A formula of two-partition Hodge integrals

Liu, Chiu-Chu Melissa^1, 2 ; Liu, Kefeng³ ; Zhou, Jian⁴

¹ Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China and Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
² Department of Mathematics, Columbia University, New York, New York, 10027
³ Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China and Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
⁴ Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China and Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China

Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 149-184

Résumé

Motivated by the Mariño-Vafa formula of Hodge integrals and physicists’ predictions on local Gromov-Witten invariants of toric Fano surfaces in a Calabi-Yau threefold, the third author conjectured a formula of certain Hodge integrals in terms of certain Chern-Simons invariants of the Hopf link. We prove this formula by virtual localization on moduli spaces of relative stable morphisms.

Détail

DOI : 10.1090/S0894-0347-06-00541-8

Bibliographie

[1] Aganagic, Mina, Klemm, Albrecht, Mariã±O, Marcos, Vafa, Cumrun The topological vertex Comm. Math. Phys. 2005 425 478

[2] Aganagic, Mina, Mariã±O, Marcos, Vafa, Cumrun All loop topological string amplitudes from Chern-Simons theory Comm. Math. Phys. 2004 467 512

[3] Bryan, Jim, Pandharipande, Rahul Curves in Calabi-Yau threefolds and topological quantum field theory Duke Math. J. 2005 369 396

[4] Diaconescu, Duiliu-Emanuel, Florea, Bogdan Localization and gluing of topological amplitudes Comm. Math. Phys. 2005 119 149

[5] Dijkgraaf, Robbert Mirror symmetry and elliptic curves 1995 149 163

[6] Ekedahl, Torsten, Lando, Sergei, Shapiro, Michael, Vainshtein, Alek Hurwitz numbers and intersections on moduli spaces of curves Invent. Math. 2001 297 327

[7] Fantechi, B., Pandharipande, R. Stable maps and branch divisors Compositio Math. 2002 345 364

[8] Goulden, I. P., Jackson, D. M., Vainshtein, A. The number of ramified coverings of the sphere by the torus and surfaces of higher genera Ann. Comb. 2000 27 46

[9] Graber, T., Pandharipande, R. Localization of virtual classes Invent. Math. 1999 487 518

[10] Graber, Tom, Vakil, Ravi Hodge integrals and Hurwitz numbers via virtual localization Compositio Math. 2003 25 36

[11] Graber, Tom, Vakil, Ravi Relative virtual localization and vanishing of tautological classes on moduli spaces of curves Duke Math. J. 2005 1 37

[12] Ionel, Eleny-Nicoleta, Parker, Thomas H. Relative Gromov-Witten invariants Ann. of Math. (2) 2003 45 96

[13] Ionel, Eleny-Nicoleta, Parker, Thomas H. The symplectic sum formula for Gromov-Witten invariants Ann. of Math. (2) 2004 935 1025

[14] Katz, Sheldon, Liu, Chiu-Chu Melissa Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc Adv. Theor. Math. Phys. 2001 1 49

[15] Kontsevich, Maxim Intersection theory on the moduli space of curves and the matrix Airy function Comm. Math. Phys. 1992 1 23

[16] Li, An-Min, Ruan, Yongbin Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds Invent. Math. 2001 151 218

[17] Li, An-Min, Zhao, Guosong, Zheng, Quan The number of ramified covering of a Riemann surface by Riemann surface Comm. Math. Phys. 2000 685 696

[18] Li, Jun Stable morphisms to singular schemes and relative stable morphisms J. Differential Geom. 2001 509 578

[19] Li, Jun A degeneration formula of GW-invariants J. Differential Geom. 2002 199 293

[20] Li, Jun Lecture notes on relative GW-invariants 2004 41 96

[21] Liu, Chiu-Chu Melissa, Liu, Kefeng, Zhou, Jian A proof of a conjecture of Mariño-Vafa on Hodge integrals J. Differential Geom. 2003 289 340

[22] Macdonald, I. G. Symmetric functions and Hall polynomials 1995

[23] Mariã±O, Marcos, Vafa, Cumrun Framed knots at large 𝑁 2002 185 204

[24] Morton, Hugh R., Lukac, Sascha G. The Homfly polynomial of the decorated Hopf link J. Knot Theory Ramifications 2003 395 416

[25] Okounkov, Andrei Toda equations for Hurwitz numbers Math. Res. Lett. 2000 447 453

[26] Okounkov, A., Pandharipande, R. Gromov-Witten theory, Hurwitz theory, and completed cycles Ann. of Math. (2) 2006 517 560

[27] Okounkov, A., Pandharipande, R. Hodge integrals and invariants of the unknot Geom. Topol. 2004 675 699

[28] Witten, Edward Quantum field theory and the Jones polynomial Comm. Math. Phys. 1989 351 399

[29] Witten, Edward Two-dimensional gravity and intersection theory on moduli space 1991 243 310