  We demonstrate that Appell-Lerch sums with higher order poles as well as
their modular covariant completions arise as partition functions in the cigar
conformal field theory with worldsheet supersymmetry. The modular covariant
derivatives of the elliptic genus of the cigar give rise to operator insertions
corresponding to (powers of) right-moving momentum, left-moving fermion number,
as well as a term corresponding to an ordinary zero mode partition sum. To show
this, we demonstrate how the right-moving supersymmetric quantum mechanics (and
in particular the Hamiltonian and spectral density) depend on the imaginary
part of the chemical potential for angular momentum. As a consequence of our
analysis we find that varying the imaginary part of the chemical potential for
angular momentum on the cigar gives rise to a wall-crossing phenomenon in the
bound state contribution to the elliptic genus, while the full elliptic genus
is a continuous function of the chemical potential.
