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

TL;DR

This paper proves that the big quantum cohomology of $ ext{P}^1$-orbifolds is generically semisimple, confirming Dubrovin's conjecture for orbi-curves and characterizing when the small quantum cohomology is semisimple.

Contribution

It establishes the generic semisimplicity of big quantum cohomology for $ ext{P}^1$-orbifolds and verifies Dubrovin's conjecture for these cases.

Findings

Big quantum cohomology of $ ext{P}^1$-orbifolds is generically semisimple.

Small quantum cohomology is semisimple iff the orbifold is Fano.

Confirmed Dubrovin's conjecture for orbi-curves.

Abstract

For a $\mathbb P^1$-orbifold $\mathscr C$, 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 $\mathscr C$ is generically semisimple iff $\mathscr C$ is Fano, i.e. it has positive orbifold Euler characteristic.