Convergence and analytic decomposition of quantum cohomology of toric bundles

Abstract

We prove that the equivariant big quantum cohomology QHT⁎(E) of the total space E of a toric bundle E→B converges provided that the big quantum cohomology QH⁎(B) converges. The proof is based on Brown's mirror theorem for toric bundles [5]. It has been observed by Coates, Givental and Tseng that the quantum connection of E splits into copies of that of B[10]. 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 QHT⁎(E). In particular, we obtain an analytic decomposition for the equivariant/non-equivariant big quantum cohomology of E.