A twisted non-compact elliptic genus

We give a detailed path integral derivation of the elliptic genus of the supersymmetric coset conformal field theory $$ {{{{\text{SL}}\left( {2,\mathbb{R}} \right)}} \left/ {{{\text{U}}(1)}} \right.} $$, 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.