We study N = 2 compactifications of E 8 × E 8 heterotic string theory on orbifolds of K3 × T 2 by g which acts as an Z N automorphism of K3 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 K3 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 K3 twisted by g . We work out in detail the case for which g belongs to the equivalence class 2B. We then investigate all the non-standard embeddings for K3 realized as a T 4 /Z ν orbifold with ν = 2, 4 and g the 2A 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.

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.

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 ⊂ 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 × T 2 and orbifolded by g which acts as a 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.

We study three properties of 1/4 BPS dyons at small charges in string compactifications which preserve $$ \mathcal{N} $$ N = 4 supersymmetry. We evaluate the non-trivial constant present in the one loop statistical entropy for $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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.

Classical single centered solutions of 1/4 BPS dyons in $$ \mathcal{N} $$ 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 × T2 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 $$ \mathcal{N} $$ N = 4 models obtained by freely acting ℤ2 and ℤ3 orbifolds of type IIB theory compactified on T6 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.