Convergence and Analytic Decomposition of Quantum Cohomology of Toric Bundles

We prove that the equivariant big quantum cohomology QH^*_T(E) of the total space of a toric bundle E \to B converges provided that the big quantum cohomology QH^*(B) converges. The proof is based on Brown's mirror theorem for toric bundles. It has been observed by Coates, Givental and Tseng that the quantum connection of E splits into copies of that of B. Under the assumption that QH^*(B) is convergent, we construct a decomposition of the quantum D-module of E into a direct sum of that of B, which is analytic with respect to parameters of QH^*_T(E). In particular, we obtain an analytic decomposition for the equivariant/non-equivariant big quantum cohomology of E.