Twisted K-theoretic Gromov-Witten invariants

TL;DR

This paper develops twisted K-theoretic Gromov-Witten invariants, characterizes their J-function ranges, and applies the D_q module structure to prove a quantum Lefschetz theorem and relate cones in toric fibrations.

Contribution

It introduces twisted K-theoretic GW invariants, characterizes their J-functions, and extends Givental's results using the D_q module structure in permutation-equivariant theory.

Findings

Characterization of J-function range for twisted invariants

Proof of a quantum Lefschetz type theorem

Relation between cones of total space and base in toric fibrations

Abstract

We introduce twisted K-theoretic Gromov-Witten invariants - in the frameworks of both "ordinary" and permutation-equivariant K-theoretic GW theory defined recently by Givental. We focus on the case when the twisting is given by the Euler class - we characterize in both cases the range of the $J$-function of the twisted theory in terms of the untwisted theory. As applications we use the D_q module structure in the permutation-equivariant case to generalize results of Givental: we prove a "quantum Lefschetz" type theorem and we relate points on the cones of the total space with those of the base of a toric fibration.