Stationary Descendents and the Discriminant Modular Form

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.