8 research outputs found

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

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

### Gravitational couplings in ${\cal N}=2$ string compactifications and Mathieu Moonshine

We evaluate the low energy gravitational couplings, $F_g$ in the heterotic
$E_8\times E_8$ string theory compactified on orbifolds of $K3\times T^2$ by
$g'$ which acts as a $\mathbb{Z}_N$ automorphisim on $K3$ together with a $1/N$
shift along $T^2$. The orbifold $g'$ corresponds to the conjugacy classes of
the Mathieu group $M_{24}$. The holomorphic piece of $F_g$ is given in terms of
a polylogarithim with index $3-2g$ and predicts the Gopakumar-Vafa invariants
in the corresponding dual type II Calabi-Yau compactifications. We show that
low lying Gopakumar-Vafa invariants for each of these compactifications
including the twisted sectors are integers. We observe that the conifold
singularity for all these compactifications occurs only when states in the
twisted sectors become massless and the strength of the singularity is
determined by the genus zero Gopakumar-Vafa invariant at this point in the
moduli space.Comment: 54 pages, Mathematica code included in the source file, minor typos
fixed, one reference adde

### Dyon degeneracies from Mathieu moonshine

We construct the Siegel modular forms associated with the theta lift of
twisted elliptic genera of $K3$ orbifolded with $g'$ corresponding to the
conjugacy classes of the Mathieu group $M_{24}$. We complete the construction
for all the classes which belong to $M_{23} \subset M_{24}$ and two other
classes outside the subgroup $M_{23}$. For this purpose we provide the explicit
expressions for all the twisted elliptic genera in all the sectors of these
classes.
We show that the Siegel modular forms satisfy the required properties for
them to be generating functions of $1/4$ BPS dyons of type II string theories
compactified on $K3\times T^2$ and orbifolded by $g'$ which acts as a
$\mathbb{Z}_N$ automorphism on $K3$ together with a $1/N$ shift on a circle of
$T^2$. In particular the inverse of these Siegel modular forms admit a Fourier
expansion with integer coefficients together with the right sign as predicted
from black hole physics. Our analysis completes the construction of the
partition function for dyons as well as the twisted elliptic genera for all the
$7$ CHL compactifications.Comment: Section on comparison with earlier literature added, References adde