Gromov-Witten Theory of Elliptic Orbifold Projective Lines.

Abstract

In this dissertation, we prove the Landau-Ginzburg/Calabi-Yau correspondence of all genera holds true for elliptic orbifold projective lines with three orbifold points. More precisely, we prove the Gromov-Witten theories of those orbifolds are related to the Fan-Jarvis-Ruan-Witten theories of some invertible simple elliptic singularities by analytic continuation and quantization of symplectic transformation. The proof is based on global mirror symmetry method. We analyze special limits which appear in Saito-Givental theories of mirror singularities. We also prove that Gromov-Witten cycles of those orbifolds are quasi-modular forms.