Quantum K-theory of flag varieties via non-abelian localization

TL;DR

This paper develops a method to explicitly compute the genus-zero K-theoretic Gromov-Witten invariants for flag varieties, extending localization techniques and establishing a K-theoretic mirror symmetry framework.

Contribution

It generalizes torus localization to handle non-isolated orbits, constructs a K-theoretic mirror, and explores dualities and Lefschetz theorems for flag varieties.

Findings

Explicit reconstruction of the big J-function for flag varieties.

Extension of localization to non-isolated toric orbits.

Construction of a K-theoretic mirror using Jackson integrals.

Abstract

In this paper, we reconstruct explicitly the generating function of genus-zero K-theoretic permutation-invariant Gromov-Witten invariants, known as the big $\mathcal{J}$-function, for any partial flag variety. The reconstruction may start with any Weyl-group-invariant value of the well-understood big $\mathcal{J}$-function of an associated toric variety. We generalize the recursive method \cite{Givental:perm2}, based on torus fixed point localization, to deal with non-isolated one-dimensional toric orbits, through incorporating ``balanced broken orbits'' into consideration and subsequently proving a vanishing result of their contribution. Furthermore, we extend the study to twisted generating functions of the flag variety, and demonstrate properties including a non-abelian quantum Lefschetz theorem and a duality between level structures. In the end, we construct a K-theoretic mirror in terms of Jackson-type integrals.