The local Gromov–Witten invariants of configurations of rational curves

Karp, Dagan¹ ; Liu, Chiu-Chu Melissa² ; Mariño, Marcos³

¹ Department of Mathematics, University of California at Berkeley, California 94720-3840, USA
² Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370, USA
³ Department of Physics, CERN, Geneva 23, CH-1211, Switzerland

Geometry & topology, Tome 10 (2006) no. 1, pp. 115-168.

Résumé

We compute the local Gromov–Witten invariants of certain configurations of rational curves in a Calabi–Yau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of ${\mathbb{P}}^1$’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov–Witten invariants of a blowup of ${\mathbb{P}}^3$ at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov–Witten invariants using the mathematical and physical theories of the topological vertex. In particular, we provide further evidence equating the vertex amplitudes derived from physical and mathematical theories of the topological vertex.

Détail

DOI : 10.2140/gt.2006.10.115

Keywords: Gromov–Witten invariants, Calabi–Yau threefolds, topological vertex

Bibliographie

[1] M Aganagic, R Dijkgraaf, A Klemm, M Mariño, C Vafa, Topological Strings and Integrable Hierarchies,

[2] M Aganagic, A Klemm, M Mariño, C Vafa, The topological vertex, Comm. Math. Phys. 254 (2005) 425

[3] P S Aspinwall, D R Morrison, Topological field theory and rational curves, Comm. Math. Phys. 151 (1993) 245

[4] J Bryan, D Karp, The closed topological vertex via the Cremona transform, J. Algebraic Geom. 14 (2005) 529

[5] J Bryan, S Katz, N C Leung, Multiple covers and the integrality conjecture for rational curves in Calabi–Yau threefolds, J. Algebraic Geom. 10 (2001) 549

[6] J Bryan, R Pandharipande, The local Gromov-Witten theory of curves,

[7] J Bryan, R Pandharipande, BPS states of curves in Calabi–Yau 3–folds, Geom. Topol. 5 (2001) 287 | DOI

[8] J Bryan, R Pandharipande, Curves in Calabi–Yau threefolds and topological quantum field theory, Duke Math. J. 126 (2005) 369

[9] D A Cox, S Katz, Mirror symmetry and algebraic geometry, 68, American Mathematical Society (1999)

[10] C Faber, R Pandharipande, Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000) 173

[11] W Fulton, Introduction to intersection theory in algebraic geometry, 54, Published for the Conference Board of the Mathematical Sciences, Washington, DC (1984)

[12] A Gathmann, Gromov–Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001) 399

[13] P Griffiths, J Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley Sons] (1978)

[14] J Hu, Gromov–Witten invariants of blow-ups along points and curves, Math. Z. 233 (2000) 709 | DOI

[15] E N Ionel, T H Parker, Relative Gromov–Witten invariants, Ann. of Math. 157 (2003) 45

[16] E N Ionel, T H Parker, The symplectic sum formula for Gromov–Witten invariants, Ann. of Math. 159 (2004) 935

[17] D Karp, The local Gromov-Witten invariants of configurations of rational curves, PhD thesis, University of British Columbia, Vancouver, British Columbia, Canada (2005)

[18] M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves (Texel Island, 1994)", Progr. Math. 129, Birkhäuser (1995) 335

[19] A M Li, Y Ruan, Symplectic surgery and Gromov–Witten invariants of Calabi–Yau 3–folds, Invent. Math. 145 (2001) 151 | DOI

[20] J Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001) 509

[21] J Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002) 199

[22] J Li, C C M Liu, K Liu, J Zhou, A mathematical theory of the topological vertex,

[23] B H Lian, K Liu, S T Yau, Mirror principle I, Asian J. Math. 1 (1997) 729

[24] C C M Liu, K Liu, J Zhou, A formula of two-partition Hodge integrals,

[25] C C M Liu, K Liu, J Zhou, A proof of a conjecture of Mariño–Vafa on Hodge integrals, J. Differential Geom. 65 (2003) 289

[26] I G Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press Oxford University Press (1979)

[27] Y I Manin, Generating functions in algebraic geometry and sums over trees, from: "The moduli space of curves (Texel Island, 1994)", Progr. Math. 129, Birkhäuser (1995) 401

[28] M Mariño, C Vafa, Framed knots at large N, from: "Orbifolds in mathematics and physics (Madison, WI, 2001)", Contemp. Math. 310, Amer. Math. Soc. (2002) 185

[29] A Okounkov, R Pandharipande, Hodge integrals and invariants of the unknot, Geom. Topol. 8 (2004) 675 | DOI

[30] A Okounkov, N Reshetikhin, C Vafa, Quantum Calabi–Yau and classical crystals, from: "The unity of mathematics", Progr. Math. 244, Birkhäuser (2006) 597

[31] R Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999) 489 | DOI

[32] C Voisin, A mathematical proof of a formula of Aspinwall and Morrison, Compositio Math. 104 (1996) 135

[33] J Zhou, A Conjecture on Hodge Integrals,

[34] J Zhou, Hodge Integrals and Integrable Hierarchies,