K-theoretic Gromov-Witten invariants of lines in homogeneous spaces

TL;DR

This paper establishes a connection between equivariant K-theoretic Gromov-Witten invariants of lines in homogeneous spaces and the K-theory of associated varieties, enabling computation and revealing positivity properties.

Contribution

It demonstrates that these invariants can be computed via the K-theory of certain homogeneous varieties, generalizing previous results and providing new identities.

Findings

Invariants equal quantities from K-theory of Z_k(X)

Computed structure constants for degree e_k

Proved vanishing and positivity properties

Abstract

Let X=G/P be a homogeneous space and e_k be the class of a simple coroot in H_2(X). A theorem of Strickland shows that for almost all X, the variety of pointed lines of degree e_k, denoted Z_k(X), is again a homogeneous space. For these X we show that the 3-point, genus 0, equivariant K-theoretic Gromov-Witten invariants of lines of degree e_k are equal to quantities obtained in the (ordinary) equivariant K-theory of Z_k(X). We apply this to compute the structure constants N_{u,v}^{w, e_k} for degree e_k from the multiplication of two Schubert classes in the equivariant quantum K-theory ring of X. Using geometry of spaces of lines through Schubert or Richardson varieties we prove vanishing and positivity properties of N_{u,v}^{w,e_k}. This generalizes many results about K-theory and quantum cohomology of X, and also gives new identities among the structure constants in equivariant K-theory of X.