Loop Space Formalism and K-Theoretic Quantum Serre Duality

In this paper, we prove the quantum Serre duality for genus-zero K-theoretic permutation-invariant Gromov-Witten theory. The formulation of the theorem relies on an extension to the formalism of loop spaces and big $\mathcal{J}$-functions more intrinsic to quantum K-theory. With the extended formalism, we also arrive at a re-interpretation of the level structures in terms of twisted quantum K-theories. We discuss the torus-equivariant theory in the end, and as an application generalize the K-theoretic quantum Serre duality to non-primitive vector bundles over flag varieties.