Orbifolded Elliptic Genera of Non-Compact Models

TL;DR

This paper analyzes the orbifolded elliptic genera of non-compact N=2 superconformal models, revealing their structure as completed mock modular forms and exploring their decomposition into discrete and continuous parts.

Contribution

It generalizes the flavored elliptic genus to real central charge and computes orbifolded genera for cigar and Liouville models, connecting them to mock modular forms.

Findings

Orbifolded elliptic genus is a completed mock modular form.

Decomposition into discrete and continuous contributions.

Connection to U(1) modular invariants at rational radius.

Abstract

We revisit the flavored elliptic genus of the N=2 superconformal cigar model and generalize the analysis of the path integral result to the case of real central charge. It gives rise to a non-holomorphic modular covariant function generalizing completed mock modular forms. We also compute the genus for angular orbifolds of the cigar and Liouville theory and decompose it in terms of discrete and continuous contributions. The orbifolded elliptic genus at fractional level is a completed mock modular form with a shadow related to U$(1)$ modular invariants at rational radius squared. We take the limit of the orbifolded genera towards a weighted ground state index and carefully interpret the contributions. We stress that the orbifold cigar and Liouville theories have a maximal and a minimal radius, respectively.