Quantum K-theory studies the K-theoretic Gromov-Witten (GW) invariants, which, for a given target space $X$, are defined as virtual holomorphic Euler characteristics of vector bundles over the moduli space $\overline{\mathcal{M}}_{g,m}(X,d)$ of stable maps. Roughly speaking, the invariants count the number of curves on $X$ satisfying specific topological and geometric constraints. Since the emergence of the theory at the beginning of this century as an analogue of quantum cohomology, these invariants have not only provided new insight into enumerative geometry, but have inspired applications in representation theory, integrable systems, and even theoretic physics as well.
In this dissertation, we study the genus-$0$ quantum K-theory of partial flag varieties using the idea of abelian/non-abelian correspondence. Regarding any flag variety as a GIT quotient of vector space by a non-abelian group, we relate its quantum K-theory to that of its associated abelian GIT quotient, a toric variety whose quantum K-theory is well-understood. As a consequence, we provide eventually for the flag variety an explicit parameterization of the image cone of a generating function commonly known as the ``big $\mathcal{J}$-function'', which encodes all of its genus-$0$ K-theoretic GW invariants. This generalizes existing results for toric varieties.
The main technical tool employed in the process is a recursive characterization of the image cone developed in a series of papers by Givental. The characterization exists for any target space equipped with a torus action with isolated fixed points as well as isolated connecting orbits, and is obtained from a brilliant observation of the fixed-point localization computations of the invariants. Since the toric fixed points on the associated abelian quotient $Y$ of a flag variety are, however, seldom isolated, a substantial part of this dissertation is dedicated to extending the characterization to non-isolated cases. We demonstrate a vanishing result of the contribution from the non-isolated fixed points to the recursive formulae.
As applications of the techniques developed in this dissertation, we consider various twisted quantum K-theories. In particular, we first derive a quantum Lefschetz theorem which characterizes genus-$0$ K-theoretic GW invariants of a complete intersection in (or a vector bundle space over) the flag variety. Various generating functions of quasi-map invariants that arise in literature are thus realized as values of twisted big $\mathcal{J}$-functions. Moreover, we study the level structures, which are twistings of determinantal type on the quantum K-theory, and explain the level correspondence phenomenon between dual flag varieties. Finally, we establish the quantum Serre duality for general target spaces, which relates two different types of twistings involving the level structures.
At the end of this dissertation, we construct the K-theoretic mirror of the flag variety as $q$-integrals on a family of tori, following the pattern in quantum cohomology. When the integration cycles are carefully chosen, the $q$-integrals are proven to recover, up to some scalars, components of the small $J$-function which is the specific value of the big $\mathcal{J}$-function with vanishing input.
