Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 048, 21 pages      arXiv:1407.1370      https://doi.org/10.3842/SIGMA.2017.048
Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond

Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

Chiu-Chu Melissa Liu a and Artan Sheshmani bc
a) Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA
b) Harvard University, Department of Mathematics (CMSA), 20 Garden Street, Cambridge, MA, 02138, USA
c) Aarhus University, Department of Mathematics, QGM, Ny Munkegade 118, 8000 Aarhus, Denmark

Received January 16, 2017, in final form June 21, 2017; Published online July 01, 2017

Abstract
An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

Key words: Gromov-Witten theory; GKM manifold; moduli space; equivariant cohomology; localization.

pdf (462 kb)   tex (25 kb)

References

  1. Atiyah M.F., Bott R., The moment map and equivariant cohomology, Topology 23 (1984), 1-28.
  2. Behrend K., Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601-617, alg-geom/9601011.
  3. Behrend K., Localization and Gromov-Witten invariants, in Quantum Cohomology (Cetraro, 1997), Lecture Notes in Math., Vol. 1776, Springer, Berlin, 2002, 3-38.
  4. Behrend K., Fantechi B., The intrinsic normal cone, Invent. Math. 128 (1997), 45-88, alg-geom/9601010.
  5. Behrend K., Manin Yu., Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), 1-60, alg-geom/9506023.
  6. Chen L., Li Y., Liu K., Localization, Hurwitz numbers and the Witten conjecture, Asian J. Math. 12 (2008), 511-518, math.AG/0609263.
  7. Deligne P., Mumford D., The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75-109.
  8. Faber C., Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians, in New Trends in Algebraic Geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge University Press, Cambridge, 1999, 93-109, alg-geom/9706006.
  9. Goresky M., Kottwitz R., MacPherson R., Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25-83.
  10. Graber T., Pandharipande R., Localization of virtual classes, Invent. Math. 135 (1999), 487-518, alg-geom/9708001.
  11. Guillemin V., Zara C., Equivariant de Rham theory and graphs, Asian J. Math. 3 (1999), 49-76, math.AG/9808135.
  12. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York - Heidelberg, 1977.
  13. Hori K., Katz S., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., Zaslow E., Mirror symmetry, Clay Mathematics Monographs, Vol. 1, Amer. Math. Soc., Providence, RI, Clay Mathematics Institute, Cambridge, MA, 2003.
  14. Kazarian M., KP hierarchy for Hodge integrals, Adv. Math. 221 (2009), 1-21, arXiv:0809.3263.
  15. Kazarian M.E., Lando S.K., An algebro-geometric proof of Witten's conjecture, J. Amer. Math. Soc. 20 (2007), 1079-1089, math.AG/0601760.
  16. Kim Y.-S., Liu K., Virasoro constraints and Hurwitz numbers through asymptotic analysis, Pacific J. Math. 241 (2009), 275-284.
  17. Knudsen F.F., Mumford D., The projectivity of the moduli space of stable curves. I. Preliminaries on ''det'' and ''Div'', Math. Scand. 39 (1976), 19-55.
  18. Knudsen F.F., The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$, Math. Scand. 52 (1983), 161-199.
  19. Knudsen F.F., The projectivity of the moduli space of stable curves. III. The line bundles on $M_{g,n}$, and a proof of the projectivity of $\overline M_{g,n}$ in characteristic $0$, Math. Scand. 52 (1983), 200-212.
  20. Kontsevich M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23.
  21. Kontsevich M., Enumeration of rational curves via torus actions, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math., Vol. 129, Birkhäuser Boston, Boston, MA, 1995, 335-368, hep-th/9405035.
  22. Li J., Tian G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119-174, alg-geom/9602007.
  23. Liu C.-C.M., Localization in Gromov-Witten theory and orbifold Gromov-Witten theory, in Handbook of Moduli, Vol. II, Adv. Lect. Math. (ALM), Vol. 25, Int. Press, Somerville, MA, 2013, 353-425, arXiv:1107.4712.
  24. Mirzakhani M., Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), 1-23.
  25. Mulase M., Zhang N., Polynomial recursion formula for linear Hodge integrals, Commun. Number Theory Phys. 4 (2010), 267-293, arXiv:0908.2267.
  26. Mumford D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry, Vol. II, Progr. Math., Vol. 36, Birkhäuser Boston, Boston, MA, 1983, 271-328.
  27. Okounkov A., Pandharipande R., Gromov-Witten theory, Hurwitz numbers, and matrix models, in Algebraic Geometry - Seattle 2005, Part 1, Proc. Sympos. Pure Math., Vol. 80, Amer. Math. Soc., Providence, RI, 2009, 325-414, math.AG/0101147.
  28. Spielberg H., A formula for the Gromov-Witten invariants of toric varieties, Ph.D. Thesis, Université Louis Pasteur (Strasbourg I), Strasbourg, 1999, math.AG/0006156.
  29. Witten E., Two-dimensional gravity and intersection theory on moduli space, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, PA, 1991, 243-310.

Previous article  Next article   Contents of Volume 13 (2017)