Counting genus-zero real curves in symplectic manifolds

Farajzadeh Tehrani, Mohammad¹

¹ Simons Center for Geometry and Physics, Stony Brook University, John S Toll Dr., Stony Brook, NY 11794, USA

Geometry & topology, Tome 20 (2016) no. 2, pp. 629-695.

Résumé

There are two types of $J$–holomorphic spheres in a symplectic manifold which are invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$–holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on ${\mathbb{P}}^{4n-1}$.

Détail

DOI : 10.2140/gt.2016.20.629

Classification : 53D45, 14N35
Keywords: Gromov-Witten theory, anti-symplectic involution, real curves

Bibliographie

[1] M Aganagic, A Klemm, C Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1

[2] M F Atiyah, R Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1

[3] R Crétois, Déterminant des opérateurs de Cauchy–Riemann réels et application á l’orientabilité d’espaces de modules de courbes réelles, preprint (2013)

[4] R Crétois, Real bundle automorphisms, Cauchy–Riemann operators and orientability of moduli spaces, preprint (2014)

[5] M Farajzadeh Tehrani, Open Gromov–Witten theory on symplectic manifolds and symplectic cutting, Adv. Math. 232 (2013) 238

[6] K Fukaya, Counting pseudo-holomorphic discs in Calabi–Yau 3–folds, Tohoku Math. J. 63 (2011) 697

[7] K Fukaya, Y G Oh, H Ohta, K Ono, Lagrangian intersection Floer theory : anomaly and obstruction, II, 46, Amer. Math. Soc. (2009)

[8] K Fukaya, Y G Oh, H Ohta, K Ono, Anti-symplectic involution and Floer cohomology, preprint (2014)

[9] K Fukaya, K Ono, Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999) 933

[10] P Georgieva, Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution, preprint (2013)

[11] P Georgieva, The orientability problem in open Gromov–Witten theory, Geom. Topol. 17 (2013) 2485

[12] P Georgieva, A Zinger, Enumeration of real curves in CP2n−1 and a WDVV relation for real Gromov–Witten invariants, preprint (2013)

[13] P Georgieva, A Zinger, The moduli space of maps with crosscaps: the relative signs of the natural automorphisms, preprint (2013)

[14] P Georgieva, A Zinger, A recursion for counts of real curves in CP2n−1 : another proof, preprint (2014)

[15] P Georgieva, A Zinger, The moduli space of maps with crosscaps : Fredholm theory and orientability, Comm. Anal. Geom. 23 (2015) 499

[16] M Gross, D Huybrechts, D Joyce, Calabi–Yau manifolds and related geometries, Springer (2003)

[17] K Hori, S Katz, A Klemm, R Pandharipande, R Thomas, C Vafa, R Vakil, E Zaslow, Mirror symmetry, 1, Amer. Math. Soc. (2003)

[18] S Katz, C C M Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001) 1

[19] J M F Labastida, M Mariño, C Vafa, Knots, links and branes at large N, J. High Energy Phys. (2000) 42

[20] J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119

[21] D Mcduff, D Salamon, J–holomorphic curves and symplectic topology, 52, Amer. Math. Soc. (2004)

[22] H Ooguri, C Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000) 419

[23] R Pandharipande, J Solomon, J Walcher, Disk enumeration on the quintic 3–fold, J. Amer. Math. Soc. 21 (2008) 1169

[24] A Popa, A Zinger, Mirror symmetry for closed, open, and unoriented Gromov–Witten invariants, Adv. Math. 259 (2014) 448

[25] Y Ruan, G Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995) 259

[26] J P Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, PhD thesis, Massachusetts Institute of Technology (2006)

[27] J Y Welschinger, Open Gromov–Witten invariants in dimension four, preprint (2013)

[28] J Y Welschinger, Open Gromov–Witten invariants in dimension six, Math. Ann. 356 (2013) 1163

[29] E Witten, Chern–Simons gauge theory as a string theory, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 637