# zbMATH — the first resource for mathematics

The Gopakumar-Vafa formula for symplectic manifolds. (English) Zbl 1459.53078
Summary: The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.

##### MSC:
 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
VFC package
Full Text:
##### References:
 [1] Bryan, Jim; Pandharipande, Rahul, B{PS} states of curves in {C}alabi-{Y}au 3-folds, Geom. Topol.. Geometry and Topology, 5, 287-318, (2001) · Zbl 1063.14068 [2] Bryan, Jim; Pandharipande, Rahul, The local {G}romov-{W}itten theory of curves, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 21, 101-136, (2008) · Zbl 1126.14062 [3] Faber, C.; Pandharipande, R., Hodge integrals and {G}romov-{W}itten theory, Invent. Math.. Inventiones Mathematicae, 139, 173-199, (2000) · Zbl 0960.14031 [4] Gopakumar, R.; Vafa, C., $${M}$$-Theory and Topological Strings {I} [5] Gopakumar, R.; Vafa, C., $${M}$$-Theory and Topological Strings {II} [6] Ionel, Eleny-Nicoleta; Parker, Thomas H., The {G}romov invariants of {R}uan-{T}ian and {T}aubes, Math. Res. Lett.. Mathematical Research Letters, 4, 521-532, (1997) · Zbl 0889.57030 [7] Ionel, Eleny-Nicoleta; Parker, Thomas H., Relative {G}romov-{W}itten Invariants, Ann. of Math., 157, 45-96, (2003) · Zbl 1039.53101 [8] Ivashkovich, S. M.; Shevchishin, V. V., Deformations of noncompact complex curves, and envelopes of meromorphy of spheres, Mat. Sb.. Rossi\u\i skaya Akademiya Nauk. Matematicheski\u\i Sbornik, 189, 23-60, (1998) · Zbl 0923.46036 [9] Author’s review (S. I.), Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls, Invent. Math.. Inventiones Mathematicae, 136, 571-602, (1999) · Zbl 0930.32017 [10] Klemm, A.; Pandharipande, R., Enumerative geometry of {C}alabi-{Y}au 4-folds, Comm. Math. Phys.. Communications in Mathematical Physics, 281, 621-653, (2008) · Zbl 1157.32022 [11] Koschorke, Ulrich, Infinite dimensional {$$K$$}-theory and characteristic classes of {F}redholm bundle maps. Global {A}nalysis, 95-133, (1970) [12] Lee, Junho, Holomorphic 2-forms and vanishing theorems for {G}romov-{W}itten invariants, Canad. Math. Bull.. Canadian Mathematical Bulletin. Bulletin Canadien de Math\'ematiques, 52, 87-94, (2009) · Zbl 1167.53072 [13] Lee, Junho; Parker, Thomas H., A structure theorem for the {G}romov-{W}itten invariants of {K}\"ahler surfaces, J. Differential Geom.. Journal of Differential Geometry, 77, 483-513, (2007) · Zbl 1130.53059 [14] Lee, Junho; Parker, Thomas H., An obstruction bundle relating {G}romov-{W}itten invariants of curves and {K}\"ahler surfaces, Amer. J. Math.. American Journal of Mathematics, 134, 453-506, (2012) · Zbl 1253.53086 [15] Lockhart, Robert B.; McOwen, Robert C., Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 12, 409-447, (1985) · Zbl 0615.58048 [16] Maulik, D.; Oblomkov, A.; Okounkov, A.; Pandharipande, R., Gromov-{W}itten/{D}onaldson-{T}homas correspondence for toric 3-folds, Invent. Math.. Inventiones Mathematicae, 186, 435-479, (2011) · Zbl 1232.14039 [17] Micallef, Mario J.; White, Brian, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2). Annals of Mathematics. Second Series, 141, 35-85, (1995) · Zbl 0873.53038 [18] McDuff, Dusa; Salamon, Dietmar, {$$J$$}-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, xiv+726 pp., (2012) · Zbl 1272.53002 [19] Pandharipande, R., Hodge integrals and degenerate contributions, Comm. Math. Phys.. Communications in Mathematical Physics, 208, 489-506, (1999) · Zbl 0953.14036 [20] Pandharipande, R.; Pixton, A., Gromov-{W}itten/{P}airs correspondence for the quintic 3-fold, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 30, 389-449, (2017) · Zbl 1360.14134 [21] Pandharipande, R.; Thomas, R. P., Curve counting via stable pairs in the derived category, Invent. Math.. Inventiones Mathematicae, 178, 407-447, (2009) · Zbl 1204.14026 [22] Pardon, John, An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves, Geom. Topol.. Geometry & Topology, 20, 779-1034, (2016) · Zbl 1342.53109 [23] Smale, S., An infinite dimensional version of {S}ard’s theorem, Amer. J. Math.. American Journal of Mathematics, 87, 861-866, (1965) · Zbl 0143.35301 [24] Taubes, Clifford Henry, Counting pseudo-holomorphic submanifolds in dimension {$$4$$}, J. Differential Geom.. Journal of Differential Geometry, 44, 818-893, (1996) · Zbl 0883.57020 [25] Zinger, Aleksey, A comparison theorem for {G}romov-{W}itten invariants in the symplectic category, Adv. Math.. Advances in Mathematics, 228, 535-574, (2011) · Zbl 1225.14046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.