K-theoretic Gromov-Witten invariants of line degrees on flag varieties

TL;DR

This paper establishes a formula linking quantum K-theoretic Gromov-Witten invariants of line degrees on flag varieties to classical intersection numbers, simplifying computations of the quantum K-theory ring.

Contribution

It proves a quantum equals classical formula and an analogue of the Peterson comparison formula for line degrees on flag varieties, connecting quantum invariants to classical geometry.

Findings

Quantum invariants equal classical intersection numbers on related flag varieties.

Invariants coincide with those of the variety of complete flags.

Simplifies computation of the quantum K-theory ring for degrees up to line degrees.

Abstract

A homology class $d \in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $\overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals classical formula stating that any $n$-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal to a classical intersection number computed on the flag variety $G/P'$. We also prove an $n$-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov-Witten invariants of the variety of complete flags $G/B$. Our formulas make it straightforward to compute the big quantum K-theory ring of $X$ modulo degrees larger than line degrees.