Gromov–Witten theory of elliptic fibrations : Jacobi forms and holomorphic anomaly equations

Oberdieck, Georg¹ ; Pixton, Aaron²

¹ Mathematisches Institut, Universität Bonn, Bonn, Germany
² Department of Mathematics, MIT, Cambridge, MA, United States

Geometry & topology, Tome 23 (2019) no. 3, pp. 1415-1489.

Résumé

We conjecture that the relative Gromov–Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice $E_8$. We also show the compatibility of the conjecture with the degeneration formula. As a corollary we deduce that the Gromov–Witten potentials of the Schoen Calabi–Yau threefold (relative to ${\mathbb{P}}^1$) are $E_8\times E_8$ quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi–Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed.

In the appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.

Détail

DOI : 10.2140/gt.2019.23.1415

Classification : 14N35
Keywords: Gromov–Witten theory, elliptic fibrations, holomorphic anomaly equation, rational elliptic surface, Schoen, Jacobi forms, quasimodular, Siegel modular forms, abelian surfaces

Bibliographie

[1] M Alim, E Scheidegger, S T Yau, J Zhou, Special polynomial rings, quasi modular forms and duality of topological strings, Adv. Theor. Math. Phys. 18 (2014) 401

[2] B Andreas, D Hernández Ruipérez, Fourier Mukai transforms and applications to string theory, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 99 (2005) 29

[3] M Bershadsky, S Cecotti, H Ooguri, C Vafa, Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994) 311

[4] S Bloch, A Okounkov, The character of the infinite wedge representation, Adv. Math. 149 (2000) 1 | DOI

[5] J Bryan, N C Leung, The enumerative geometry of K3 surfaces and modular forms, J. Amer. Math. Soc. 13 (2000) 371 | DOI

[6] J Bryan, G Oberdieck, R Pandharipande, Q Yin, Curve counting on abelian surfaces and threefolds, Algebr. Geom. 5 (2018) 398

[7] C H Conley, M Westerholt-Raum, Harmonic Maaß–Jacobi forms of degree 1 with higher rank indices, Int. J. Number Theory 12 (2016) 1871 | DOI

[8] M Eichler, D Zagier, The theory of Jacobi forms, 55, Birkhäuser (1985) | DOI

[9] C Faber, R Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. 7 (2005) 13 | DOI

[10] C Faber, R Pandharipande, Tautological and non-tautological cohomology of the moduli space of curves, from: "Handbook of moduli, I" (editors G Farkas, I Morrison), Adv. Lect. Math. 24, Int. (2013) 293

[11] W Fulton, R Pandharipande, Notes on stable maps and quantum cohomology, from: "Algebraic geometry" (editors J Kollár, R Lazarsfeld, D R Morrison), Proc. Sympos. Pure Math. 62, Amer. Math. Soc. (1997) 45 | DOI

[12] E Getzler, R Pandharipande, Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B 530 (1998) 701 | DOI

[13] T Graber, R Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005) 1 | DOI

[14] S Hosono, BCOV ring and holomorphic anomaly equation, from: "New developments in algebraic geometry, integrable systems and mirror symmetry" (editors M H Saito, S Hosono, K Yoshioka), Adv. Stud. Pure Math. 59, Math. Soc. Japan (2010) 79 | DOI

[15] S Hosono, M H Saito, J Stienstra, On the mirror symmetry conjecture for Schoen’s Calabi–Yau 3–folds, from: "Integrable systems and algebraic geometry" (editors M H Saito, Y Shimizu, K Ueno), World Sci. (1998) 194

[16] S Hosono, M H Saito, A Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177 | DOI

[17] M X Huang, S Katz, A Klemm, Topological string on elliptic CY 3–folds and the ring of Jacobi forms, J. High Energy Phys. (2015) | DOI

[18] F Janda, Gromov–Witten theory of target curves and the tautological ring, Michigan Math. J. 66 (2017) 683 | DOI

[19] F Janda, Frobenius manifolds near the discriminant and relations in the tautological ring, Lett. Math. Phys. 108 (2018) 1649 | DOI

[20] M Kaneko, D Zagier, A generalized Jacobi theta function and quasimodular forms, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 165 | DOI

[21] Y H Kiem, J Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025 | DOI

[22] A Klemm, D Maulik, R Pandharipande, E Scheidegger, Noether–Lefschetz theory and the Yau–Zaslow conjecture, J. Amer. Math. Soc. 23 (2010) 1013 | DOI

[23] M Kool, R Thomas, Reduced classes and curve counting on surfaces, I : Theory, Algebr. Geom. 1 (2014) 334 | DOI

[24] M Krauel, G Mason, Jacobi trace functions in the theory of vertex operator algebras, Commun. Number Theory Phys. 9 (2015) 273 | DOI

[25] Y P Lee, F Qu, A product formula for log Gromov–Witten invariants, J. Math. Soc. Japan 70 (2018) 229 | DOI

[26] H Lho, R Pandharipande, Stable quotients and the holomorphic anomaly equation, Adv. Math. 332 (2018) 349 | DOI

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

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

[29] A Libgober, Elliptic genera, real algebraic varieties and quasi-Jacobi forms, from: "Topology of stratified spaces" (editors G Friedman, E Hunsicker, A Libgober, L Maxim), Math. Sci. Res. Inst. Publ. 58, Cambridge Univ. Press (2011) 95

[30] D Maulik, R Pandharipande, A topological view of Gromov–Witten theory, Topology 45 (2006) 887 | DOI

[31] D Maulik, R Pandharipande, R P Thomas, Curves on K3 surfaces and modular forms, J. Topol. 3 (2010) 937 | DOI

[32] T Milanov, Y Ruan, Y Shen, Gromov–Witten theory and cycle-valued modular forms, J. Reine Angew. Math. 735 (2018) 287 | DOI

[33] G Oberdieck, A Pixton, Holomorphic anomaly equations and the Igusa cusp form conjecture, Invent. Math. 213 (2018) 507 | DOI

[34] G Oberdieck, J Shen, Curve counting on elliptic Calabi–Yau threefolds via derived categories, preprint (2016)

[35] A Okounkov, R Pandharipande, Gromov–Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. 163 (2006) 517 | DOI

[36] R Pandharipande, A Pixton, Gromov–Witten/Pairs correspondence for the quintic 3–fold, J. Amer. Math. Soc. 30 (2017) 389 | DOI

[37] K Sakai, En Jacobi forms and Seiberg–Witten curves, preprint (2017)

[38] K Sakai, Topological string amplitudes for the local 1/2K3 surface, Prog. Theor. Exp. Phys. (2017) | DOI

[39] N R Scheithauer, The Weil representation of SL2(Z) and some applications, Int. Math. Res. Not. 2009 (2009) 1488 | DOI

[40] C Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988) 177 | DOI

[41] T Schürg, B Toën, G Vezzosi, Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes, J. Reine Angew. Math. 702 (2015) 1 | DOI

[42] T Shioda, On the Mordell–Weil lattices, Comment. Math. Univ. St. Paul. 39 (1990) 211

[43] N P Skoruppa, Jacobi forms of critical weight and Weil representations, from: "Modular forms on Schiermonnikoog" (editors B Edixhoven, G van der Geer, B Moonen), Cambridge Univ. Press (2008) 239 | DOI

[44] C Teleman, The structure of 2D semi-simple field theories, Invent. Math. 188 (2012) 525 | DOI

[45] C Ziegler, Jacobi forms of higher degree, Abh. Math. Sem. Univ. Hamburg 59 (1989) 191 | DOI