Elliptic Trace Map on Chiral Algebras

TL;DR

This paper develops a two-dimensional chiral analogue of the algebraic index theorem using chiral algebras, constructing trace maps on elliptic chiral homology and computing the formal Witten genus.

Contribution

It introduces a novel trace map on elliptic chiral homology and extends the algebraic index theorem to a chiral, two-dimensional setting.

Findings

Constructed a trace map on elliptic chiral homology of free beta gamma-bc system.

Computed the trace on the unit constant chiral chain, obtaining the formal Witten genus.

Developed a family of elliptic trace maps on coset models.

Abstract

Trace map on deformation quantized algebra leads to the algebraic index theorem. In this paper, we investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras developed by Beilinson and Drinfeld. We construct a trace map on the elliptic chiral homology of the free beta gamma-bc system using the BV quantization framework. As an example, we compute the trace evaluated on the unit constant chiral chain and obtain the formal Witten genus in the Lie algebra cohomology. We also construct a family of elliptic trace maps on coset models.