Abstract:

Quantum cohomology can be studied by a mirror theorem, that is, by finding an $I$-function, which is a certain generating function of genus zero Gromov-Witten invariants. In general, it is a very difficult problem explicitly describing an $I$-function. For instance, an $I$-function of a smooth semi-projective toric variety is a hypergeometric series (given by Givental, Coates, Corti, Iritani and Tseng), and that of a toric bundle constructed as a GIT quotient of a direct sum of line bundles is described as a hypergeometric modification of the $J$-function of the base space. It is natural to consider an $I$-function of a toric bundle that is a GIT quotient of a vector bundle, which is called a non-split toric bundle. However, Brown's method cannot be applied to the non-split case, so we need a completely different strategy. In this talk, I will explain how to establish a mirror theorem for non-split toric bundles. This work is partially based on joint work with Iritani.