Scaling black holes are solutions of supergravity with multiple black hole singularities, which can be adiabatically connected to a single center black hole solution. We develop techniques to determine partition functions for such scaling black holes, if each constituent carries a non-vanishing magnetic charge corresponding to a D4-brane in string theory, or equivalently M5-brane in M-theory. For three constituents, we demonstrate that the partition function is a mock modular form of depth two, and we determine the appropriate non-holomorphic completion using generalized error functions. From the four-dimensional perspective, the modular parameter is the axion-dilaton, and our results show that S-duality leaves this subset of the spectrum invariant. From the five-dimensional perspective, the modular parameter is the complex structure of a torus T 2, and the scaling black holes are dual to states in the dimensional reduction of the M5-brane worldvolume theory to T 2. As a case study, we specialize the compactification manifold to a K3 fibration, and explicitly evaluate holomorphic parts of partition functions.

Ireland

Aradhita Chattopadhyaya

Jan Manschot

Swapnamay Mondal

Scaling black holes and modularity

Classical single centered solutions of 1/4 BPS dyons in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 theories are usually constructed in duality frames which contain non-trivial hair degrees of freedom localized outside the horizon. These modes are in addition to the fermionic zero modes associated with broken supersymmetry. Identifying and removing the hair from the 1/4 BPS index allows us to isolate the degrees of freedom associated with the horizon. The spherical symmetry of the horizon then ensures that index of the horizon states has to be positive. We verify that this is indeed the case for the canonical example of dyons in type IIB theory on K3 × T 2 and prove this property holds for a class of states. We generalise this observation to all CHL orbifolds, this involves identifying the hair and isolating the horizon degrees of freedom. We then identify the horizon states for 1/4 BPS dyons in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 models obtained by freely acting ℤ2 and ℤ3 orbifolds of type IIB theory compactified on T 6 and observe that the index is again positive for single centred black holes. This observation coupled with the fact the 1/4 BPS index of single centred solutions without removal of the hair violates positivity indicates that there exists no duality frame in these models without non-trivial hair.

India

Justin David

Horizon states and the sign of their index in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 dyons

In this paper we compute the gravitational couplings of the heterotic string compactified on (K3 × T 2)/ℤ N  and E 8 × E 8 and predict the Gopakumar Vafa invariants of the dual Calabi Yau manifold in presence of Wilson lines. Here ℤ N  acts as an automorphism on K3 associated with the conjugacy classes of M 23 and a shift of 1/N on one of the S 1 of T 2. We study in detail the cases N = 2, 3 for standard and several non-standard embeddings where K3 is realized as toroidal orbifolds T 4 /ℤ4 and T 4 /ℤ3. From these computations we extract the polynomial term in perturbative pre-potential for these orbifold models in presence of a single Wilson line. We also show for standard embeddings the integrality of the Gopakumar Vafa invariants depend on the integrality of Fourier coefficients of Fourier transform of the twisted elliptic genus of K3 in presence of n &lt; 8 Wilson lines.

Gravitational couplings in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 heterotic compactifications with Wilson lines

In this paper we study compactifications of the $$ \mathcal{N} $$ = 2 heterotic string on ( 3 )/ℤ with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Yau geometries are 3 fibrations that are also genus one fibered with three- sections. Moreover, we show that the intersection form on the polarization lattice of the 3 fibration has to be three times the intersection form on the Narain lattice Γ .

Austria

Andreas Banlaki

United States

Abhiram Kidambi

Thorsten Schimannek

Maria Schimpf

Heterotic strings on (

We study three properties of 1/4 BPS dyons at small charges in string compactifications which preserve   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetry. We evaluate the non-trivial constant present in the one loop statistical entropy for   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 compactifications of type IIB theory on K3 × T 2 orbifolded by an order 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 Γ0(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   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 models obtained by freely acting ℤ2 and ℤ3 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.

Properties of dyons in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 theories at small charges

We evaluate the low energy gravitational couplings, F g in the heterotic E 8 ×E 8 string theory compactified on orbifolds of K3 × T 2 by g ′ which acts as a ℤ N  automorphism 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 polylogarithm 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.

Gravitational couplings in  <math><mi>N</mi><mo>=</mo><mn>2</mn></math> $$ \mathcal{N}=2 $$ string compactifications and Mathieu Moonshine

We study N $$ \mathcal{N} $$ = 2 compactifications of E 8 × E 8 heterotic string theory on orbifolds of K 3 × T 2 by g ′ which acts as an ℤ N $$ {\mathbb{Z}}_N $$ automorphism of K 3 together with a 1 /N shift on a circle of T 2 . The orbifold action g ′ corresponds to the 26 conjugacy classes of the Mathieu group M 24 . We show that for the standard embedding the new supersymmetric index for these compactifications can always be decomposed into the elliptic genus of K 3 twisted by g ′ . The difference in one-loop corrections to the gauge couplings are captured by automorphic forms obtained by the theta lifts of the elliptic genus of K 3 twisted by g ′ . We work out in detail the case for which g ′ belongs to the equivalence class 2 B . We then investigate all the non-standard embeddings for K 3 realized as a T 4 / ℤ ν $$ {T}^4/{\mathbb{Z}}_{\nu } $$ orbifold with ν = 2 , 4 and g ′ the 2 A involution. We show that for non-standard embeddings the new supersymmetric index as well as the difference in one-loop corrections to the gauge couplings are completely characterized by the instanton numbers of the embeddings together with the difference in number of hypermultiplets and vector multiplets in the spectrum.