A Twisted Non-compact Elliptic Genus

TL;DR

This work provides a detailed path-integral derivation of the twisted elliptic genus for the non-compact axial coset $SL(2,\mathbb{R})/U(1)$, twisted by a global $U(1)$ symmetry. The authors show that the resulting object is a real Jacobi form in three variables, whose holomorphic part corresponds to discrete (short) $N=2$ characters (mock modular contributions) and whose non-holomorphic remainder arises from a spectral asymmetry between right-moving bosons and fermions in the continuous spectrum. They derive the full partition function as a product of bosonic, compact-boson, fermionic, and ghost sectors, perform a Poisson resummation to separate holomorphic and non-holomorphic pieces, and connect these to generalized Appell functions. An orbifold generalization to a $\mathbb{Z}_k$ twist is analyzed, clarifying the spectrum and modular properties, with a physical interpretation of the spectral-density mismatch via reflection amplitudes, linking the mathematical completion to physical densities and potential applications to mirror symmetry and black-hole entropy counting.

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.