Properties of dyons in ${\cal N}=4$ theories at small charges

TL;DR

This paper investigates properties of 1/4 BPS dyons in ${ m N}=4$ theories at small charges, focusing on entropy contributions, modular form decompositions, and sign behavior of dyon indices in orbifold compactifications.

Contribution

It provides new insights into the entropy constants, modular form structures, and sign properties of dyon indices in ${ m N}=4$ string compactifications with orbifolds.

Findings

The constant in the one-loop entropy is crucial for orbifold models.

Decomposition of the Jacobi form into Appell-Lerch sum and mock Jacobi form generalizes previous results.

Sign of the dyon index violates positivity conjecture at low charges, indicating non-trivial hair.

Abstract

We study three properties of $1/4$ BPS dyons at small charges in string compactifications which preserve ${\cal N}=4$ supersymmetry. We evaluate the non-trivial constant present in the one loop statistical entropy for ${\cal N}=4$ compactifications of type IIB theory on $K3\times T^2$ orbifolded by an order $\mathbb{Z}_N$ freely acting orbifold $g'$ including all CHL compactifications. This constant is trivial for the un-orbifolded model but we show that it contributes crucially to the entropy of low charge dyons in all the orbifold models. We then show that the meromorphic Jacobi form which captures the degeneracy of $1/4$ BPS states for the first two non-trivial magnetic charges can be decomposed into an Appell-Lerch sum and a mock Jacobi form transforming under $\Gamma_0(N)$. This generalizes the earlier observation of Dabholkar-Murthy-Zagier to the orbifold models. Finally we study the sign of the Fourier coefficients of the inverse Siegel modular form which counts the index of $1/4$ BPS dyons in ${\cal N}=4$ models obtained by freely acting $\mathbb{Z}_2$ and $\mathbb{Z}_3$ orbifolds of type II theory compactified on $T^6$. We show that sign of the index for sufficiently low charges and ensuring that it counts single centered black holes, violates the positivity conjecture of Sen which indicates that these states posses non-trivial hair.