Higher Poles and Crossing Phenomena from Twisted Genera

TL;DR

This paper links higher pole Appell-Lerch sums and their modular completions to the cigar conformal field theory's partition functions, revealing how chemical potential variations induce wall-crossing phenomena in bound state contributions.

Contribution

It demonstrates the emergence of higher pole Appell-Lerch sums from the cigar CFT and explores the effects of chemical potential variations on elliptic genus wall-crossing.

Findings

Higher pole Appell-Lerch sums arise in cigar CFT partition functions.

Varying the imaginary chemical potential causes wall-crossing in bound states.

Full elliptic genus remains continuous despite wall-crossing phenomena.

Abstract

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.