Relations in Twisted Quantum K-Rings

We introduce twisted quantum K-rings, defined via twisted K-theoretic Gromov-Witten invariants. We develop a toolkit for computing relations by adapting some results about ordinary quantum K rings to our setting, and discuss some applications, including Ruan-Zhang's quantum K-theory with level structure, and complete intersections inside projective space, confirming some predictions coming from physics. In addition, we formulate a ring-theoretic abelian/non-abelian correspondence conjecture, relating the quantum K-ring of a GIT quotient $X//G$ to a certain twist of the quantum K-ring of $X//T$, the quotient by the maximal torus. We prove this conjecture for the case of Grassmanians, and use this to give another proof of the Whitney relations of Mihalcea-Gu-Sharpe-Zhou.