Degeneracy of Decadent Dyons

TL;DR

This paper shows that the degeneracies of quarter-BPS dyons in ${ m N}=4$ and ${ m N}=2$ supersymmetric gauge theories can be derived from a simple semiclassical picture near the surface of decadence, where a dyon splits into multi-centered constituents. The authors derive a general degeneracy formula based on the angular-momentum produced by long-range electromagnetic fields and the degeneracies of the decay products, with additional factors accounting for supersymmetry-specific zero modes; in particular, ${ m N}=4$ introduces a prefactor of 4 and a center-of-mass contribution, yielding results that precisely match established index computations. They explicitly work through ${ m SU}(3)$ and general ${ m SU}(N)$ Stern–Yi dyons, obtaining degeneracies that align with previous determinations, and extend the analysis to ${ m N}=2$ cases. The work also clarifies how these field-theory dyons relate (or fail to relate) to the dyon partition function in heterotic string theory via the Igusa cusp form ${ m }_{10}$, showing that only a narrow subset (notably SU(3) with certain invariants) lie in the corresponding duality orbit. Overall, the method provides a transparent, broadly applicable check on degeneracy computations and wall-crossing intuition across gauge theories and their string-theoretic avatars.

Abstract

A quarter-BPS dyon in $\mathcal{N}=4$ super Yang-Mills theory is generically `decadent' in that it is stable only in some regions of the moduli space and decays on submanifolds in the moduli space. Using this fact, and from the degeneracy of the system close to the decay, a new derivation for the degeneracy of such dyons is given. The degeneracy obtained from these very simple physical considerations is in precise agreement with the results obtained from index computations in all known cases. Similar considerations apply to dyons in $\mathcal{N}=2$ gauge theories. The relation between the $\mathcal{N} =4 $ field theory dyons and those counted by the Igusa cusp form in toroidally compactified heterotic string is elucidated.