A Twisted Non-compact Elliptic Genus

TL;DR

This paper derives a twisted elliptic genus for a supersymmetric coset conformal field theory, revealing its structure as a Jacobi form and providing physical insights into its non-holomorphic components.

Contribution

It offers a detailed path integral derivation of the twisted elliptic genus, connecting spectral asymmetry to reflection amplitudes and generalizing previous examples through orbifolding.

Findings

Elliptic genus expressed as a Jacobi form in three variables.

Non-holomorphic part linked to spectral density differences.

Generalization of existing models via orbifold construction.

Abstract

We give a detailed path integral derivation of the elliptic genus of a supersymmetric coset conformal field theory, further twisted by a global U(1) symmetry. It gives rise to a Jacobi form in three variables, which is the modular completion of a mock modular form. The derivation provides a physical interpretation to the non-holomorphic part as arising from a difference in spectral densities for the continuous part of the right-moving bosonic and fermionic spectrum. The spectral asymmetry can also be read off directly from the reflection amplitudes of the theory. By performing an orbifold, we show how our twisted elliptic genus generalizes an existing example.