  We clarify three aspects of non-compact elliptic genera. Firstly, we give a
path integral derivation of the elliptic genus of the cigar conformal field
theory from its non-linear sigma-model description. The result is a manifestly
modular sum over a lattice. Secondly, we discuss supersymmetric quantum
mechanics with a continuous spectrum. We regulate the theory and analyze the
dependence on the temperature of the trace weighted by the fermion number. The
dependence is dictated by the regulator. From a detailed analysis of the
dependence on the infrared boundary conditions, we argue that in non-compact
elliptic genera right-moving supersymmetry combined with modular covariance is
anomalous. Thirdly, we further clarify the relation between the flat space
elliptic genus and the infinite level limit of the cigar elliptic genus.
