The First Chiral Homology Group

We study the first chiral homology group of elliptic curves with coefficients in vacuum insertions of a conformal vertex algebra V . We find finiteness conditions on V guaranteeing that these homologies are finite dimensional, generalizing the C 2 -cofinite, or quasi-lisse condition in the degree 0 case. We determine explicitly the flat connections that these homologies acquire under smooth variation of the elliptic curve, as insertions of the conformal vector and the Weierstrass ζ function. We construct linear functionals associated to self-extensions of V -modules and prove their convergence under said finiteness conditions. These linear functionals turn out to be degree 1 analogs of the n -point functions in the degree 0 case. As a corollary we prove the vanishing of the first chiral homology group of an elliptic curve with values in several rational vertex algebras, including affine sl 2 at non-negative integral level, the ( 2 , 2 k + 1 ) -minimal models and arbitrary simple affine vertex algebras at level 1. Of independent interest, we prove a Fourier space version of the Borcherds formula.