Higher Poles and Crossing Phenomena from Twisted Genera

Understand

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.