Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities

Satake, Ikuo; Takahashi, Atsushi

Annales de l'Institut Fourier, Tome 61 (2011) no. 7, pp. 2885-2907.

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.

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.

Classification : 14J33, 14N35, 32S25
Keywords: a mirror symmetry, simple elliptic singularities, Frobenius manifolds, Gromov–Witten theory, weighted projective line, primitive forms, the invariant theory, an elliptic Weyl group, an eta product
Mot 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

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

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

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

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

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

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

[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, Progress in Math., 28, Birkhäuser, 1983 | MR | Zbl

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

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

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

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

[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., Volume 28 (1982) no. 3, pp. 775-792 | MR | Zbl

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

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

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

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

[20] Satake, I. Flat Structure and the Prepotential for the Elliptic Root System of Type $D_4^{(1,1)}$, Topological field theory, primitive forms and related topics ((ed. by M. Kashiwara, et al.), Progress in Math.), Volume 160, Birkhäuser, 1998, pp. 427-452 | MR | Zbl

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

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

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

[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, Volume 269 (1991), pp. 96-102 | MR