Quantum cohomology of partial flag manifolds

We compute the quantum cohomology rings of the partial flag manifolds F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive computation uses the idea of Givental and Kim. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle F_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found.Comment: 33 page