No More Walls! A Tale of Modularity, Symmetry, and Wall Crossing for 1/4 BPS Dyons

TL;DR

This paper determines the generating functions for 1/4 BPS dyons in certain string vacua, showing they are uniquely fixed by physical consistency conditions and connecting them to moonshine phenomena through worldsheet indices.

Contribution

It introduces a method to fix the generating functions of 1/4 BPS dyons using wall crossing and consistency conditions, linking them to moonshine-related functions.

Findings

Generating functions obey wall crossing constraints.

Explicit derivation of twining genera matches moonshine functions.

Physical consistency conditions often suffice to determine the functions.

Abstract

We determine the generating functions of 1/4 BPS dyons in a class of 4d $\mathcal{N}=4$ string vacua arising as CHL orbifolds of $K3 \times T^2$, a classification of which has been recently completed. We show that all such generating functions obey some simple physical consistency conditions that are very often sufficient to fix them uniquely. The main constraint we impose is the absence of unphysical walls of marginal stability: discontinuities of 1/4 BPS degeneracies can only occur when 1/4 BPS dyons decay into pairs of 1/2 BPS states. Formally, these generating functions in spacetime can be described as multiplicative lifts of certain supersymmetric indices (twining genera) on the worldsheet of the corresponding nonlinear sigma model on K3. As a consequence, our procedure also leads to an explicit derivation of almost all of these twining genera. The worldsheet indices singled out in this way match precisely a set of functions of interest in moonshine, as predicted by a recent conjecture.