Skip to main content
SpringerLink
Log in

QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

This is the first in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology QH G (X) of a smooth polarized complex projective variety X with the action of a connected complex reductive group G to the orbifold quantum cohomology QH(X//G) of its geometric invariant theory quotient X//G, and prove that it intertwines the genus zero gauged Gromov–Witten potential of X with the genus zero Gromov–Witten graph potential of X//G. In this part we introduce the moduli spaces used in the construction of the quantum Kirwan morphism.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Abramovich, T. Graber, A. Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.

    Article  MATH  MathSciNet  Google Scholar 

  2. E. Arbarello, M. Cornalba, P. A. Griffiths, Geometry of Algebraic Curves, Vol. II, Grundlehren der Mathematischen Wissenschaften, Vol. 268, Springer, Heidelberg, 2011. With a contribution by Joseph D. Harris.

  3. K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Behrend, Yu. Manin, Stacks of stable maps and Gromov–Witten invariants, Duke Math. J. 85 (1996), no. 1, 1–60.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Bertram, I. Ciocan-Fontanine, B. Kim, Gromov–Witten invariants for abelian and nonabelian quotients, J. Algebraic Geom. 17 (2008), no. 2, 275–294.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. B. Bradlow, Vortices in holomorphic line bundles over closed Kähler manifolds, Comm. Math. Phys. 135 (1990), no. 1, 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Chen, Y. Ruan, Orbifold Gromov–Witten theory, in: Orbifolds in Mathematics and Physics (Madison, WI, 2001), Contemp. Math., Vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25–85.

  8. K. Cieliebak, A. Rita Gaio, I. Mundet i Riera, D. A. Salamon, The symplectic vortex equations and invariants of Hamiltonian group actions, J. Symplectic Geom. 1 (2002), no. 3, 543–645.

  9. I. Ciocan-Fontanine, B. Kim, Big I-functions, arXiv:1401.7417.

  10. I. Ciocan-Fontanine, B. Kim, Wall-crossing in genus zero quasimap theory and mirror maps, Algebr. Geom. 1 (2014), 400–448.

    Article  MathSciNet  Google Scholar 

  11. I. Ciocan-Fontanine, B. Kim, Higher genus quasimap wall-crossing for semi-positive targets, arXiv:1308.6377.

  12. I. Ciocan-Fontanine, B. Kim, D. Maulik, Stable quasimaps to GIT quotients, J. Geom. Phys. 75 (2014), 17–47.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Coates, Y.-P. Lee, A. Corti, H. H. Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009), no. 2, 139–193.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Coates, A. Corti, H. Iritani, Hsian-Hua Tseng, A mirror theorem for toric stacks, arXiv:1310.4163.

  15. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.

    Article  MATH  MathSciNet  Google Scholar 

  16. K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction., AMS/IP Studies in Advanced Mathematics, Vol. 46, American Mathematical Society, Providence, RI, 2009.

  17. W. Fulton, R. MacPherson, A compactification of configuration spaces, Ann. of Math. (2) 139 (1994), no. 1, 183–225.

  18. A. R. P. Gaio, D. A. Salamon, Gromov–Witten invariants of symplectic quotients and adiabatic limits, J. Symplectic Geom. 3 (2005), no. 1, 55–159.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. Getzler, M. M. Kapranov, Modular operads, Compositio Math. 110 (1998), no. 1, 65–126.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. B. Givental, Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices 13 (1996), 613–663.

    Article  MathSciNet  Google Scholar 

  21. E. Gonzalez, C. Woodward, Quantum Witten localization and abelianization for qde solutions, arXiv:0811.3358.

  22. V. Guillemin, S. Sternberg, Geometric quantization and multiplicities of group representations, Invent. Math. 67 (1982), 515–538.

    Article  MATH  MathSciNet  Google Scholar 

  23. K. Hori, C. Vafa, Mirror symmetry, arXiv:hep-th/0002222.

  24. H. Iritani, Quantum D-modules and generalized mirror transformations, Topology 47 (2008), no. 4, 225–276.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Jaffe, C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Progress in Physics, Vol. 2, Birkhäuser Boston, Mass., 1980.

  26. T. Kawasaki, The Riemann–Roch theorem for complex V-manifolds, Osaka J. Math. 16 (1979), 151–159.

    MATH  MathSciNet  Google Scholar 

  27. F. C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Mathematical Notes, Vol. 31, Princeton Univ. Press, Princeton, 1984.

  28. S.-C. Lau, N. C. Leung, B. Wu, Mirror maps equal SYZ maps for toric Calabi–Yau surfaces, Bull. Lond. Math. Soc. 44 (2012), 255–270.

  29. J. Li and G. Tian, Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds, in: Topics in Symplectic 4-Manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 1998, pp. 47–83.

  30. B. H. Lian, K. Liu, S.-T. Yau, Mirror principle. I, Asian J. Math. 1 (1997), no. 4, 729–763.

    MATH  MathSciNet  Google Scholar 

  31. Y. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, American Mathematical Society Colloquium Publications, Vol. 47, American Mathematical Society, Providence, RI, 1999.

  32. M. Markl, Simplex, associahedron, and cyclohedron, in: Higher Homotopy Structures in Topology and Mathematical Physics (Poughkeepsie, NY, 1996), Contemp. Math., Vol. 227, Amer. Math. Soc., Providence, RI, 1999, pp. 235–265.

  33. M. Markl, Free loop spaces and cyclohedra, in: Proceedings of the 22nd Winter School “Geometry and Physics” (Srní, 2002), no. 71, 2003, pp. 151–157.

  34. M. Markl, S. Shnider, J. Stasheff, Operads in Algebra, Topology and Physics, Mathematical Surveys and Monographs, Vol. 96, American Mathematical Society, Providence, RI, 2002.

  35. S. Ma’u, C. Woodward, Geometric realizations of the multiplihedra, Compos. Math. 146 (2010), no. 4, 1002–1028.

    Article  MATH  MathSciNet  Google Scholar 

  36. D. McDuff, D. Salamon, J-Holomorphic Curves and Symplectic Topology, American Mathematical Society Colloquium Publications, Vol. 52, American Mathematical Society, Providence, RI, 2004.

  37. D. Mumford, Boundary points of moduli schemes, in: Lecture notes prepared in connection with the seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate, Woods Hole, Massachusetts, July 6–July 31, 1964, http://www.jmilne.org/math/Documents/woodshole.pdfwoodshole.pdf.

  38. I. Mundet i Riera, A Hitchin–Kobayashi correspondence for Kähler fibrations, J. Reine Angew. Math. 528 (2000), 41–80.

  39. I. Mundet i Riera, Hamiltonian Gromov–Witten invariants, Topology 42 (2003), no. 3, 525–553.

  40. K. Nguyen, C. Woodward, F. Ziltener, Morphisms of CohFT algebras and quantization of the Kirwan map, in: Symplectic, Poisson, and Noncommutative Geometry, MSRI Publications, Vol. 62, Cambridge University Press, New York, 2014, pp. 131–170.

  41. A. Ott, Removal of singularities and Gromov compactness for symplectic vortices, J. Symplectic Geom. 12 (2014), 257–311.

    Article  MATH  MathSciNet  Google Scholar 

  42. J. Preskill, Vortices and monopoeles, in: Architecture of Fundamental Interactions at Short Distances, P. Ramond, R. Stora, eds., 1987.

  43. A. Schmitt, A universal construction for moduli spaces of decorated vector bundles over curves, Transform. Groups 9 (2004), no. 2, 167–209.

    Article  MATH  MathSciNet  Google Scholar 

  44. A. H. W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.

  45. J. Stasheff, H-Spaces from a Homotopy Point of View, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin, 1970.

  46. C. Teleman, Topological field theories in 2 dimensions, in: European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 197–210.

  47. S. Venugopalan, Vortices on surfaces with cylindrical ends, arXiv:1201.1933.

  48. S. Venugopalan, Yang–Mills heat ow on gauged holomorphic maps, arXiv:1201 1933.

  49. S. Venugopalan, C. Woodward, Classification of affine vortices, arXiv:1301.7052.

  50. E. Witten, Phases of N = 2 theories in two dimensions, Nuclear Phys. B 403 (1993), no. 1–2, 159–222.

    Article  MATH  MathSciNet  Google Scholar 

  51. C. Woodward, Quantum Kirwan morphism and Gromov–Witten invariants of quotients II, preprint.

  52. C. Woodward, Quantum Kirwan morphism and Gromov–Witten invariants of quotients III, preprint.

  53. F. Ziltener, Symplectic vortices on the complex plane, ETH Zurich PhD thesis, 2006.

  54. F. Ziltener, A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane, Mem. Amer. Math. Soc. 230, no. 1082, 2014.

  55. F. Ziltener, The invariant symplectic action and decay for vortices, J. Symplectic Geom. 7 (209), no. 3, 357–376.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    CHRIS T. WOODWARD

Authors
  1. CHRIS T. WOODWARD
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to CHRIS T. WOODWARD.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

WOODWARD, C.T. QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I. Transformation Groups 20, 507–556 (2015). https://doi.org/10.1007/s00031-015-9313-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-015-9313-1

Keywords

Search

Navigation