Stationary Descendents and the Discriminant Modular Form

TL;DR

This paper explores the linear relations among stationary descendent Gromov-Witten invariants of elliptic curves, revealing connections to the discriminant modular form and Ramanujan tau values, with computational aid from Sage.

Contribution

It introduces the descendent matroid of weight k and uses it to express the discriminant modular form and Ramanujan tau values in terms of Gromov-Witten invariants.

Findings

All relations among stationary descendents at weight 12 are classified.

A closed-form expression for Ramanujan tau values in terms of Gromov-Witten invariants.

Computational methods using Sage are developed and discussed.

Abstract

The generating functions of stationary descendent Gromov-Witten invariants of an elliptic curve are known to be Fourier expansions of quasimodular forms. When one restricts to the subspace of forms of a fixed weight $k$, there is an abundance of linear relations among these generating functions. This naturally leads one to study the resulting linear matroid, which we refer to as the descendent matroid of weight $k$. In the case of weight 12, we use this matroid to compute and organize all of the ways to express the discriminant modular form in terms of stationary descendents. As a consequence, we find a closed-form expression of Ramanujan tau values in terms of Gromov-Witten invariants of an elliptic curve. All computations were aided with the use of Sage, and the classes and functions written in Sage are discussed in the appendix.