Stokes Matrices for the Quantum Cohomologies of Orbifold Projective Lines

TL;DR

This paper proves Dubrovin's conjecture linking Stokes matrices of quantum cohomology of orbifold projective lines with Euler matrices of exceptional collections, using mirror symmetry and Picard-Lefschetz theory.

Contribution

It provides a proof of Dubrovin's conjecture for orbifold projective lines' quantum cohomology, connecting Stokes matrices with derived category data.

Findings

Confirmed the conjecture for orbifold projective lines

Established the correspondence between Stokes matrices and Euler matrices

Utilized homological mirror symmetry and primitive forms

Abstract

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.