  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 $\zeta$ 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 (2,2k+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.
