A mirror theorem for non-split toric bundles

Abstract We construct an I -function for toric bundles obtained as a fiberwise GIT quotient of a (not necessarily split) vector bundle. This is a generalization of Brown’s I -function for split toric bundles (Brown in Int Math Res Not IMRN 19:5437–5482, 2014) and the I -function for non-split projective bundles (Iritani and Koto in arXiv:2307.03696 [math.AG], 2023). In order to prove the mirror theorem, we establish a characterization of points on the Givental Lagrangian cones of toric bundles and prove a mirror theorem for the twisted Gromov–Witten theory of a fiber product of projective bundles. The former result generalizes Brown’s characterization for split toric bundles [6] to the non-split case.