Abstract:

Quantum cohomology $QH^*(X)$ of a smooth projective variety $X$ is a Frobenius manifold defined by genus zero Gromov-Witten invariants. A priori, the structure constants of $QH^*(X)$ are formal power series, but are expected to be convergent series in the context of mirror symmetry. It was proved by Coates, Corti, Iritani and Tseng that torus-equivariant quantum cohomology of a toric variety is convergent. It was observed by Coates, Givental and Tseng that torus-equivariant quantum cohomology $QH^*_T(E)$ is formally decomposed into a direct sum of $QH^*(B)$. In this talk, I will explain that, under the assumption that $QH^*(B)$ is convergent, $QH^*_T(E)$ is convergent and the decomposition becomes analytic. As an application, we can take the non-equivariant limit of the decomposition.