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.