Relations in Twisted Quantum K-Rings

TL;DR

This paper introduces twisted quantum K-rings, develops computational tools, explores applications including quantum K-theory with level structure, and proposes a conjecture relating quantum K-rings of GIT quotients, proving it for Grassmannians.

Contribution

It defines twisted quantum K-rings, adapts computational methods from ordinary quantum K-theory, and formulates and proves a conjecture relating quantum K-rings of GIT quotients.

Findings

Developed a toolkit for computing relations in twisted quantum K-rings.

Confirmed predictions from physics regarding quantum K-theory with level structure.

Proved the abelian/non-abelian conjecture for Grassmannians and re-derived Whitney relations.

Abstract

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 in that case.