On semisimplicity of quantum cohomology of ${\mathbb{P}}^1$-orbifolds

Abstract

For a ${\mathbb{P}}^1$-orbifold $\mathcal{C}$, we prove that its big quantum cohomology is generically semisimple. As a corollary, we verify a conjecture of Dubrovin for orbi-curves. We also show that the small quantum cohomology of $\mathcal{C}$ is generically semisimple iff $\mathcal{C}$ is Fano, i.e. it has positive orbifold Euler characteristic.

16 pages. Comments are welcome!