MIRROR SYMMETRY FOR ORBIFOLD PROJECTIVE LINES YUUKI SHIRAISHI (JOINT WORK WITH YOSHIHISA ISHIBASHI AND ATSUSHI TAKAHASHI)

Abstract There are three theories coming from different origins; the Gromov–Witten theory, primitive forms for isolated hypersurface singurality and the invariant theory of extended affine Weyl groups. These theories seems, at a glance, to have no relations each other. However, it is well-known that each theory provides a certain complex manifold with special structure called Frobenius structure or Frobenius manifold. In this talk, we will deal with following three theories; • the Gromov–Witten theory for orbifold projective line P1A with three orbifold points whose oders are a1 , a2 , a3 respectively, • a primitive form for cusp polynomial fA = xa11 + xa22 + xa33 − s−1 µA x1 x2 x3 , c • an extended affine Weyl group WA , where A is a triplet of positive integers (a1 , a2 , a3 ) such that χA := 1/a1 +1/a2 +1/a3 −1 > 0. We will also show that Frobenius structures from them are isomorphic each other under a suitable choice of primitive form ζA , i.e., Theorem. If χA > 0, there are isomorphisms of Frobenius manifolds: MP1A ' M(fA ,ζA ) ' MW cA . Our result simplifies and generalizes the results of Milanov–Tseng [4] and Rossi [5]. References [1] Y. Ishibashi, Y. Shiraishi, A. Takahashi, A Uniqueness Theorem for Frobenius Manifolds and Gromov–Witten Theory for Orbifold Projective Lines, arXiv:1209.4870. [2] Y. Ishibashi, Y. Shiraishi, A. Takahashi, Primitive Forms for Affine Cusp Polynomials, arXiv:1211.1128. [3] Y. Ishibashi, Y. Shiraishi, A. Takahashi, On the Frobenius Manifolds for Cusp Singularities, in preparation. [4] Todor E. Milanov, Hsian-Hua Tseng, The spaces of Laurent polynomials, P1 -orbifolds, and integrable hierarchies, Journal f¨ ur die reine und angewandte Mathematik (Crelle’s Journal), Volume 2008, Issue 622, Pages189–235.

[5] P. Rossi, Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of Seifert fibrations, arXiv:0808.2626. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka Osaka, 560-0043, Japan E-mail address: [email protected]