against the virtual fundamental class. These are more general than the twisted invariants introduced by Tseng. We express the generating series of the twisted invariants in terms of the generating series of the untwisted ones. We derive the corollaries which are used in a paper with Givental about the quantum K-theory of a complex compact manifold X.

Keywords

14N35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 213 , March 2014 , pp. 141 - 187
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Abramovich, D., “Lectures on Gromov–Witten invariants of orbifolds” in Enumerative Invariants in Algebraic Geometry and String Theory, Lecture Notes in Math. 1947, Springer, Berlin, 2008, 1–48. MR 2493583. DOI 10.1007/978-3-540-79814-9 1.Google Scholar
[2] Abramovich, D., Graber, T., Olsson, M., and Tseng, H.-H., On the global quotient structure of the space of twisted stable maps to a quotient stack, J. Algebraic Geom. 16 (2007), 731–751. MR 2357688. DOI 10.1090/S1056-3911-07-00443-2.CrossRefGoogle Scholar
[3] Abramovich, D., Graber, T., and Vistoli, A., “Algebraic orbifold quantum products” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 1–24. MR 1950940. DOI 10.1090/conm/ 310/05397.Google Scholar
[4] Abramovich, D., Graber, T., and Vistoli, A., Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008), 1337–1398. MR 2450211. DOI 10.1353/ajm.0.0017.CrossRefGoogle Scholar
[5] Abramovich, D. and Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 27–75. MR 1862797. DOI 10.1090/S0894-0347-01-00380-0.CrossRefGoogle Scholar
[6] Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 45–88. MR 1437495. DOI 10.1007/s002220050136.CrossRefGoogle Scholar
[7] Chen, W. and Ruan, Y., “Orbifold Gromov–Witten theory” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 25–85. MR 1950941. DOI 10.1090/conm/310/05398.Google Scholar
[8] Coates, T. H., Riemann–Roch theorems in Gromov–Witten theory, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2003. MR 2705177.Google Scholar
[9] Coates, T. and Givental, A., Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53. MR 2276766. DOI 10.4007/annals.2007.165.15.Google Scholar
[10] Givental, A. B., Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551–568, 645. MR 1901075.CrossRefGoogle Scholar
[11] Givental, A. B., “Symplectic geometry of Frobenius structures” in Frobenius Manifolds, Aspects Math. E36, Friedr. Vieweg, Wiesbaden, 2004, 91–112. MR 2115767.Google Scholar
[12] Givental, A. and Tonita, V., The Hirzebruch–Riemann–Roch theorem in true genus 0 quantum K-theory, preprint, arXiv:1106.3136v1 [math.AG].Google Scholar
[13] Jarvis, T. J. and Kimura, T., “Orbifold quantum cohomology of the classifying space of a finite group” in Orbifolds in Mathematics and Physics (Madison Wis., 2001), Contemp. Math. 310, Amer. Math. Soc., Providence, 2002, 123–134. MR 1950944. DOI 10.1090/conm/310/05401.Google Scholar
[14] Kabanov, A. and Kimura, T., A change of coordinates on the large phase space of quantum cohomology, Comm. Math. Phys. 217 (2001), 107–126. MR 1815027. DOI 10.1007/s002200000359.Google Scholar
[15] Lee, Y.-P., A formula for Euler characteristics of tautological line bundles on the Deligne–Mumford moduli spaces, Int. Math. Res. Not. IMRN 1997, 393–400. MR 1443318. DOI 10.1155/S1073792897000263.Google Scholar
[16] McDuff, D. and Salamon, D., J-Holomorphic Curves and Quantum Cohomology, Univ. Lecture Ser. 6, Amer. Math. Soc., Providence, 1994. MR 1286255.Google Scholar
[17] Teleman, C., The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), 525–588. MR 2917177. DOI 10.1007/s00222-011-0352-5.Google Scholar
[18] Toën, B., Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford, K- Theory 18 (1999), 33–76. MR 1710187. DOI 10.1023/A:1007791200714.Google Scholar
[19] Tseng, H.-H., Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010), 1–81. MR 2578300. DOI 10.2140/gt.2010.14.1.Google Scholar