On the descendent Gromov-Witten theory of a K3 surface

TL;DR

This paper investigates the reduced descendent Gromov-Witten theory of K3 surfaces, proposing a conjectural formula for stationary invariants, a recursion to simplify computations, and conjecturing an underlying polynomial structure.

Contribution

It introduces a conjectural closed formula for stationary invariants and a new recursion to compute descendent Gromov-Witten invariants of K3 surfaces.

Findings

Proposed a conjectural closed formula for stationary Gromov-Witten invariants.

Developed a recursion to remove descendent insertions of 1.

Conjectured a polynomial structure underlying the invariants.

Abstract

We study the reduced descendent Gromov-Witten theory of K3 surfaces in primitive curve classes. We present a conjectural closed formula for the stationary theory, which generalizes the Bryan-Leung formula. We also prove a new recursion that allows to remove descendent insertions of $1$ in many instances. Together this yields an efficient way to compute a large class of invariants (modulo the conjecture on the stationary part). As a corollary we conjecture a surprising polynomial structure which underlies the Gromov-Witten invariants of the K3 surface.