The First Chiral Homology Group

TL;DR

This paper investigates the first chiral homology group of elliptic curves with vertex algebra coefficients, establishing finiteness conditions, explicit flat connections, and applications to rational vertex algebras.

Contribution

It introduces new finiteness conditions for chiral homology, explicitly describes flat connections, and constructs linear functionals related to module extensions, extending previous degree 0 results.

Findings

Finiteness conditions guarantee finite dimensionality of homologies.

Explicit description of flat connections under elliptic curve variation.

Proved vanishing of the first chiral homology for several rational vertex algebras.

Abstract

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.