In this talk I will review part of an ongoing project aiming to understand Dubrovin conjecture from the perspective of Gauged Linear Sigma Models (GLSM). One important part of the Dubrovin conjecture relates the geometry of the (bounded) derived category of coherent sheaves of a Fano manifold X and the asymptotic behaviour of its quantum differential equation around infinity. I will explain the role of the hemisphere partition function introduced by Hori and Romo in this story by analysing one simple (but non trivial) case, namely: the CP^{k-1}-model. If time permits, I will also discuss the CP^{k-1}-model with twisted masses (equivariant model). This is a joint work with Jin Chen and Mauricio Romo.