Self-duality in quantum K-theory

We describe an attempt to make quantum K-theory (of stable maps) more amenable to the self-duality/rigidity arguments of arXiv:1512.07363 in quasimap theory, by twisting the virtual structure sheaf. For $\mathbb{P}^n$ this twist produces invariants which are self-dual rational functions, but asymptotic analysis shows this is no longer the case for general GKM manifolds such as flag varieties. Such analysis is done via an explicit combinatorial description of localization for quantum K-theory on GKM manifolds, and Givental's adelic characterization.