Moduli of $J$-holomorphic curves with Lagrangian boundary conditions ‎and open Gromov-Witten invariants for an $S^1$-equivariant pair

Document Type: Original Article


‎Department of Mathematics‎, ‎Columbia University‎, ‎2990 Broadway‎, ‎New York‎, ‎NY 10027.


Let $(X,\omega)$ be a symplectic manifold‎, ‎$J$ be an $\omega$-tame‎ ‎almost complex structure‎, ‎and $L$ be a Lagrangian submanifold‎. ‎The stable compactification of the moduli space of parametrized $J$-holomorphic‎ ‎curves in $X$ with boundary in $L$ (with prescribed topological data)‎ is compact and Hausdorff in Gromov's $C^\infty$-topology‎. ‎We construct a Kuranishi structure with corners in the sense of Fukaya and‎ ‎Ono‎. ‎This Kuranishi structure is orientable if $L$ is spin‎. ‎In the special case where the expected dimension of the moduli space‎ ‎is zero‎, ‎and there is an $S^1$-action on the pair $(X,L)$ which‎ ‎preserves $J$ and has no fixed points on $L$‎, ‎we define the ‎Euler number for this $S^1$-equivariant pair and the prescribed‎ ‎topological data‎. ‎We conjecture that this rational number is‎ ‎the one computed by localization techniques using the given $S^1$-action‎.


W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, 820, Springer-Verlag, Berlin-New York, 1980.
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957) 235--249.
N. L. Alling and N. Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics, 219, Springer-Verlag, Berlin-New York, 1971.
M. Aganagic, A. Klemm, and C. Vafa, Disk instantons, mirror symmetry, and the duality web, Z. Naturforsch. 57 (2002), no. 1-2, 1--28
M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041.
E. Bujalance, J. J. Etayo, J. M. Gamboa, and G. Gromadzki, Automorphism groups of compact bordered Klein surfaces, Lecture Notes in Mathematics, 1439, Springer-Verlag, Berlin-New York, 1990.
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45--88.
P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble supercon formal field theory, Nuclear Phys. B 359 (1991), no.1, 21--74.
K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933--1048.
K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory--anomaly and obstruction, preprint 2000.
A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 1996, no. 13, 613--663.
R. Gopakumar and C. Vafa, M-theory and topological strings--II, hep-th/9812127.
T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487-518
T. Graber, E. Zaslow, Open-string Gromov-Witten invariants: Calculations and a Mirror theorem", Orbifolds in mathematics and physics (Madison, WI, 2001), 107--121, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002.
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307--347.
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977.
M. W. Hirsch Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg, 1976.
J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, 187, Springer-Verlag, New York, 1998.
S. Ivashkovich and V. Shevchishin, Gromov compactness theorem for J-complex curves with boundary, Internat.Math. Res. Notices 2000, no. 22, 1167--1206.
S. Ivashkovich and V. Shevchishin, Holomorphic structure on the space of Riemann surfaces with marked boundary, Tr. Mat. Inst. Steklova 235 (2001), Anal. i Geom. Vopr. Kompleks. Analiza, 98--109; reprinted in in Proc. Steklov Inst. Math. 2001 (235), no. 4, 91--102.
S. Ivashkovich, V. Shevchishin, Reection principle and J-complex curves with boundary on totally real immersions, Commun. Contemp. Math. 4 (2002), no. 1, 65--106.
F. Klein, Uber Realitatsverhaltnisse bei der einem beliebigen Geschlechte zugehorigen Normalkurve der φ, Math. Ann. 42 (1893), no. 1, 1-29.
K. Kodaira, Complex manifolds and deformation of complex structures, Grundlehrender Mathematischen Wissenschaften 283. Springer-Verlag, New York, 1986.
S. Katz and C. C. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), no. 1, 1--49.
B. Lian, K. Liu and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), no. 4, 729--763.
J. Li and Y. S. Song, Open string instantons and relative stable morphisms, Adv. Theor. Math. Phys. 5 (2001), no. 1, 67--91.
J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119--174.
J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47--83, First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 1998.
G. Liu and G. Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998), no. 1, 1--74.
J. M. F. Labastida, M. Marino and C. Vafa Knots, links and branes at large N, Jour. High En. Phys. (2000), no. 11, Paper 7, 42 pp.
D. McDuff and D. Salamon, J-holomorphic curves and quantum cohomology, University Lecture Series, 6, American Mathematical Society, Providence, RI, 1994.
M. Mari~no and C. Vafa Framed knots at large N, Orbifolds in mathematics and physics (Madison, WI, 2001), 185--204, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002.
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), no. 3, 69--108.
T. Parker and J. G. Wolfson, Pseudo-holomorphic maps and bubble trees, J. Geom. Anal. 3 (1993), no. 1, 63--98.
I. Satake, The Gauss-Bonnet theorem for V -manifolds, J. Math. Soc. Japan 9 (1957) 464-492.
M. Seppälä, Moduli spaces of stable real algebraic curves, Ann. Sci. École norm.Sup. (4) 24 (1991), no. 5, 519--544.
B. Siebert, Gromov-Witten invariants of general symplectic manifolds, dg-ga/9608005
B. Siebert, Symplectic Gromov-Witten invariants, New trends in algebraic geometry (Warwick, 1996), 375--424,
London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge, 1999.
R. Silhol, Real algebraic surfaces, Lecture Notes in Mathematics, 1392, Springer-Verlag, Berlin, 1989
J.D. Stasheff, Homotopy associativity of H-spaces I, II, Trans. Amer. Math. Soc. 108 (1963) 275--292, 293--312.
M. Seppälä and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties, Math. Z. 201 (1989), no. 2, 151-165.
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1--24.
W.P. Thurston, The geometry and topology of three-manifolds, Princeton mimeographed lecture notes, 1979.
G. Weichhold, Uber symmetrische Riemannsche Flachen und die Periodizitatsmodulen der zugerhorigen Abelschen Normalintegrale erstes Gattung, Leipziger dissertation, 1883.
J. G. Wolfson, Gromov's compactness of pseudo-holomorphic curves and symplectic geometry, J. Differential Geom. 28 (1988), no. 3, 383--405.
S. Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969--997.
R. Ye, Gromov's compactness theorem for pseudo holomorphic curves, Trans. Amer. Math. Soc. 342 (1994), no. 2, 671--694.