Horizon states and the sign of their index in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 dyons

Classical single centered solutions of 1/4 BPS dyons in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 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 N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 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.


Introduction
Counting microscopic degrees of freedom for extremal black holes in string theory is a useful probe into aspects of quantum gravity [1]. For supersymmetric black holes, one should in principle be able identify the degrees of freedom both from the macroscopic solution as well as count them from the microscopic description of these black holes. The 1/4 BPS dyonic black holes in N = 4 theory is a system which has been extensively studied in this context, see [2,3] for reviews. The identification of the degrees of freedom is complicated by the fact that classical solutions of black holes are multi-centered and usually they contain hair degrees of freedom localized outside the horizon [4][5][6]. The microscopic analysis counts all these configurations together. Let us make this precise, let d micro ( q) be the degeneracy or in the case of the extremal supersymmetric black holes the appropriate supersymmetric index evaluated from the microscopic description of a BPS state with charge q. Similarly let d macro ( q) be the corresponding macroscopic index. Then (1.1) Each term on the right hand side of (1.1) is the contribution to the index of say, the ncentered black hole configuration. d hor ( q i ) is the contribution to the index from the horizon JHEP03(2021)106 degrees of freedom with charge q i and d hair ( q hair ; { q i }) is the index of the hair carrying total charge q hair of a n-centered black hole whose horizons carry charges q 1 , · · · q n . We expect d macro ( q) = d micro ( q). (1.2) It would simplify matters if we can restrict our attention to single centred black hole configurations. Then (1.1) indicates that we would need to identify the hair to isolate the horizon degrees of freedom. Since we are dealing with 1/4 BPS states in N = 4 theories, which break 12 supersymmetries, the degeneracy d( q) will refer to the index where J is the component of the angular momentum in say, the 3 direction. The factorized form of the Hilbert space corresponding to the hair degrees of freedom and the horizon degrees of freedom follows from the fact the these are well separated due the presence of an infinite throat [4]. The utility of identifying the horizon degrees of freedom lies in the fact that the horizon is spherically symmetric and therefore carries zero momentum J = 0. The index taken over the horizon states reduces to (−1) 2J d hor = d hor , where d hor is the total number of states associated with the horizon. Therefore the index of the horizon states must be a positive number. This leads to an important check on the microscopic counting and the equality (1.2). Once one determines the hair degrees of freedom for a given macroscopic black hole and factors them out of the index, what must remain is a positive number which counts the index of the horizon states. This argument clearly relies on what are the hair degrees of freedom and this in turn depends on the duality frame of the macroscopic solution. This prediction was tested in [7] with the assumption that there exists a frame in which the only hair degrees of freedom are the fermionic zero modes associated with the broken supersymmetry generators. For black holes in N = 8 there is evidence towards this fact in [8,9]. These authors worked in a frame in which the black hole configuration reduced to a system of only D-branes and showed the only hair degrees of freedom were the fermionic zero modes and the BPS configuration indeed had zero angular momentum. However such a frame has not yet been shown to exist for black holes in N = 4 theory.
Given this situation, one way of proceeding is to evaluate the partition functions corresponding to the hair degrees of freedom and isolate the horizon degrees of freedom in a given frame. This has already been done in [5,6], for 1/4 BPS dyons in the type IIB frame, but a test of positivity of the index for the resulting horizon degrees of freedom has not been done. We perform this analysis in this paper and indeed demonstrate that the d hor is indeed positive. This is quite remarkable as we will see, since factorizing the hair degrees of freedom naively seems to introduce terms with negative contributions to the index. We adapt the proof of [10] for configurations with magnetic charge P 2 = 2 and demonstrate that the index is positive. We then extend this observation to all the CHL models and to other orbifolds associated with Mathieu moonshine introduced in [11,12].
In [11,13] it was observed that for N = 4 models obtained by freely acting Z 2 , Z 3 orbifolds of type IIB on T 6 , the index for single centered configurations after factorising JHEP03(2021)106 the sign due to the fermionic zero modes did not obey the expectation d hor is positive. 1 But as the above discussion shows, a possible reason for this could be that the assumption that there exists a frame in which the fermionic zero modes are the only hair degrees of freedom might not be true. Therefore we re-examine this question in this paper. Following the same procedure used in the CHL models we isolate the hair degrees of freedom in the type IIB frame. Then on examining the sign of the index for single centered black holes we observe that d hor is positive.
The organisation of the paper is as follows. In the section 2 we briefly review the statements about the hair and the partition function for the horizon degrees of freedom for the 1/4 BPS dyonic black hole in type IIB compactified on K3 × T 2 . We then generalise this to all the CHL orbifolds as well as other orbifolds associated with Mathieu Moonshine. Finally we construct the partition function for the horizon states for the toroidal models obtained by freely acting Z 2 , Z 3 orbifolds of type IIB on T 6 . In section 3 we perform a consistency check on the d hor obtained. This check relies on the fact that the 5-dimensional BMPV black hole has the same near horizon geometry [5,6]. Therefore d hor for the BMPV black hole should agree with that of the 1/4 BPS dyon. We show that this is indeed the case for all the examples. Finally in section 4, armed with the d hor for all the models we study the positivity of the index for single centered black holes for all the models. We have evaluated numerically the indices of horizon states for several charges in all the N = 4 models for which dyon partition functions are known which confirm that the index is positive. We adapt the proof of [10] to show that the index is positive for charge configurations with P 2 = 2. Section 5 contains our conclusions.

Horizon states for the 1/4 BPS dyon
In this section we construct the partition function for the horizon states for 1/4 BPS dyons in N = 4 compactifications. This is done by identifying the 'hair' degrees of freedom which are localized outside the horizon. Such a partition function for the horizon states was constructed for the canonical N = 4 theory obtained by compactifying type IIB string theory K3 × T 2 in [5,6] in the type IIB frame. We review this in section and then extend the analysis for other N = 4 models.
The N = 4 compactifications of interest are type IIB theory on K3 × T 2 /Z N where Z N acts as an automorphism g on K3 along with a shift of 1/N units on one of the circles of T 2 . The action of g can be labelled by the 26 conjugacy classes of the Mathieu group M 23 . The classes pA with p = 2, 3, 5, 6, 7, 8 and the class 4B are called as Nikulin's automorphism of K3. They were first introduced in [14,15] as models dual to heterotic string theory with N = 4 superysmmetry but with gauge groups with reduced from the maximal rank of 28. All these compactifications admit 1/4 BPS dyons, let (Q, P ) be the electric and magnetic charge vector of these dyons, then the 1/4 BPS index B 6 is given by [16][17][18][19][20]

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where C is a contour in the complex 3-plane defined by Here 2) implies that we first expand in powers or e 2πiρ , e 2πiσ and at the end perform the expansion in e 2πiv .
The Siegel modular form of weight k given byΦ k (ρ, σ, v) transforming under Sp(2, Z), or its subgroups for N > 1 admits an infinite product representation given bỹ The coefficients c (r,s) are determined from the expansion of the twisted elliptic genera for the various order N orbifolds g of K 3 . The twisted elliptic genus of K3 is defined by The trace is performed over the Ramond-Ramond sector of the N = (4, 4) super conformal field theory of K3 with (c,c) = (6, 6), F is the Fermion number and j is the left moving U(1) charge of the SU(2) R-symmetry of K3. The twisted elliptic genera for the g corresponding to conjugacy classes of M 23 ∪ M 24 have been evaluated in [11]. These take the form where we can write A(τ, z) and B(τ, z) in terms of Jacobi theta functions θ i and Dedekind eta functions as follows We have used the following notation for all cases:

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The coefficients α (r,s) g in (2.5) are numerical constants, while β (r,s) g (τ ) are modular forms that transform under Γ 0 (N ). For g corresponding to conjugacy classes of M 23 , they can be read out from appendix E of [11]. For example, in the case of the pA orbifolds with p = 1, 2, 3, 5, 7, they are given by [18].
For N composite corresponding to the classes 4B, 6A, 8A, the strategy for construction of the twisted elliptic genus was first given in [21] and it was worked out explicitly for the 4B example. 2 The papers [22][23][24] contain the twining characters, F (0,s) and [25] also contains the strategy to construct the twisted elliptic genera for other conjugacy classes of M 23 and a Mathematica code for generating the elliptic genera. The weight of the Siegel modular formΦ(ρ, σ, v) is given by For the classes pA, p = 1, 2, 3, 5, 7, 11 we have for 4B, 6A, 8A we have k = 3, 2, 1 respectively and for 14A, 15A k = 0. Finally, as discussed in the introduction the study of horizon states would be much simpler if one could focus on single centered dyons. Such a system would have only one horizon. The choice of the contour chosen in (2.2) together with some kinematic constraints on charges such as (4.3) ensures that we are in the attractor region of the axion-dilaton moduli and the index given by (2.1) is that of single centred dyons [7,26]. All the indices evaluated in this section paper is done using the contour (2.2).

The canonical example: K3 × T 2
In the work of [5,6] the hair modes of the 1/4 BPS dyonic black hole in type IIB theory compactified on K3×T 2 were constructed. Here we briefly review this construction. These modes were shown to be deformations localized outside the horizon and they preserved supersymmetry. Let us first recall that the dyonic black hole in 4-dimensions is constructed by placing the 5 dimensional BMPV black hole or the rotating D1-D5 system [27] in Taub-Nut space [28]. The Taub-Nut space has the geometry which at the origin is R 4 but at 2 Suresh Govindarajan informed us that the authors of [21] also explicitly constructed all the sectors of the 6A and 8A twisted elliptic genera though it was not reported in the paper.

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infinity it is R 3 ×S. The isometry along S 1 coincides with the angular direction the BMPV rotates. The hair modes arise from the collective modes of the D1-D5 system thought of as an effective string along say the x 5 and the time t directions. Therefore these modes are oscillations of the effective string, they are left moving since they have to preserve supersymmetry 3 After allowing the fermionic zero modes associated with the 12 broken susy generators to saturate (2J) 6 /6! in the helicity trace given in (1.3), the non-trivial hair modes consist of • 4 left moving fermionic modes arising from the deformations of the gravitino giving rise to the contribution (2.10) • 3 left moving bosonic modes associated with the oscillation of the effective string in the 3 transverse directions R 3 as Taub-NUT is asymptotically (2.11) • 21 left moving bosonic modes, these arise from the deformations of the 21 anti-selfdual forms of type IIB on K3. These deformations involve 21 scalar functions folded with the 2 form dω T N on the Taub-Nut given by Counting these oscillations we obtain (2.13) The 21 anti-self dual forms arise from compactifying the RR 4-form on the 19 anti-self dual 2 form of the K3 together with the NS 2-form and the RR 2-form of type IIB.
Note that in the partition function we labelled the chemical potential to count the oscillations by ρ, this is because exciting these left moving momentum modes correspond to exciting the electric charge of the dyon [19]. Now combining these partition functions we obtain (2.14) The Bosonic hair partition function is given by

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this is identical to that of the counting the degeneracy of purely electric states in this model without the zero point energy. This observation will help in the generalizations to CHL models.
To obtain the partition function of horizon states we factor out the hair degrees of freedom resulting in (2.16) The index for the horizon states can be then be obtained by extracting the Fourier coefficients using the expression given by Here the contour C is same as that defined in (2.2).

Orbifolds of K3 × T 2
2A orbfiold. Before we present the analysis for the most general orbifold, let us examine in detail the analysis for the 2A orbifold. In this case, the orbifold acts by exchanging 8 pairs of anti-self dual (1, 1) forms out of the 19 anti-self dual forms of K3 with the 1/2 shift on S 1 [15]. Note that because of the 1/2 shift, the natural unit of momentum on S 1 is N = 2. With this input we are ready to repeat the analysis for the partition function of the hair modes Note that due to the fact that the periodicity is now 2π N , the unit of momentum is doubled.
• The 3 transverse bosonic deformations along R 3 of the effective string results in • The action of the orbifold projects out 8 anti-self dual forms. The analysis for 13 = 11 + 2. 4 invariant anti-self dual forms proceeds as before except for the fact that the unit of momentum is 2 (2.20) 4 The 2 arises from the anti-self dual component of the RR 2-form and the NS 2-form.

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Consider the following boundary conditions of the function h(s) in (2.12) for the 8 projected anti-self dual forms These deformation pick up sign when one move by 1/2 unit on S 1 . The partition function corresponding to these modes is given by Note that these modes are twisted for the circle of radius 2π/N, N = 2, they obey antiperiodic boundary conditions. However in supergravity periodicities are measured over the circle of radius 2π and they are periodic for this radius, therefore these modes can be counted as hair modes. Together, the contribution of the anti-self dual forms to the partition function is given by Observe that the partition function of the bosonic hair modes is given by This is the partition function of the fundamental string in the N = 2 CHL orbifold of the heterotic theory with the zero point energy removed [19,29].
pA orbifolds p = 2, 3, 5, 7. The construction of the hair modes for the case of orbifolds of prime order, the method proceeds as discussed in detail for the 2A orbifold. In each case we need to count the number of 2-forms which are left invariant and which pick up phases and evaluate the partition function. The result for the bosonic hair modes is given by Note that this is the partition of the states containing only the electric charges or the fundamental string without the zero point energy [19]. Now including the 4 fermionic deformations we obtain It is useful to rewrite this expression as follows The sum is on the range of s = 0 to N − 1 and N l implies N does not divide l.
Orbifolds of composite order: 4B, 6A, 8A. One can count the hair modes in a similar fashion to the case of orbifolds with prime order. The only difference would arise for the bosonic modes Z bosons hair , which needs to be replaced by the fundamental string in these theories without the zero point energy. Including the 4 fermionic hairs, we see that the answer can be written in the same form as that seen for orbifolds with prime order The sum ranges from s = 0 to N − 1. This can be rewritten as (1 − e 2πi(4lρ) ) 4 (1 − e 2πi(4lρ) ) −4 (1 − e 2πi(2lρ) ) −2 (1 − e 2πi(lρ) ) −4 (2.32) Horizon states. We factor out the hair degrees of freedom to obtain the horizon states, this is given by (1 − e 2πi(lρ) ) s e −2πisl/N c (0,s) (0) . (2.33)

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It is useful to use the product form ofΦ k given in (2.3) to rewrite the partition function of the horizon states as follows Z 4d hor:CHL = −e −2πi(ρ+σ/N +v) This form of the horizon partition function is useful in the next section. The index for the horizon states is given by Non-geometric orbifolds: 11A, 14A, 15A, 23A. For completeness we note that we can extend the counting of hair modes to g orbifolds of K3 where g corresponds all the remaining conjugacy classes of M 23 . The CHL orbifolds also form a part of these, however the ones discussed in this section are non-geometric. The hair modes in these cases can also be written as: (1 − e 2πi(lρ) ) − e −2πisl/N c (0,s) (0) . (2.36) To be explicit, we list the of values of N −1 s=0 e −2πisl/N c (0,s) (−b 2 ) for different N for N = 11, 14, 15, 23 in table 3. Using the results from table 3 we write: (1 − e 2πi(23lρ) ) 4 (1 − e 2πi(23lρ) ) −1 (1 − e 2πi(lρ) ) −1 . The partition function of the horizon states in these models are given by the same expressions as in (2.34) with N replaced by the order of the conjugacy class and the coefficients c (r,s) read out from the respective twisted elliptic genus. Let us conclude by writing the general formula for the horizon states as Z 4d hor: g = −e −2πi(ρ+σ/N ) (2.41)

Toroidal orbifolds
In this section we construct the hair for N = 4 theories obtained by freely acting Z 2 , Z 3 involutions on T 6 [30]. Let us first briefly recall how these are constructed. In the type IIB frame, they are obtained by an inversion of 4 of the co-ordinates of T 4 together with a half shift along one of the S 1 . The type IIA description of the theory is that of a freely acting orbifold with the action of (−1) F L and a 1/2 shift along one of the circles of T 6 . 5 A JHEP03(2021)106 similar compactification of order 3 given by a 2π/3 rotation along one 2D plane of T 4 and a −2π/3 rotation along another plus an 1/3 shift along one of the circles of T 2 was also discussed in [18]. We call these models Z 2 and Z 3 toroidal orbifolds.
One key property of these models to keep in mind which will be important is that the breaking of the 32 supersymmetries of type IIB to 16 is determined by the size of S 1 . This was not the case for the orbifolds of K3 × T 2 , where supersymmetry was broken by the K3. For the toroidal models if the size of S 1 is infinite, the theory effectively behaves as though the theory has 32 supersymmetries. We will use this fact to propose certain fermionic zero modes which were present for the CHL models will become singular at the horizon.
The dyon partition function for the toroidal models is given by [31]: The coefficients c (r,s) are read out from the following twisted elliptic genus for Z 2 orbifold: The corresponding Siegel form of weight k = 2 can be written as , (2.44) whereΦ 6 is the weight 6 Siegel modular form associated with the order 2 CHL orbifold. For the Z 3 toroidal case the twisted elliptic genus is given by The Siegel modular form associated with the Z 3 toroidal orbifold has weight k = 1 and is given byΦ (2.46) whereΦ 4 is the weight 4 Siegel modular form associated with the order 3 CHL orbifold. Let us construct the hair modes and horizon states for these models.
• Just as in the case of the CHL models, we have 4 left moving fermions. This gives rise to Z 4d:f hair: • The deformations corresponding to the motion of the effective string in the 3 transverse directions of R 3 ×S 1 of the Taub-Nut space together with the fluctuations of the anti-self dual forms can be determined easily by examining the partition function of the fundamental string in this theory and removing the zero point energy. This partition function was determined in [18], using this result we obtain 6 • Contribution of the zero modes: The quantum mechanics of the bosonic zero modes describing the motion of the D1-D5 system in the Taub-Nut result in the following partition function [19] Z 4d: zeromodes For orbifolds of K3, this contribution from the bosonic zero modes was cancelled by the zero modes of 4 fermions from the right moving sector carrying angular momentum J = ± 1 2 whose partition function is given by −(e πiv − e −πiv ) 2 [5]. However for the toroidal model, we propose that these zero modes do not form part of the hair. They are either singular at the horizon or they are not localized outside the horizon. This is possible due to the fact that we are in a theory with 16 supersymmetries is tied to the radius of S 1 . Verification of this proposal would involve a detailed study of the zero mode wave functions which we leave for the future. However we will perform consistency checks of this proposal in section 4. by evaluating the index of the horizon states.
Thus the hair modes of the Z 2 toroidal model is given by The partition function of the horizon states of this model are given by (2.51) whereΦ 2 (ρ, σ, v) is given in (2.44) or (2.42). 6 One can also obtain this by counting the number of invariant 2-forms and the forms which pick up a phase as done in [20].

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The toroidal model has another special feature, they admit Wilson lines along T 4 [31], their partition function is given by (2.52) It is possible that the Wilson lines might also be part of the hair modes. In section 4 we will see that including the Wilson lines as hair modes instead of the bosonic zero modes given in (2.49) does not preserve the positivity of the index of the horizon states. T 6 /Z 3 model. Performing the same analysis as done for the Z 2 orbfiold we obtain the following partition function for the hair modes The horizon states is given by (2.54) whereΦ 1 is given by (2.46) or (2.42). For reference we also provide the partition function of the Wilson lines in this model From the expression for the Wilson lines and the infinite product representation given forΦ k given in (2.42) we obtain the following useful expression for the partition function for the horizon modes for both the toroidal orbifolds. (2.56)

Horizon states for the BMPV black hole
We now examine the BMPV black hole in 5 dimensions, that is the transverse space now does not have the Taub-Nut solution. The main reason for studying the problem in 5 JHEP03(2021)106 dimensions is that the near horizon geometry of the BMPV black hole in 5 dimensions is same as the of the 1/4 BPS dyon in 4 dimensions. This implies that the partition function of the horizon states of these 2 systems should be identical. In this section we construct the partition function of the hair and the horizon states for the BMPV black hole in type IIB on K3 × S 1 /g as well as toroidal orbifolds of T 5 . Here g corresponds to all the conjugacy classes of M 23 .

Partition function of BMPV black holes
The partition function for these black holes in the canonical compactification K3 × S 1 , was constructed in [5]. The same analysis can be extended to all the CHL models. The partition function receives contributions from the following sectors.
• The bound states of the D1-D5 system, this is given by the elliptic genus of the symmetric product of K3/g . This contribution was evaluated in [19]. It is given by Note that the only difference from the canonical model is that the unit of momentum on S 1 is N due to the 1/N shift.
• A shift of e −2πiρ to ensure to take into account of the difference in the electric charge measured at infinity and the horizon [5].
Combining all the sectors we obtain the following expression for the partition function for BMPV black hole for all orbifolds of K3 × S 1 . (3.4)

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Here the coefficients c (r,s) have to be read out from the twisted elliptic genus of K3 by g corresponding to the conjugacy classes of M 23 .
Using the counting of states for the dyon partition function done in [31] we can extend the analysis to the toroidal models. We present the analysis in some detail for the T 6 /Z 2 model Here the contributions arise from the following: • The bound state of the D1-D5 system on the T 4 /Z 2 orbifold is given by Here the coefficients c (r,s) are read out from the expansion of the functions given in (2.43).
• The contribution of the Wilson lines on T 4 /Z 2 which is given by (3.6) • The partition function corresponding to the centre of mass motion of the D1-D5 system in the transverse space (3.7) • The contribution of the zero modes Z 5d zeromodes = −(e πiv − e −πiv ) 2 . (3.8) • The shift in the electric charge accounted for by the factor e −2πiρ .
Combining all the contributions we obtain The partition function of the BMPV black hole in the T 5 /Z 3 is obtained model is given by the same expression as in (3.9) except that the coefficients c (r,s) must be read out from the functions given in (2.45) and N → 3.

Orbifolds of K3 × S 1
We now construct the hair modes in 5 dimensions for the K3 × S 1 /g where the quotient by g is associated with any conjugacy classes of the Mathieu group M 23 . The analysis proceeds identical to that done in [6], the only difference being that the unit of momentum on S 1 is N . Here we briefly state the contributions.
• The contribution of the 4 real left moving gravitino deformations of the BMPV black hole 7 (1 − e 2πilN ρ ) 4 . (3.10) • The contribution of the 8 real gravitino zero modes among the 12 modes due to broken supersymmetries which carry angular momentum J = ± 1 2 Z 5d; zero modes hair: g = (e πiv − e −πiv ) 4 . (3.11) Combining these contributions we obtain The partition function for the horizon states is given by Now comparing the horizon states of the 4d dyons from (2.41) and using (3.4) and (3.12) in (3.13) we can easily conclude Z 5d hor: g = Z 4d hor: g . (3.14)

Toroidal models
For the toroidal models the contributions of the hair are as follows.
• The contribution of the 4 left moving gravitino modes which result in • The contirbution of the zero modes. As we discussed earlier, supersymmetry in these models is tied to the radius of S 1 . We propose that due to this, out of 8 gravitino zero modes arising from broken supersymmetries which has angular momentum J = ± 1 2 , the wave functions of 4 of them either become singular at the horizon or they not localized outside the horizon. These 4 modes should not be counted as hair modes. Therefore the contribution of the zero modes in these models are given by Z 5d; zero modes hair: (3.16) As we will see consistency checks for this proposal will be done in section 4. 7 The bosonic deformations were shown to be singular at the horizon in [6].

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Combining these contributions we obtain Z 5d hair: The horizon partition function from the 5d perspective is given by Comparing the 4d horizon partition function given in (2.56) and using (3.9) and (3.17) in (3.18) we see that Z 5d hor:

The sign of the index for horizon states
In this section we will address the main goal of the paper. We observe that the index of horizon states is always positive.

Canonical example: K3 × T 2
For the un-orbifolded model recall that the hair in 4d is given by (4.1) The partition function of the horizon states is obtained by . (4.2) It was observed in [7] that the index −B 6 or the Fourier coefficients of 1/Φ 10 extracted using the contour in (2.2) subject to the kinematic restrictions were positive. The contour together with the above kinematic constraints ensures that the index counts single centred dyons. Furthermore [10] proved that the index of all single centered dyons with P 2 = 2, 4 is positive. These works assumed that there existed a frame in which the fermionic zero modes associated with broken supersymmetries were the only hair. We have seen that the type IIB frame the hair degrees of freedom is given by (4.1). Now naively it seems from the expression for the horizon states in (4.2) there are negative terms introduced due to the factor in the numerator and the observation of positivity seen in [7] and [10] might be violated once the hair in the type IIB frame is factored out. However we will show by adapting the proof of [10] that single centred dyons with P 2 = 2 do have positive index. For other values of charges we evaluate the index numerically, our results are presented in table 4. We observe that for single centered dyons the index is indeed positive.
is a weak Jacobi form of weight 2 and index m. In [32] it was shown that ψ m (τ, z) admits the following decomposition where, ψ F m (τ, z) has no poles in z. The polar part is given by an Appell-Lerch sum:
At P 2 = 2 we have m = 1 and we can write We need to show that ψ h z)) has the positivity property. Here H is the simplest mock Jacobi form defined by the Hurwitz-Kronecker class numbers H(n) (4.8)
Let us observe the expression: θ 2 (2τ ) η 6 E 4 + 216h 0 (τ ) . The only negative Fourier coefficient appears at q 0 . Using these observations we can prove the following lemma: Proof. We prove this for 1 (1−q) k and then the rest can be similarly proved by using q → q r and taking f r+1 (q) = fr(q) (1−q r+1 ) k . For f 2 the coefficient of q 1 is, a(1) − k > 0 and the coefficient of q N for N > 1 is given by, Q.E.D. 9 Explicit formula and detailed properties of H(n), h0(τ ), h1(τ ) can be found in [33].
In the series asociated with θ 2 (2τ, 2z) in equation (4.13) the Fourier coefficient of q n−1/4 is bounded from below by, Its positivity is ensured starting from n = 2 using the following bounds: 2. H(n) < n [10].
For n = 1 the positivity still holds as H(3) = 1/3. So the complete q series expansion of θ 3 (2τ ) η 6 E 4 − 216h 1 (τ ) contains no negative Fourier coefficient. This could also be seen from the Fourier expansion of θ 3 (2τ ) This ensures the positivity of −B 6 for Q · P = odd and hence the coefficients of ψ h 1 obey positivity as expected for P 2 = 2.

Orbifolds of K3 × T 2
For the 2A orbifold we extract the index of single centred dyons by using the contour in (2.2) together with the following kinematic constraints on the charges [7] The index of the horizon states for the 2A orbifold is given in table 5. The kinematic constraints on the charges for the 3B orbifold so that the dyons are single centered are given by The index for the horizon states is then obtained using contour (2.2) and is listed in table 6.
For an orbifold of order N > 3 there are infinite set of constraints for the charges to ensure that the index corresponds to single centered dyons [7]. However we see as long as the norms of electric and magnetic charges are positive and Q · P ≥ 0 together with Q 2 P 2 − (Q · P ) 2 > 0, the index −B 6 remains positive for the orbifolds of K3 (see the JHEP03(2021)106   tables 7-15). These orbifolds maybe geometric like that of CHL or even non-geometric where g ∈ [M 23 ].
It is interesting to see that the index for horizon states even in non-geometric orbifolds of K3 retains positivity of the index in the domain N Q 2 ≥ Q · P, P 2 ≥ Q · P , Q 2 P 2 − (Q · P ) 2 > 0.

JHEP03(2021)106
of the zero modes of the gravitino in the toroidal models to check the proposal in (2.50) and (2.53). In [13] it was noticed that the index of single centred dyons in these models were not positive when one assumed that the only hair modes are the Fermionic zero modes associated with broken supersymmetry generators. Since hair modes are frame dependent, the observations in this paper indicates that there is possibly no duality frame for these models which contains only the Fermionic zero modes as the hair. It will be interesting to verify this explicitly by an study similar to that done in [8,9] for the N = 8 theory.
The observation that the index of horizon states in the canonical compactification on K3 × T 2 is positive is worth further study. It should be possible to extend the proof of [10] to higher values of P 2 .
Note added: as this work was nearing completion, we became aware of the work done in [34]. The analysis of the hair modes done for the CHL orbifolds of K3 in section 2 and 3 overlaps with parts of [34].