Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities

[Invariants de Gromov–Witten pour les orbifolds miroir des singularités elliptiques simples]

Satake, Ikuo ; Takahashi, Atsushi

Annales de l'Institut Fourier, Tome 61 (2011), p. 2885-2907 / Harvested from Numdam

Résumé

Nous considérons une symétrie miroir des singularités elliptiques simples. En particulier, nous construisons des isomorphismes de variétés de Frobenius entre celui de la théorie de Gromov–Witten d’une droite projective à poids, celui de la théorie des formes primitives pour un déploiement universel d’une singularité elliptique simple et celui de la théorie des invariants pour un groupe de Weyl elliptique. Comme conséquence, nous donnons une interprétation géométrique des coefficients de Fourier d’un produit eta considéré par K. Saito.

Abstract

We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

Détail

Publié le : 2011-01-01

MR 3112511 | Zbl pre06193030

DOI : https://doi.org/10.5802/aif.2797
Classification:  14J33,  14N35,  32S25
Mots clés: symétrie miroir, singularité elliptique simple, variété de Frobenius, théorie de Gromov–Witten, droite projective à poids, forme primitive, théorie des invariants, groupe de Weyl elliptique, produit eta

Bibliographie

[1] Abramovich, D.; Graber, T.; Vistoli, A. Gromov–Witten theory of Deligne–Muford stacks, Amer. J. Math., Tome 130 (2008) no. 5, pp. 1337-1398 | MR 2450211 | Zbl 1193.14070

[2] Chen, W.; Ruan, Y. Orbifold Gromov–Witten Theory, Orbifolds in mathematics and physics (Madison, WI, 2001), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 310 (2002), pp. 25-85 | MR 1950939 | Zbl 1091.53058

[3] Dubrovin, B. Geometry of 2d topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993), Springer, Berlin (Lecture Notes in Math.) Tome 1620 (1996), pp. 120-348 | MR 1397274 | Zbl 0841.58065

[4] Dubrovin, B.; Zhang, Y. Bihamiltonian hierarchies in 2D topological field theory at one-loop approximation, Commun. Math. Phys., Tome 198 (1998), pp. 311-361 | MR 1672512 | Zbl 0923.58060

[5] Dubrovin, B.; Zhang, Y. Frobenius manifolds and Virasoro constraints, Selecta Math. (N.S.), Tome 5 (1999) no. 4, pp. 423-466 | MR 1740678 | Zbl 0963.81066

[6] Ebeling, W.; Takahashi, A. Strange duality of weighted homogeneous polynomials, Compositio Math., Tome 147 (2011), pp. 1413-1433 | MR 2834726 | Zbl 1238.14029

[7] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory (arXiv:0712.4021)

[8] Krawitz, M.; Shen, Y. Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold $\mathbb{P}^1$ (arXiv:1106.6270)

[9] Milanov, T.; Ruan, Y. Gromov-Witten theory of elliptic orbifold $\mathbb{P}^1$ and quasimodular forms (arXiv:1106.2321)

[10] Mumford, D. Tata Lectures on Theta I, Birkhäuser, Progress in Math., Tome 28 (1983) | MR 688651 | Zbl 0509.14049

[11] Noumi, M.; Yamada, Y. Notes on the flat structures associated with simple and simply elliptic singularities, Integrable Systems and Algebraic Geometry, World-Scientific ((eds. M.-H. Saito, Y. Shimuzu, K. Ueno), Proceedings of the Taniguchi Symposium 1997) (1998), pp. 373-383 | MR 1672069 | Zbl 0964.32025

[12] Ohyama, Y. Differential relations of theta functions, Osaka J. Math., Tome 32 (1995) no. 2, pp. 431-450 | MR 1355752 | Zbl 0864.34001

[13] Ohyama, Y. Differential equations for modular forms of level three, Funkcial. Ekvac., Tome 44 (2001) no. 3, pp. 377-389 | MR 1893938 | Zbl 1145.11310

[14] Rossi, P. Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations (arXiv:0808.2626) | Zbl 1235.14053

[15] Saito, K. Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Tome 28 (1982) no. 3, pp. 775-792 | MR 656053 | Zbl 0523.32015

[16] Saito, K. Period mapping associated to a primitive form, Publ. RIMS, Kyoto Univ., Tome 19 (1983), pp. 1231-1264 | MR 723468 | Zbl 0539.58003

[17] Saito, K. Extended Affine Root System II, Publ. RIMS, Kyoto Univ., Tome 26 (1990), pp. 15-78 | MR 1053908 | Zbl 0713.17014

[18] Saito, K. Duality for regular systems of weights, Asian J. Math., Tome 2 (1998), pp. 983-1047 | MR 1734136 | Zbl 0963.32023

[19] Saito, K.; Takahashi, A. From Primitive Forms to Frobenius manifolds (Proceedings of Symnposia in Pure Mathematics) Tome 78 (2008), pp. 31-48 | MR 2483747 | Zbl 1161.32013

[20] Satake, I. Flat Structure and the Prepotential for the Elliptic Root System of Type $D^{(1,1)}_4$, Topological field theory, primitive forms and related topics, Birkhäuser ((ed. by M. Kashiwara, et al.), Progress in Math.) Tome 160 (1998), pp. 427-452 | MR 1653034 | Zbl 0933.11026

[21] Satake, I. Frobenius manifolds for elliptic root systems, Osaka J. Math., Tome 17 (2010), pp. 301-330 | MR 2666136 | Zbl 1191.53057

[22] Strachan, I. Simple elliptic singularities: A note on their G-function (arXiv:1004.2140) | Zbl 1256.53056

[23] Takahashi, A. Weighted projective lines associated to regular systems of weights of dual type, Adv. Stud. Pure Math., Tome 59 (2010), pp. 371-388 | MR 2683215 | Zbl 1213.14075

[24] Ueda, K. Homological mirror symmetry and simple elliptic singularities (arXiv:math/0604361)

[25] Verlinde, E.; Warner, N. P. Topological Landau-Ginzburg matter at $c=3$, Physics Letters B, Tome 269 (1991), pp. 96-102 | MR 1134056