Generalities of Quarter BPS Dyon Partition Function and Dyons of Torsion Two

TL;DR

This work develops a universal framework for constraining the partition function of quarter-BPS dyons in ${\cal N}=4$ theories by leveraging duality symmetries, wall-crossing, and black hole entropy. It formalizes the dyon partition function through a Siegel modular form ${1\over \check\Phi}$ whose contour-dependent Fourier expansion captures domain-specific indices, and derives stringent S-duality and wall-crossing constraints that link non-primitive decay channels to products of half-BPS degeneracies. The authors apply the framework to several cases, including dyons of torsion two in heterotic on ${\cal T}^6$ and CHL models, and propose explicit partition functions that satisfy all consistency conditions, reproducing known results in gauge theories in appropriate limits. A key outcome is a new wall-crossing formula for quarter-BPS decays into non-primitive states and a concrete proposal for the torsion-two dyon partition function, along with discussion of higher-torsion extensions and reverse utility for half-BPS spectra extraction.

Abstract

We propose a general set of constraints on the partition function of quarter BPS dyons in any N=4 supersymmetric string theory by drawing insight from known examples, and study the consequences of this proposal. The main ingredients of our analysis are duality symmetries, wall crossing formula and black hole entropy. We use our analysis to constrain the dyon partition function for two hitherto unknown cases -- the partition function of dyons of torsion two (i.e. gcd(Q\wedge P)=2) in heterotic string theory on T^6 and the partition function of dyons carrying untwisted sector electric charge in Z_2 CHL model. With the help of these constraints we propose a candidate for the partition function of dyons of torsion two in heterotic string theory on T^6. This leads to a novel wall crossing formula for decay of quarter BPS dyons into half BPS dyons with non-primitive charge vectors. In an appropriate limit the proposed formula reproduces the known result for the spectrum of torsion two dyons in gauge theory.