2021/03/10 by Jethro van Ekeren, Reimundo Heluani, van Ekeren, Jethro +1 · 1 citation
Mathematics · #14H52 #17B69 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2103.06322
openalex publication_date 2021/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C2-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 sl2 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.