# An Elliptic Triptych

## TL;DR

This paper clarifies key aspects of non-compact elliptic genera, including a path integral derivation, analysis of supersymmetric quantum mechanics with continuous spectra, and the relation to flat space elliptic genus, revealing anomalies and modular properties.

## Contribution

It provides a path integral derivation of the cigar elliptic genus, analyzes supersymmetric quantum mechanics with continuous spectra, and clarifies the relation to flat space elliptic genus, highlighting anomalies.

## Key findings

- Path integral derivation yields a modular sum over a lattice.
- Supersymmetric quantum mechanics with continuous spectrum exhibits regulator-dependent temperature dependence.
- Right-moving supersymmetry combined with modular covariance is anomalous in non-compact elliptic genera.

## Abstract

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.