Stokes Matrices for the Quantum Cohomologies of Orbifold Projective Lines

We prove the Dubrovin's conjecture for the Stokes matrices for the quantum cohomology of orbifold projective lines. The conjecture states that the Stokes matrix of the first structure connection of the Frobenius manifold constructed from the Gromov-Witten theory coincides with the Euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves. Our proof is based on the homological mirror symmetry, primitive forms of affine cusp polynomials and the Picard-Lefschetz theory.