The purpose of this minicurse is to introduce the audience to a conjecture by Beilinson and Drinfeld regarding the higher chiral homology groups of the affine Kac-Moody algebra at positive integral level.

In this last lecture we will show recent advances in proving the Beilinson and Drinfeld conjecture in the case when $X$ is an elliptic curve. We will define conformal blocks and $1$-point functions on the torus, describe Zhu's algebra attached to a vertex algebra and describe the explicit complex that computes the first chiral homology of $X$ with coefficients in a vertex algebra. As an independent result we will prove a lemma regaring Koszul complexes on arc spaces.

This is joint work with Jethro Van Ekeren (UFF)