F-Bundles and the Mirror Symmetry of Flag Varieties

This thesis consists of three projects related to enumerative geometry and mirror symmetry, with particular emphasis on the mirror symmetry of flag varieties. The first project introduces and develops the theory of F-bundles, a framework for formulating mirror symmetry type results. We prove a spectral decomposition theorem for maximal F-bundles, and use it to obtain reconstruction and uniqueness results for certain decompositions of quantum cohomology related to birational geometry, complementing existing the existence theorem in the literature. In the third project, we further extend the F-bundle formalism to the equivariant setting and establish an unfolding theorem strengthening mirror symmetry statements from small to big quantum cohomology. As an application, we derive the equivariant mirror symmetry of general flag varieties in the third project, extending previous results which were previously known only at the level of small quantum cohomology. The second project focuses on the mirror symmetry of flag varieties. For complex partial flag varieties, we provide an explicit Plücker coordinate superpotential formula that is sufficient to recover their small quantum cohomology on A-side, and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues of the action of the first Chern class on quantum cohomology are equal to the critical values of the superpotential.