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

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 $G/P^{\prime }$. 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 $G/P^{\prime }$. 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.

To appear in the proceedings of the conference GLSM@30 (Stony Brook, May 2023)